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3
Task/Problem-of-Apollonius/00-META.yaml
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3
Task/Problem-of-Apollonius/00-META.yaml
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---
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from: http://rosettacode.org/wiki/Problem_of_Apollonius
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note: arithmetic operations
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11
Task/Problem-of-Apollonius/00-TASK.txt
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11
Task/Problem-of-Apollonius/00-TASK.txt
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;Task:
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Implement a solution to the Problem of Apollonius ([[wp:Problem_of_Apollonius|description on Wikipedia]]) which is the problem of finding the circle that is tangent to three specified circles (colored black in the diagram below to the right).
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There is an [[wp:Problem_of_Apollonius#Algebraic_solutions|algebraic solution]] which is pretty straightforward.
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The solutions to the example in the code are shown in the diagram (below and to the right).
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The red circle is "internally tangent" to all three black circles, and the green circle is "externally tangent" to all three black circles.
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<br><br>
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54
Task/Problem-of-Apollonius/11l/problem-of-apollonius.11l
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54
Task/Problem-of-Apollonius/11l/problem-of-apollonius.11l
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T Circle
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Float x, y, r
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F (x, y, r)
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.x = x
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.y = y
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.r = r
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F String()
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R ‘Circle(x=#., y=#., r=#.)’.format(.x, .y, .r)
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F solveApollonius(c1, c2, c3, s1, s2, s3)
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V (x1, y1, r1) = c1
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V (x2, y2, r2) = c2
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V (x3, y3, r3) = c3
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V v11 = 2 * x2 - 2 * x1
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V v12 = 2 * y2 - 2 * y1
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V v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2
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V v14 = 2 * s2 * r2 - 2 * s1 * r1
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V v21 = 2 * x3 - 2 * x2
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V v22 = 2 * y3 - 2 * y2
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V v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3
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V v24 = 2 * s3 * r3 - 2 * s2 * r2
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V w12 = v12 / v11
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V w13 = v13 / v11
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V w14 = v14 / v11
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V w22 = v22 / v21 - w12
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V w23 = v23 / v21 - w13
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V w24 = v24 / v21 - w14
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V P = -w23 / w22
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V Q = w24 / w22
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V M = -w12 * P - w13
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V n = w14 - w12 * Q
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V a = n * n + Q * Q - 1
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V b = 2 * M * n - 2 * n * x1 + 2 * P * Q - 2 * Q * y1 + 2 * s1 * r1
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V c = x1 * x1 + M * M - 2 * M * x1 + P * P + y1 * y1 - 2 * P * y1 - r1 * r1
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V D = b * b - 4 * a * c
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V rs = (-b - sqrt(D)) / (2 * a)
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V xs = M + n * rs
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V ys = P + Q * rs
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R Circle(xs, ys, rs)
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V (c1, c2, c3) = (Circle(0, 0, 1), Circle(4, 0, 1), Circle(2, 4, 2))
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print(solveApollonius(c1, c2, c3, 1, 1, 1))
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print(solveApollonius(c1, c2, c3, -1, -1, -1))
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17
Task/Problem-of-Apollonius/Ada/problem-of-apollonius-1.ada
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Task/Problem-of-Apollonius/Ada/problem-of-apollonius-1.ada
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package Apollonius is
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type Point is record
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X, Y : Long_Float := 0.0;
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end record;
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type Circle is record
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Center : Point;
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Radius : Long_Float := 0.0;
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end record;
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type Tangentiality is (External, Internal);
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function Solve_CCC
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(Circle_1, Circle_2, Circle_3 : Circle;
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T1, T2, T3 : Tangentiality := External)
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return Circle;
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end Apollonius;
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86
Task/Problem-of-Apollonius/Ada/problem-of-apollonius-2.ada
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86
Task/Problem-of-Apollonius/Ada/problem-of-apollonius-2.ada
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with Ada.Numerics.Generic_Elementary_Functions;
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package body Apollonius is
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package Math is new Ada.Numerics.Generic_Elementary_Functions
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(Long_Float);
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function Solve_CCC
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(Circle_1, Circle_2, Circle_3 : Circle;
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T1, T2, T3 : Tangentiality := External)
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return Circle
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is
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S1 : Long_Float := 1.0;
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S2 : Long_Float := 1.0;
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S3 : Long_Float := 1.0;
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X1 : Long_Float renames Circle_1.Center.X;
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Y1 : Long_Float renames Circle_1.Center.Y;
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R1 : Long_Float renames Circle_1.Radius;
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X2 : Long_Float renames Circle_2.Center.X;
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Y2 : Long_Float renames Circle_2.Center.Y;
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R2 : Long_Float renames Circle_2.Radius;
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X3 : Long_Float renames Circle_3.Center.X;
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Y3 : Long_Float renames Circle_3.Center.Y;
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R3 : Long_Float renames Circle_3.Radius;
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begin
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if T1 = Internal then
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S1 := -S1;
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end if;
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if T2 = Internal then
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S2 := -S2;
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end if;
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if T3 = Internal then
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S3 := -S3;
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end if;
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declare
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V11 : constant Long_Float := 2.0 * X2 - 2.0 * X1;
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V12 : constant Long_Float := 2.0 * Y2 - 2.0 * Y1;
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V13 : constant Long_Float :=
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X1 * X1 - X2 * X2 + Y1 * Y1 - Y2 * Y2 - R1 * R1 + R2 * R2;
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V14 : constant Long_Float := 2.0 * S2 * R2 - 2.0 * S1 * R1;
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V21 : constant Long_Float := 2.0 * X3 - 2.0 * X2;
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V22 : constant Long_Float := 2.0 * Y3 - 2.0 * Y2;
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V23 : constant Long_Float :=
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X2 * X2 - X3 * X3 + Y2 * Y2 - Y3 * Y3 - R2 * R2 + R3 * R3;
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V24 : constant Long_Float := 2.0 * S3 * R3 - 2.0 * S2 * R2;
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W12 : constant Long_Float := V12 / V11;
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W13 : constant Long_Float := V13 / V11;
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W14 : constant Long_Float := V14 / V11;
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W22 : constant Long_Float := V22 / V21 - W12;
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W23 : constant Long_Float := V23 / V21 - W13;
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W24 : constant Long_Float := V24 / V21 - W14;
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P : constant Long_Float := -W23 / W22;
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Q : constant Long_Float := W24 / W22;
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M : constant Long_Float := -W12 * P - W13;
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N : constant Long_Float := W14 - W12 * Q;
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A : constant Long_Float := N * N + Q * Q - 1.0;
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B : constant Long_Float :=
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2.0 * M * N -
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2.0 * N * X1 +
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2.0 * P * Q -
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2.0 * Q * Y1 +
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2.0 * S1 * R1;
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C : constant Long_Float :=
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X1 * X1 +
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M * M -
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2.0 * M * X1 +
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P * P +
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Y1 * Y1 -
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2.0 * P * Y1 -
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R1 * R1;
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D : constant Long_Float := B * B - 4.0 * A * C;
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RS : constant Long_Float := (-B - Math.Sqrt (D)) / (2.0 * A);
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begin
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return (Center => (X => M + N * RS, Y => P + Q * RS), Radius => RS);
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end;
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end Solve_CCC;
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end Apollonius;
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25
Task/Problem-of-Apollonius/Ada/problem-of-apollonius-3.ada
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25
Task/Problem-of-Apollonius/Ada/problem-of-apollonius-3.ada
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with Ada.Text_IO;
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with Apollonius;
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procedure Test_Apollonius is
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use Apollonius;
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package Long_Float_IO is new Ada.Text_IO.Float_IO (Long_Float);
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C1 : constant Circle := (Center => (X => 0.0, Y => 0.0), Radius => 1.0);
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C2 : constant Circle := (Center => (X => 4.0, Y => 0.0), Radius => 1.0);
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C3 : constant Circle := (Center => (X => 2.0, Y => 4.0), Radius => 2.0);
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R1 : Circle := Solve_CCC (C1, C2, C3, External, External, External);
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R2 : Circle := Solve_CCC (C1, C2, C3, Internal, Internal, Internal);
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begin
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Ada.Text_IO.Put_Line ("R1:");
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Long_Float_IO.Put (R1.Center.X, Aft => 3, Exp => 0);
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Long_Float_IO.Put (R1.Center.Y, Aft => 3, Exp => 0);
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Long_Float_IO.Put (R1.Radius, Aft => 3, Exp => 0);
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Ada.Text_IO.New_Line;
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Ada.Text_IO.Put_Line ("R2:");
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Long_Float_IO.Put (R2.Center.X, Aft => 3, Exp => 0);
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Long_Float_IO.Put (R2.Center.Y, Aft => 3, Exp => 0);
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Long_Float_IO.Put (R2.Radius, Aft => 3, Exp => 0);
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Ada.Text_IO.New_Line;
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end Test_Apollonius;
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define :circle [x y r][]
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solveApollonius: function [c1 c2 c3 s1 s2 s3][
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v11: sub 2*c2\x 2*c1\x
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v12: sub 2*c2\y 2*c1\y
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v13: (sub (sub c1\x*c1\x c2\x*c2\x) + (sub c1\y*c1\y c2\y*c2\y) c1\r*c1\r) + c2\r*c2\r
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v14: sub 2*s2*c2\r 2*s1*c1\r
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v21: sub 2*c3\x 2*c2\x
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v22: sub 2*c3\y 2*c2\y
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v23: (sub (sub c2\x*c2\x c3\x*c3\x) + (sub c2\y*c2\y c3\y*c3\y) c2\r*c2\r) + c3\r*c3\r
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v24: sub 2*s3*c3\r 2*s2*c2\r
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w12: v12/v11
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w13: v13/v11
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w14: v14/v11
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w22: sub v22/v21 w12
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w23: sub v23/v21 w13
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w24: sub v24/v21 w14
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p: neg w23/w22
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q: w24/w22
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m: sub (neg w12)*p w13
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n: sub w14 w12*q
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a: dec add n*n q*q
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b: add (sub 2*m*n 2*n*c1\x) + (sub 2*p*q 2*q*c1\y) 2*s1*c1\r
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c: sub sub (sub (c1\x*c1\x) + m*m 2*m*c1\x) + (p*p) + c1\y*c1\y 2*p*c1\y c1\r*c1\r
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d: (b*b)-4*a*c
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rs: ((neg b)-sqrt d )/(2*a)
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xs: m+n*rs
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ys: p+q*rs
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return @[xs ys rs]
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]
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c1: to :circle [0.0 0.0 1.0]
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c2: to :circle [4.0 0.0 1.0]
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c3: to :circle [2.0 4.0 2.0]
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print solveApollonius c1 c2 c3 1.0 1.0 1.0
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print solveApollonius c1 c2 c3 neg 1.0 neg 1.0 neg 1.0
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105
Task/Problem-of-Apollonius/AutoHotkey/problem-of-apollonius.ahk
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105
Task/Problem-of-Apollonius/AutoHotkey/problem-of-apollonius.ahk
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@ -0,0 +1,105 @@
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#NoEnv
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#SingleInstance, Force
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SetBatchLines, -1
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; Uncomment if Gdip.ahk is not in your standard library
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;#Include, Gdip.ahk
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; Start gdi+
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If !pToken := Gdip_Startup()
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{
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MsgBox, 48, gdiplus error!, Gdiplus failed to start. Please ensure you have gdiplus on your system
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ExitApp
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}
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OnExit, Exit
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; I've added a simple new function here, just to ensure if anyone is having any problems then to make sure they are using the correct library version
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If (Gdip_LibraryVersion() < 1.30)
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{
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MsgBox, 48, version error!, Please download the latest version of the gdi+ library
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ExitApp
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}
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x1:=300,y1:=500,r1:=50,x2:=200,y2:=200,r2:=150,x3:=600,y3:=400,r3:=100,s1:=-1,s2:=-1,s3:=-1,xs:=0,ys:=0,rs:=0
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, Apollonius(x1,y1,r1,x2,y2,r2,x3,y3,r3,s1,s2,s3,xs,ys,rs)
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, Width:=max(x1+r1 "," x2+r2 "," x3+r3 "," xs+rs)*1.1
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, Height:=max(y1+r1 "," y2+r2 "," y3+r3 "," ys+rs)*1.1
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Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs
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Gui, Show
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hwnd1 := WinExist()
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, hbm := CreateDIBSection(Width, Height)
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, hdc := CreateCompatibleDC()
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, obm := SelectObject(hdc, hbm)
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, G := Gdip_GraphicsFromHDC(hdc)
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, Gdip_SetSmoothingMode(G, 4)
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, bWhite := Gdip_BrushCreateSolid(0xffffffff)
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, Gdip_FillRectangle(G, bWhite, 0, 0, Width, Height)
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, pRed := Gdip_CreatePen(0x88ff0000, 3)
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, pGreen := Gdip_CreatePen(0x8800ff00, 3)
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, pBlue := Gdip_CreatePen(0x880000ff, 3)
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, pBlack := Gdip_CreatePen(0x88000000, 3)
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, Gdip_DrawCircle(G, pRed, x1, y1, r1)
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, Gdip_DrawCircle(G, pGreen, x2, y2, r2)
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, Gdip_DrawCircle(G, pBlue, x3, y3, r3)
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, Gdip_DrawCircle(G, pBlack, xs, ys, rs)
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, Gdip_DeletePen(pRed)
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, Gdip_DeletePen(pGreen)
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, Gdip_DeletePen(pBlue)
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, Gdip_DeletePen(pBlack)
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, UpdateLayeredWindow(hwnd1, hdc, 0, 0, Width, Height)
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Return
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GuiEscape:
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GuiClose:
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Exit:
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SelectObject(hdc, obm)
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, DeleteObject(hbm)
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, DeleteDC(hdc)
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, Gdip_DeleteGraphics(G)
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, Gdip_Shutdown(pToken)
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ExitApp
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Apollonius(x1=300,y1=500,r1=50,x2=200,y2=200,r2=150,x3=600,y3=400,r3=100,s1=1,s2=1,s3=1,ByRef xs=0, ByRef ys=0, ByRef rs=0) {
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v11 := 2*x2 - 2*x1
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v12 := 2*y2 - 2*y1
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v13 := x1**2 - x2**2 + y1**2 - y2**2 - r1**2 + r2**2
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v14 := 2*s2*r2 - 2*s1*r1
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v21 := 2*x3 - 2*x2
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v22 := 2*y3 - 2*y2
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v23 := x2**2 - x3**2 + y2**2 - y3**2 - r2**2 + r3**2
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v24 := 2*s3*r3 - 2*s2*r2
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w12 := v12/v11
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w13 := v13/v11
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w14 := v14/v11
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w22 := v22/v21 - w12
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w23 := v23/v21 - w13
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w24 := v24/v21 - w14
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p := -w23/w22
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q := w24/w22
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m := -w12*p - w13
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n := w14 - w12*q
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a := n**2 + q**2 - 1
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b := 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1
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c := x1**2 + m**2 - 2*m*x1 + p**2 + y1**2 - 2*p*y1 - r1**2
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d := b**2 - 4*a*c
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rs := (-b - d**0.5)/(2*a)
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xs := m + n*rs
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ys := p + q*rs
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||||
}
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|
||||
; from http://rosettacode.org/wiki/Greatest_element_of_a_list#AutoHotkey
|
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max(list) {
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Loop Parse, list, `,
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||||
x := x < A_LoopField ? A_LoopField : x
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||||
Return x
|
||||
}
|
||||
|
||||
; Gdip helper function
|
||||
Gdip_DrawCircle(G, pPen, x, y, r) {
|
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Return Gdip_DrawEllipse(G, pPen, x-r, y-r, r*2, r*2)
|
||||
}
|
||||
|
|
@ -0,0 +1,53 @@
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circle1$ = " 0.000, 0.000, 1.000"
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||||
circle2$ = " 4.000, 0.000, 1.000"
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circle3$ = " 2.000, 4.000, 2.000"
|
||||
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subroutine ApolloniusSolver(c1$, c2$, c3$, s1, s2, s3)
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x1 = int(mid(c1$, 3, 1)): y1 = int(mid(c1$, 11, 1)): r1 = int(mid(c1$, 19, 1))
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x2 = int(mid(c2$, 3, 1)): y2 = int(mid(c2$, 11, 1)): r2 = int(mid(c2$, 19, 1))
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x3 = int(mid(c3$, 3, 1)): y3 = int(mid(c3$, 11, 1)): r3 = int(mid(c3$, 19, 1))
|
||||
|
||||
v11 = 2 * x2 - 2 * x1
|
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v12 = 2 * y2 - 2* y1
|
||||
v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2
|
||||
v14 = 2 * s2 * r2 - 2 * s1 * r1
|
||||
|
||||
v21 = 2 * x3 - 2 * x2
|
||||
v22 = 2 * y3 - 2 * y2
|
||||
v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3
|
||||
v24 = 2 * s3 * r3 - 2 * s2 * r2
|
||||
|
||||
w12 = v12 / v11
|
||||
w13 = v13 / v11
|
||||
w14 = v14 / v11
|
||||
|
||||
w22 = v22 / v21 - w12
|
||||
w23 = v23 / v21 - w13
|
||||
w24 = v24 / v21 - w14
|
||||
|
||||
P = 0 - w23 / w22
|
||||
Q = w24 / w22
|
||||
M = 0 - w12 * P - w13
|
||||
N = w14 - w12 * Q
|
||||
|
||||
a = N * N + Q * Q - 1
|
||||
b = 2 * M * N - 2 * N * x1 + 2 * P * Q - 2 * Q * y1 + 2 * s1 * r1
|
||||
c = x1 * x1 + M * M -2 * M * x1 + P * P + y1 * y1 - 2 * P * y1 - r1 * r1
|
||||
|
||||
D = b * b - 4 * a * c
|
||||
|
||||
Radius = (0 - b - sqr(D)) / (2 * a)
|
||||
XPos = M + N * Radius
|
||||
YPos = P + Q * Radius
|
||||
|
||||
print " "; XPos; ", " ; YPos; ", " ; Radius
|
||||
end subroutine
|
||||
|
||||
print " x_pos y_pos radius"
|
||||
print circle1$
|
||||
print circle2$
|
||||
print circle3$
|
||||
print
|
||||
print "R1: " : call ApolloniusSolver(circle1$, circle2$, circle3$, 1, 1, 1)
|
||||
print "R2: " : call ApolloniusSolver(circle1$, circle2$, circle3$, -1, -1, -1)
|
||||
end
|
||||
|
|
@ -0,0 +1,37 @@
|
|||
DIM Circle{x, y, r}
|
||||
DIM Circles{(2)} = Circle{}
|
||||
Circles{(0)}.x = 0 : Circles{(0)}.y = 0 : Circles{(0)}.r = 1
|
||||
Circles{(1)}.x = 4 : Circles{(1)}.y = 0 : Circles{(1)}.r = 1
|
||||
Circles{(2)}.x = 2 : Circles{(2)}.y = 4 : Circles{(2)}.r = 2
|
||||
|
||||
@% = &2030A
|
||||
REM Solution for internal circle:
|
||||
PROCapollonius(Circle{}, Circles{()}, -1, -1, -1)
|
||||
PRINT "Internal: x = ";Circle.x ", y = ";Circle.y ", r = ";Circle.r
|
||||
REM Solution for external circle:
|
||||
PROCapollonius(Circle{}, Circles{()}, 1, 1, 1)
|
||||
PRINT "External: x = ";Circle.x ", y = ";Circle.y ", r = ";Circle.r
|
||||
END
|
||||
|
||||
DEF PROCapollonius(c{}, c{()}, s0, s1, s2)
|
||||
LOCAL x0, x1, x2, y0, y1, y2, r0, r1, r2, a, b, c
|
||||
LOCAL u(), v(), w() : DIM u(2), v(2), w(2)
|
||||
x0 = c{(0)}.x : y0 = c{(0)}.y : r0 = c{(0)}.r
|
||||
x1 = c{(1)}.x : y1 = c{(1)}.y : r1 = c{(1)}.r
|
||||
x2 = c{(2)}.x : y2 = c{(2)}.y : r2 = c{(2)}.r
|
||||
|
||||
u() = 2*y1-2*y0, x0*x0-x1*x1+y0*y0-y1*y1-r0*r0+r1*r1, 2*s1*r1-2*s0*r0
|
||||
v() = 2*y2-2*y1, x1*x1-x2*x2+y1*y1-y2*y2-r1*r1+r2*r2, 2*s2*r2-2*s1*r1
|
||||
w() = u() / (2*x1 - 2*x0)
|
||||
u() = v() / (2*x2 - 2*x1) - w()
|
||||
u() /= u(0)
|
||||
w(1) -= w(0)*u(1)
|
||||
w(2) -= w(0)*u(2)
|
||||
a = w(2)*w(2) + u(2)*u(2) - 1
|
||||
b = -2*w(1)*w(2) - 2*w(2)*x1 - 2*u(1)*u(2) - 2*u(2)*y1 + 2*s1*r1
|
||||
c = x1*x1 + w(1)*w(1) + 2*w(1)*x1 + u(1)*u(1) + y1*y1 + 2*u(1)*y1 - r1*r1
|
||||
|
||||
c.r = (-b - SQR(b^2 - 4*a*c)) / (2*a)
|
||||
c.x = c.r * w(2) - w(1)
|
||||
c.y = c.r * u(2) - u(1)
|
||||
ENDPROC
|
||||
156
Task/Problem-of-Apollonius/C-sharp/problem-of-apollonius.cs
Normal file
156
Task/Problem-of-Apollonius/C-sharp/problem-of-apollonius.cs
Normal file
|
|
@ -0,0 +1,156 @@
|
|||
using System;
|
||||
|
||||
namespace ApolloniusProblemCalc
|
||||
{
|
||||
class Program
|
||||
{
|
||||
static float rs = 0;
|
||||
static float xs = 0;
|
||||
static float ys = 0;
|
||||
|
||||
public static void Main(string[] args)
|
||||
{
|
||||
float gx1;
|
||||
float gy1;
|
||||
float gr1;
|
||||
float gx2;
|
||||
float gy2;
|
||||
float gr2;
|
||||
float gx3;
|
||||
float gy3;
|
||||
float gr3;
|
||||
|
||||
//----------Enter values for the given circles here----------
|
||||
gx1 = 0;
|
||||
gy1 = 0;
|
||||
gr1 = 1;
|
||||
gx2 = 4;
|
||||
gy2 = 0;
|
||||
gr2 = 1;
|
||||
gx3 = 2;
|
||||
gy3 = 4;
|
||||
gr3 = 2;
|
||||
//-----------------------------------------------------------
|
||||
|
||||
for (int i = 1; i <= 8; i++)
|
||||
{
|
||||
SolveTheApollonius(i, gx1, gy1, gr1, gx2, gy2, gr2, gx3, gy3, gr3);
|
||||
|
||||
|
||||
if (i == 1)
|
||||
{
|
||||
Console.WriteLine("X of point of the " + i + "st solution: " + xs.ToString());
|
||||
Console.WriteLine("Y of point of the " + i + "st solution: " + ys.ToString());
|
||||
Console.WriteLine(i + "st Solution circle's radius: " + rs.ToString());
|
||||
}
|
||||
else if (i == 2)
|
||||
{
|
||||
Console.WriteLine("X of point of the " + i + "ed solution: " + xs.ToString());
|
||||
Console.WriteLine("Y of point of the " + i + "ed solution: " + ys.ToString());
|
||||
Console.WriteLine(i + "ed Solution circle's radius: " + rs.ToString());
|
||||
}
|
||||
else if(i == 3)
|
||||
{
|
||||
Console.WriteLine("X of point of the " + i + "rd solution: " + xs.ToString());
|
||||
Console.WriteLine("Y of point of the " + i + "rd solution: " + ys.ToString());
|
||||
Console.WriteLine(i + "rd Solution circle's radius: " + rs.ToString());
|
||||
}
|
||||
else
|
||||
{
|
||||
Console.WriteLine("X of point of the " + i + "th solution: " + xs.ToString());
|
||||
Console.WriteLine("Y of point of the " + i + "th solution: " + ys.ToString());
|
||||
Console.WriteLine(i + "th Solution circle's radius: " + rs.ToString());
|
||||
}
|
||||
|
||||
Console.WriteLine();
|
||||
}
|
||||
|
||||
|
||||
Console.ReadKey(true);
|
||||
}
|
||||
|
||||
private static void SolveTheApollonius(int calcCounter, float x1, float y1, float r1, float x2, float y2, float r2, float x3, float y3, float r3)
|
||||
{
|
||||
float s1 = 1;
|
||||
float s2 = 1;
|
||||
float s3 = 1;
|
||||
|
||||
if (calcCounter == 2)
|
||||
{
|
||||
s1 = -1;
|
||||
s2 = -1;
|
||||
s3 = -1;
|
||||
}
|
||||
else if (calcCounter == 3)
|
||||
{
|
||||
s1 = 1;
|
||||
s2 = -1;
|
||||
s3 = -1;
|
||||
}
|
||||
else if (calcCounter == 4)
|
||||
{
|
||||
s1 = -1;
|
||||
s2 = 1;
|
||||
s3 = -1;
|
||||
}
|
||||
else if (calcCounter == 5)
|
||||
{
|
||||
s1 = -1;
|
||||
s2 = -1;
|
||||
s3 = 1;
|
||||
}
|
||||
else if (calcCounter == 6)
|
||||
{
|
||||
s1 = 1;
|
||||
s2 = 1;
|
||||
s3 = -1;
|
||||
}
|
||||
else if (calcCounter == 7)
|
||||
{
|
||||
s1 = -1;
|
||||
s2 = 1;
|
||||
s3 = 1;
|
||||
}
|
||||
else if (calcCounter == 8)
|
||||
{
|
||||
s1 = 1;
|
||||
s2 = -1;
|
||||
s3 = 1;
|
||||
}
|
||||
|
||||
//This calculation to solve for the solution circles is cited from the Java version
|
||||
float v11 = 2 * x2 - 2 * x1;
|
||||
float v12 = 2 * y2 - 2 * y1;
|
||||
float v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2;
|
||||
float v14 = 2 * s2 * r2 - 2 * s1 * r1;
|
||||
|
||||
float v21 = 2 * x3 - 2 * x2;
|
||||
float v22 = 2 * y3 - 2 * y2;
|
||||
float v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3;
|
||||
float v24 = 2 * s3 * r3 - 2 * s2 * r2;
|
||||
|
||||
float w12 = v12 / v11;
|
||||
float w13 = v13 / v11;
|
||||
float w14 = v14 / v11;
|
||||
|
||||
float w22 = v22 / v21 - w12;
|
||||
float w23 = v23 / v21 - w13;
|
||||
float w24 = v24 / v21 - w14;
|
||||
|
||||
float P = -w23 / w22;
|
||||
float Q = w24 / w22;
|
||||
float M = -w12 * P - w13;
|
||||
float N = w14 - w12 * Q;
|
||||
|
||||
float a = N * N + Q * Q - 1;
|
||||
float b = 2 * M * N - 2 * N * x1 + 2 * P * Q - 2 * Q * y1 + 2 * s1 * r1;
|
||||
float c = x1 * x1 + M * M - 2 * M * x1 + P * P + y1 * y1 - 2 * P * y1 - r1 * r1;
|
||||
|
||||
float D = b * b - 4 * a * c;
|
||||
|
||||
rs = (-b - float.Parse(Math.Sqrt(D).ToString())) / (2 * float.Parse(a.ToString()));
|
||||
xs = M + N * rs;
|
||||
ys = P + Q * rs;
|
||||
}
|
||||
}
|
||||
}
|
||||
110
Task/Problem-of-Apollonius/C/problem-of-apollonius.c
Normal file
110
Task/Problem-of-Apollonius/C/problem-of-apollonius.c
Normal file
|
|
@ -0,0 +1,110 @@
|
|||
#include <stdio.h>
|
||||
#include <tgmath.h>
|
||||
|
||||
#define VERBOSE 0
|
||||
#define for3 for(int i = 0; i < 3; i++)
|
||||
|
||||
typedef complex double vec;
|
||||
typedef struct { vec c; double r; } circ;
|
||||
|
||||
#define re(x) creal(x)
|
||||
#define im(x) cimag(x)
|
||||
#define cp(x) re(x), im(x)
|
||||
#define CPLX "(%6.3f,%6.3f)"
|
||||
#define CPLX3 CPLX" "CPLX" "CPLX
|
||||
|
||||
double cross(vec a, vec b) { return re(a) * im(b) - im(a) * re(b); }
|
||||
double abs2(vec a) { return a * conj(a); }
|
||||
|
||||
int apollonius_in(circ aa[], int ss[], int flip, int divert)
|
||||
{
|
||||
vec n[3], x[3], t[3], a, b, center;
|
||||
int s[3], iter = 0, res = 0;
|
||||
double diff = 1, diff_old = -1, axb, d, r;
|
||||
|
||||
for3 {
|
||||
s[i] = ss[i] ? 1 : -1;
|
||||
x[i] = aa[i].c;
|
||||
}
|
||||
|
||||
while (diff > 1e-20) {
|
||||
a = x[0] - x[2], b = x[1] - x[2];
|
||||
diff = 0;
|
||||
axb = -cross(a, b);
|
||||
d = sqrt(abs2(a) * abs2(b) * abs2(a - b));
|
||||
|
||||
if (VERBOSE) {
|
||||
const char *z = 1 + "-0+";
|
||||
printf("%c%c%c|%c%c|",
|
||||
z[s[0]], z[s[1]], z[s[2]], z[flip], z[divert]);
|
||||
printf(CPLX3, cp(x[0]), cp(x[1]), cp(x[2]));
|
||||
}
|
||||
|
||||
/* r and center represent an arc through points x[i]. Each step,
|
||||
we'll deform this arc by pushing or pulling some point on it
|
||||
towards the edge of each given circle. */
|
||||
r = fabs(d / (2 * axb));
|
||||
center = (abs2(a)*b - abs2(b)*a) / (2 * axb) * I + x[2];
|
||||
|
||||
/* maybe the "arc" is actually straight line; then we have two
|
||||
choices in defining "push" and "pull", so try both */
|
||||
if (!axb && flip != -1 && !divert) {
|
||||
if (!d) { /* generally means circle centers overlap */
|
||||
printf("Given conditions confused me.\n");
|
||||
return 0;
|
||||
}
|
||||
|
||||
if (VERBOSE) puts("\n[divert]");
|
||||
divert = 1;
|
||||
res = apollonius_in(aa, ss, -1, 1);
|
||||
}
|
||||
|
||||
/* if straight line, push dir is its norm; else it's away from center */
|
||||
for3 n[i] = axb ? aa[i].c - center : a * I * flip;
|
||||
for3 t[i] = aa[i].c + n[i] / cabs(n[i]) * aa[i].r * s[i];
|
||||
|
||||
/* diff: how much tangent points have moved since last iteration */
|
||||
for3 diff += abs2(t[i] - x[i]), x[i] = t[i];
|
||||
|
||||
if (VERBOSE) printf(" %g\n", diff);
|
||||
|
||||
/* keep an eye on the total diff: failing to converge means no solution */
|
||||
if (diff >= diff_old && diff_old >= 0)
|
||||
if (iter++ > 20) return res;
|
||||
|
||||
diff_old = diff;
|
||||
}
|
||||
|
||||
printf("found: ");
|
||||
if (axb) printf("circle "CPLX", r = %f\n", cp(center), r);
|
||||
else printf("line "CPLX3"\n", cp(x[0]), cp(x[1]), cp(x[2]));
|
||||
|
||||
return res + 1;
|
||||
}
|
||||
|
||||
int apollonius(circ aa[])
|
||||
{
|
||||
int s[3], i, sum = 0;
|
||||
for (i = 0; i < 8; i++) {
|
||||
s[0] = i & 1, s[1] = i & 2, s[2] = i & 4;
|
||||
|
||||
/* internal or external results of a zero-radius circle are the same */
|
||||
if (s[0] && !aa[0].r) continue;
|
||||
if (s[1] && !aa[1].r) continue;
|
||||
if (s[2] && !aa[2].r) continue;
|
||||
sum += apollonius_in(aa, s, 1, 0);
|
||||
}
|
||||
return sum;
|
||||
}
|
||||
|
||||
int main()
|
||||
{
|
||||
circ a[3] = {{0, 1}, {4, 1}, {2 + 4 * I, 1}};
|
||||
circ b[3] = {{-3, 2}, {0, 1}, {3, 2}};
|
||||
circ c[3] = {{-2, 1}, {0, 1}, {2 * I, 1}};
|
||||
//circ c[3] = {{0, 1}, {0, 2}, {0, 3}}; <-- a fun one
|
||||
|
||||
puts("set 1"); apollonius(a);
|
||||
puts("set 2"); apollonius(b);
|
||||
puts("set 3"); apollonius(c);
|
||||
}
|
||||
|
|
@ -0,0 +1,51 @@
|
|||
class Circle
|
||||
constructor: (@x, @y, @r) ->
|
||||
|
||||
apollonius = (c1, c2, c3, s1=1, s2=1, s3=1) ->
|
||||
[x1, y1, r1] = [c1.x, c1.y, c1.r]
|
||||
[x2, y2, r2] = [c2.x, c2.y, c2.r]
|
||||
[x3, y3, r3] = [c3.x, c3.y, c3.r]
|
||||
|
||||
sq = (n) -> n*n
|
||||
|
||||
v11 = 2*x2 - 2*x1
|
||||
v12 = 2*y2 - 2*y1
|
||||
v13 = sq(x1) - sq(x2) + sq(y1) - sq(y2) - sq(r1) + sq(r2)
|
||||
v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
v21 = 2*x3 - 2*x2
|
||||
v22 = 2*y3 - 2*y2
|
||||
v23 = sq(x2) - sq(x3) + sq(y2) - sq(y3) - sq(r2) + sq(r3)
|
||||
v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21 - w12
|
||||
w23 = v23/v21 - w13
|
||||
w24 = v24/v21 - w14
|
||||
|
||||
p = -w23/w22
|
||||
q = w24/w22
|
||||
m = -w12*p - w13
|
||||
n = w14 - w12*q
|
||||
|
||||
a = sq(n) + sq(q) - 1
|
||||
b = 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1
|
||||
c = sq(x1) + sq(m) - 2*m*x1 + sq(p) + sq(y1) - 2*p*y1 - sq(r1)
|
||||
|
||||
d = sq(b) - 4*a*c
|
||||
rs = (-b - Math.sqrt(d)) / (2*a)
|
||||
xs = m + n*rs
|
||||
ys = p + q*rs
|
||||
|
||||
new Circle(xs, ys, rs)
|
||||
|
||||
|
||||
console.log c1 = new Circle(0, 0, 1)
|
||||
console.log c2 = new Circle(2, 4, 2)
|
||||
console.log c3 = new Circle(4, 0, 1)
|
||||
|
||||
console.log apollonius(c1, c2, c3)
|
||||
console.log apollonius(c1, c2, c3, -1, -1, -1)
|
||||
|
|
@ -0,0 +1,6 @@
|
|||
> coffee foo.coffee
|
||||
{ x: 0, y: 0, r: 1 }
|
||||
{ x: 2, y: 4, r: 2 }
|
||||
{ x: 4, y: 0, r: 1 }
|
||||
{ x: 2, y: 2.1, r: 3.9 }
|
||||
{ x: 2, y: 0.8333333333333333, r: 1.1666666666666667 }
|
||||
80
Task/Problem-of-Apollonius/D/problem-of-apollonius.d
Normal file
80
Task/Problem-of-Apollonius/D/problem-of-apollonius.d
Normal file
|
|
@ -0,0 +1,80 @@
|
|||
import std.stdio, std.math;
|
||||
|
||||
immutable struct Circle { double x, y, r; }
|
||||
enum Tangent { externally, internally }
|
||||
|
||||
/**
|
||||
Solves the Problem of Apollonius (finding a circle tangent to three
|
||||
other circles in the plane).
|
||||
|
||||
Params:
|
||||
c1 = First circle of the problem.
|
||||
c2 = Second circle of the problem.
|
||||
c3 = Third circle of the problem.
|
||||
t1 = How is the solution tangent (externally or internally) to c1.
|
||||
t2 = How is the solution tangent (externally or internally) to c2.
|
||||
t3 = How is the solution tangent (externally or internally) to c3.
|
||||
|
||||
Returns: The Circle that is tangent to c1, c2 and c3.
|
||||
*/
|
||||
Circle solveApollonius(in Circle c1, in Circle c2, in Circle c3,
|
||||
in Tangent t1, in Tangent t2, in Tangent t3)
|
||||
pure nothrow @safe @nogc {
|
||||
alias Imd = immutable(double);
|
||||
Imd s1 = (t1 == Tangent.externally) ? 1.0 : -1.0;
|
||||
Imd s2 = (t2 == Tangent.externally) ? 1.0 : -1.0;
|
||||
Imd s3 = (t3 == Tangent.externally) ? 1.0 : -1.0;
|
||||
|
||||
Imd v11 = 2 * c2.x - 2 * c1.x;
|
||||
Imd v12 = 2 * c2.y - 2 * c1.y;
|
||||
Imd v13 = c1.x ^^ 2 - c2.x ^^ 2 +
|
||||
c1.y ^^ 2 - c2.y ^^ 2 -
|
||||
c1.r ^^ 2 + c2.r ^^ 2;
|
||||
Imd v14 = 2 * s2 * c2.r - 2 * s1 * c1.r;
|
||||
|
||||
Imd v21 = 2 * c3.x - 2 * c2.x;
|
||||
Imd v22 = 2 * c3.y - 2 * c2.y;
|
||||
Imd v23 = c2.x ^^ 2 - c3.x ^^ 2 +
|
||||
c2.y ^^ 2 - c3.y ^^ 2 -
|
||||
c2.r ^^ 2 + c3.r ^^ 2;
|
||||
Imd v24 = 2 * s3 * c3.r - 2 * s2 * c2.r;
|
||||
|
||||
Imd w12 = v12 / v11;
|
||||
Imd w13 = v13 / v11;
|
||||
Imd w14 = v14 / v11;
|
||||
|
||||
Imd w22 = v22 / v21 - w12;
|
||||
Imd w23 = v23 / v21 - w13;
|
||||
Imd w24 = v24 / v21 - w14;
|
||||
|
||||
Imd P = -w23 / w22;
|
||||
Imd Q = w24 / w22;
|
||||
Imd M = -w12 * P - w13;
|
||||
Imd N = w14 - w12 * Q;
|
||||
|
||||
Imd a = N * N + Q ^^ 2 - 1;
|
||||
Imd b = 2 * M * N - 2 * N * c1.x +
|
||||
2 * P * Q - 2 * Q * c1.y +
|
||||
2 * s1 * c1.r;
|
||||
Imd c = c1.x ^^ 2 + M ^^ 2 - 2 * M * c1.x +
|
||||
P ^^ 2 + c1.y ^^ 2 - 2 * P * c1.y - c1.r ^^ 2;
|
||||
|
||||
// find a root of a quadratic equation.
|
||||
// This requires the circle centers not to be e.g. colinear
|
||||
Imd D = b ^^ 2 - 4 * a * c;
|
||||
Imd rs = (-b - D.sqrt) / (2 * a);
|
||||
|
||||
return Circle(M + N * rs, P + Q * rs, rs);
|
||||
}
|
||||
|
||||
void main() {
|
||||
immutable c1 = Circle(0.0, 0.0, 1.0);
|
||||
immutable c2 = Circle(4.0, 0.0, 1.0);
|
||||
immutable c3 = Circle(2.0, 4.0, 2.0);
|
||||
|
||||
alias Te = Tangent.externally;
|
||||
solveApollonius(c1, c2, c3, Te, Te, Te).writeln;
|
||||
|
||||
alias Ti = Tangent.internally;
|
||||
solveApollonius(c1, c2, c3, Ti, Ti, Ti).writeln;
|
||||
}
|
||||
|
|
@ -0,0 +1,56 @@
|
|||
type Circle(Array center, Float radius) with Lookup
|
||||
func Circle.ToString() =>
|
||||
"Circle[x=\(this.center[0]),y=\(this.center[1]),r=\(this.radius)]"
|
||||
|
||||
func solveApollonius(Circle c1, Circle c2, Circle c3, Float s1, Float s2, Float s3) {
|
||||
let x1 = c1.center[0]
|
||||
let y1 = c1.center[1]
|
||||
let r1 = c1.radius
|
||||
let x2 = c2.center[0]
|
||||
let y2 = c2.center[1]
|
||||
let r2 = c2.radius
|
||||
let x3 = c3.center[0]
|
||||
let y3 = c3.center[1]
|
||||
let r3 = c3.radius
|
||||
|
||||
let v11 = 2.0 * x2 - 2.0 * x1
|
||||
let v12 = 2.0 * y2 - 2.0 *y1
|
||||
let v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2
|
||||
let v14 = 2.0 * s2 * r2 - 2.0 * s1 * r1
|
||||
|
||||
let v21 = 2.0 * x3 - 2.0 * x2
|
||||
let v22 = 2.0 * y3 - 2.0 * y2
|
||||
let v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3
|
||||
let v24 = 2.0 * s3 * r3 - 2 * s2 * r2
|
||||
|
||||
let w12 = v12 / v11
|
||||
let w13 = v13 / v11
|
||||
let w14 = v14 / v11
|
||||
|
||||
let w22 = v22 / v21-w12
|
||||
let w23 = v23 / v21-w13
|
||||
let w24 = v24 / v21-w14
|
||||
|
||||
let p = -w23 / w22
|
||||
let q = w24 / w22
|
||||
let m = -w12 * p - w13
|
||||
let n = w14 - w12 * q
|
||||
|
||||
let a = n * n + q * q - 1.0
|
||||
let b = 2.0 * m * n - 2.0 * n * x1 + 2 * p * q - 2.0 * q * y1 + 2.0 * s1 * r1
|
||||
let c = x1 * x1 + m * m - 2.0 * m * x1 + p * p + y1 * y1 - 2.0 * p * y1 - r1 * r1
|
||||
|
||||
let d = b * b - 4.0 * a * c
|
||||
let rs = (-b - sqrt(d)) / (2.0 * a)
|
||||
let xs = m + n * rs
|
||||
let ys = p + q * rs
|
||||
|
||||
Circle(center: [xs,ys], radius: rs)
|
||||
}
|
||||
|
||||
let c1 = Circle(center: [0.0, 0.0], radius: 1.0)
|
||||
let c2 = Circle(center: [4.0, 0.0], radius: 1.0)
|
||||
let c3 = Circle(center: [2.0, 4.0], radius: 2.0)
|
||||
|
||||
print(solveApollonius(c1, c2, c3, 1.0, 1.0, 1.0))
|
||||
print(solveApollonius(c1, c2, c3, -1.0, -1.0, -1.0))
|
||||
|
|
@ -0,0 +1,33 @@
|
|||
defmodule Circle do
|
||||
def apollonius(c1, c2, c3, s1, s2, s3) do
|
||||
{x1, y1, r1} = c1
|
||||
{w12, w13, w14} = calc(c1, c2, s1, s2)
|
||||
{u22, u23, u24} = calc(c2, c3, s2, s3)
|
||||
{w22, w23, w24} = {u22 - w12, u23 - w13, u24 - w14}
|
||||
|
||||
p = -w23 / w22
|
||||
q = w24 / w22
|
||||
m = -w12 * p - w13
|
||||
n = w14 - w12 * q
|
||||
|
||||
a = n*n + q*q - 1
|
||||
b = 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1
|
||||
c = x1*x1 + m*m - 2*m*x1 + p*p + y1*y1 - 2*p*y1 - r1*r1
|
||||
|
||||
d = b*b - 4*a*c
|
||||
rs = (-b - :math.sqrt(d)) / (2*a)
|
||||
{m + n*rs, p + q*rs, rs}
|
||||
end
|
||||
|
||||
defp calc({x1, y1, r1}, {x2, y2, r2}, s1, s2) do
|
||||
v1 = x2 - x1
|
||||
{(y2 - y1) / v1, (x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2) / (2*v1), (s2*r2 - s1*r1) / v1}
|
||||
end
|
||||
end
|
||||
|
||||
c1 = {0, 0, 1}
|
||||
c2 = {2, 4, 2}
|
||||
c3 = {4, 0, 1}
|
||||
|
||||
IO.inspect Circle.apollonius(c1, c2, c3, 1, 1, 1)
|
||||
IO.inspect Circle.apollonius(c1, c2, c3, -1, -1, -1)
|
||||
67
Task/Problem-of-Apollonius/F-Sharp/problem-of-apollonius.fs
Normal file
67
Task/Problem-of-Apollonius/F-Sharp/problem-of-apollonius.fs
Normal file
|
|
@ -0,0 +1,67 @@
|
|||
type point = { x:float; y:float }
|
||||
type circle = { center: point; radius: float; }
|
||||
|
||||
let new_circle x y r =
|
||||
{ center = { x=x; y=y }; radius = r }
|
||||
|
||||
let print_circle c =
|
||||
printfn "Circle(x=%.2f, y=%.2f, r=%.2f)"
|
||||
c.center.x c.center.y c.radius
|
||||
|
||||
let xyr c = c.center.x, c.center.y, c.radius
|
||||
|
||||
let solve_apollonius c1 c2 c3
|
||||
s1 s2 s3 =
|
||||
|
||||
let x1, y1, r1 = xyr c1
|
||||
let x2, y2, r2 = xyr c2
|
||||
let x3, y3, r3 = xyr c3
|
||||
|
||||
let v11 = 2. * x2 - 2. * x1
|
||||
let v12 = 2. * y2 - 2. * y1
|
||||
let v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
let v14 = (2. * s2 * r2) - (2. * s1 * r1)
|
||||
|
||||
let v21 = 2. * x3 - 2. * x2
|
||||
let v22 = 2. * y3 - 2. * y2
|
||||
let v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
let v24 = (2. * s3 * r3) - (2. * s2 * r2)
|
||||
|
||||
let w12 = v12 / v11
|
||||
let w13 = v13 / v11
|
||||
let w14 = v14 / v11
|
||||
|
||||
let w22 = v22 / v21 - w12
|
||||
let w23 = v23 / v21 - w13
|
||||
let w24 = v24 / v21 - w14
|
||||
|
||||
let p = - w23 / w22
|
||||
let q = w24 / w22
|
||||
let m = - w12 * p - w13
|
||||
let n = w14 - w12 * q
|
||||
|
||||
let a = n*n + q*q - 1.
|
||||
let b = 2.*m*n - 2.*n*x1 + 2.*p*q - 2.*q*y1 + 2.*s1*r1
|
||||
let c = x1*x1 + m*m - 2.*m*x1 + p*p + y1*y1 - 2.*p*y1 - r1*r1
|
||||
|
||||
let d = b * b - 4. * a * c
|
||||
let rs = (- b - (sqrt d)) / (2. * a)
|
||||
|
||||
let xs = m + n * rs
|
||||
let ys = p + q * rs
|
||||
|
||||
new_circle xs ys rs
|
||||
|
||||
|
||||
[<EntryPoint>]
|
||||
let main argv =
|
||||
let c1 = new_circle 0. 0. 1.
|
||||
let c2 = new_circle 4. 0. 1.
|
||||
let c3 = new_circle 2. 4. 2.
|
||||
|
||||
let r1 = solve_apollonius c1 c2 c3 1. 1. 1.
|
||||
print_circle r1
|
||||
|
||||
let r2 = solve_apollonius c1 c2 c3 (-1.) (-1.) (-1.)
|
||||
print_circle r2
|
||||
0
|
||||
72
Task/Problem-of-Apollonius/Fortran/problem-of-apollonius.f
Normal file
72
Task/Problem-of-Apollonius/Fortran/problem-of-apollonius.f
Normal file
|
|
@ -0,0 +1,72 @@
|
|||
program Apollonius
|
||||
implicit none
|
||||
|
||||
integer, parameter :: dp = selected_real_kind(15)
|
||||
|
||||
type circle
|
||||
real(dp) :: x
|
||||
real(dp) :: y
|
||||
real(dp) :: radius
|
||||
end type
|
||||
|
||||
type(circle) :: c1 , c2, c3, r
|
||||
|
||||
c1 = circle(0.0, 0.0, 1.0)
|
||||
c2 = circle(4.0, 0.0, 1.0)
|
||||
c3 = circle(2.0, 4.0, 2.0)
|
||||
|
||||
write(*, "(a,3f12.8))") "External tangent:", SolveApollonius(c1, c2, c3, 1, 1, 1)
|
||||
write(*, "(a,3f12.8))") "Internal tangent:", SolveApollonius(c1, c2, c3, -1, -1, -1)
|
||||
|
||||
contains
|
||||
|
||||
function SolveApollonius(c1, c2, c3, s1, s2, s3) result(res)
|
||||
type(circle) :: res
|
||||
type(circle), intent(in) :: c1, c2, c3
|
||||
integer, intent(in) :: s1, s2, s3
|
||||
|
||||
real(dp) :: x1, x2, x3, y1, y2, y3, r1, r2, r3
|
||||
real(dp) :: v11, v12, v13, v14
|
||||
real(dp) :: v21, v22, v23, v24
|
||||
real(dp) :: w12, w13, w14
|
||||
real(dp) :: w22, w23, w24
|
||||
real(dp) :: p, q, m, n, a, b, c, det
|
||||
|
||||
x1 = c1%x; x2 = c2%x; x3 = c3%x
|
||||
y1 = c1%y; y2 = c2%y; y3 = c3%y
|
||||
r1 = c1%radius; r2 = c2%radius; r3 = c3%radius
|
||||
|
||||
v11 = 2*x2 - 2*x1
|
||||
v12 = 2*y2 - 2*y1
|
||||
v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
v21 = 2*x3 - 2*x2
|
||||
v22 = 2*y3 - 2*y2
|
||||
v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21-w12
|
||||
w23 = v23/v21-w13
|
||||
w24 = v24/v21-w14
|
||||
|
||||
p = -w23/w22
|
||||
q = w24/w22
|
||||
m = -w12*P - w13
|
||||
n = w14 - w12*q
|
||||
|
||||
a = n*n + q*q - 1
|
||||
b = 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1
|
||||
c = x1*x1 + m*m - 2*m*x1 + p*p + y1*y1 - 2*p*y1 - r1*r1
|
||||
|
||||
det = b*b - 4*a*c
|
||||
res%radius = (-b-sqrt(det)) / (2*a)
|
||||
res%x = m + n*res%radius
|
||||
res%y = p + q*res%radius
|
||||
|
||||
end function
|
||||
end program
|
||||
|
|
@ -0,0 +1,58 @@
|
|||
Dim As String circle1 = " 0.000, 0.000, 1.000"
|
||||
Dim As String circle2 = " 4.000, 0.000, 1.000"
|
||||
Dim As String circle3 = " 2.000, 4.000, 2.000"
|
||||
|
||||
Sub ApolloniusSolver(c1 As String, c2 As String, c3 As String, s1 As Single, s2 As Single, s3 As Single)
|
||||
Dim As Single x1, x2, x3, y1, y2, y3, r1, r2, r3
|
||||
Dim As Single v11, v12, v13, v14, v21, v22, v23, v24, w12, w13, w14
|
||||
Dim As Single w22, w23, w24,P, Q, M, N, a, b, c, D
|
||||
Dim As Single Radius, XPos, YPos
|
||||
|
||||
x1 = Val(Mid(c1, 3, 1)): y1 = Val(Mid(c1, 11, 1)): r1 = Val(Mid(c1, 19, 1))
|
||||
x2 = Val(Mid(c2, 3, 1)): y2 = Val(Mid(c2, 11, 1)): r2 = Val(Mid(c2, 19, 1))
|
||||
x3 = Val(Mid(c3, 3, 1)): y3 = Val(Mid(c3, 11, 1)): r3 = Val(Mid(c3, 19, 1))
|
||||
|
||||
v11 = 2 * x2 - 2 * x1
|
||||
v12 = 2 * y2 - 2* y1
|
||||
v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2
|
||||
v14 = 2 * s2 * r2 - 2 * s1 * r1
|
||||
|
||||
v21 = 2 * x3 - 2 * x2
|
||||
v22 = 2 * y3 - 2 * y2
|
||||
v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3
|
||||
v24 = 2 * s3 * r3 - 2 * s2 * r2
|
||||
|
||||
w12 = v12 / v11
|
||||
w13 = v13 / v11
|
||||
w14 = v14 / v11
|
||||
|
||||
w22 = v22 / v21 - w12
|
||||
w23 = v23 / v21 - w13
|
||||
w24 = v24 / v21 - w14
|
||||
|
||||
P = 0 - w23 / w22
|
||||
Q = w24 / w22
|
||||
M = 0 - w12 * P - w13
|
||||
N = w14 - w12 * Q
|
||||
|
||||
a = N * N + Q * Q - 1
|
||||
b = 2 * M * N - 2 * N * x1 + 2 * P * Q - 2 * Q * y1 + 2 * s1 * r1
|
||||
c = x1 * x1 + M * M -2 * M * x1 + P * P + y1 * y1 - 2 * P * y1 - r1 * r1
|
||||
|
||||
D = b * b - 4 * a * c
|
||||
|
||||
Radius = (0 - b - Sqr(D)) / (2 * a)
|
||||
XPos = M + N * Radius
|
||||
YPos = P + Q * Radius
|
||||
|
||||
Print Using " ##.###, ##.###, ##.###"; XPos; YPos; Radius
|
||||
End Sub
|
||||
|
||||
Print " x_pos y_pos radius"
|
||||
Print circle1
|
||||
Print circle2
|
||||
Print circle3
|
||||
Print
|
||||
Print "R1: " : ApolloniusSolver(circle1, circle2, circle3, 1, 1, 1)
|
||||
Print "R2: " : ApolloniusSolver(circle1, circle2, circle3, -1, -1, -1)
|
||||
Sleep
|
||||
|
|
@ -0,0 +1,88 @@
|
|||
Problem of Apollonius
|
||||
|
||||
include "NSLog.incl"
|
||||
|
||||
begin record Circle
|
||||
CGPoint center
|
||||
double radius
|
||||
CFStringRef locator
|
||||
end record
|
||||
|
||||
local fn CircleToString( c as Circle ) as CFStringRef
|
||||
end fn = fn StringWithFormat( @"%@ Circle( x = %0.3f, y = %0.3f, radius = %0.3f )", c.locator, c.center.x, c.center.y, c.radius )
|
||||
|
||||
local fn SolveApollonius( c1 as Circle, c2 as Circle, c3 as Circle, s1 as Double, s2 as Double, s3 as Double ) as Circle
|
||||
'~'1
|
||||
Circle result
|
||||
|
||||
double x1 = c1.center.x
|
||||
double y1 = c1.center.y
|
||||
double r1 = c1.radius
|
||||
|
||||
double x2 = c2.center.x
|
||||
double y2 = c2.center.y
|
||||
double r2 = c2.radius
|
||||
|
||||
double x3 = c3.center.x
|
||||
double y3 = c3.center.y
|
||||
double r3 = c3.radius
|
||||
|
||||
double v11 = 2*x2 - 2*x1
|
||||
double v12 = 2*y2 - 2*y1
|
||||
double v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
double v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
double v21 = 2*x3 - 2*x2
|
||||
double v22 = 2*y3 - 2*y2
|
||||
double v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
double v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
double w12 = v12/v11
|
||||
double w13 = v13/v11
|
||||
double w14 = v14/v11
|
||||
|
||||
double w22 = v22/v21-w12
|
||||
double w23 = v23/v21-w13
|
||||
double w24 = v24/v21-w14
|
||||
|
||||
double P = -w23/w22
|
||||
double Q = w24/w22
|
||||
double M = -w12*P-w13
|
||||
double N = w14 - w12*Q
|
||||
|
||||
double a = N*N + Q*Q - 1
|
||||
double b = 2*M*N - 2*N*x1 + 2*P*Q - 2*Q*y1 + 2*s1*r1
|
||||
double c = x1*x1 + M*M - 2*M*x1 + P*P + y1*y1 - 2*P*y1 - r1*r1
|
||||
|
||||
double D = b*b-4*a*c
|
||||
|
||||
double rs = (-b - sqr(D)) / (2*a)
|
||||
double xs = M + N * rs
|
||||
double ys = P + Q * rs
|
||||
|
||||
result.center.x = xs
|
||||
result.center.y = ys
|
||||
result.radius = rs
|
||||
|
||||
if ( s1 < 1 )
|
||||
result.locator = @"Internal Tangent:"
|
||||
else
|
||||
result.locator = @"External Tangent:"
|
||||
end if
|
||||
end fn = result
|
||||
|
||||
Circle c1, c2, c3, c
|
||||
|
||||
c1.center.x = 0.0 : c1.center.y = 0.0 : c1.radius = 1.0
|
||||
c2.center.x = 4.0 : c2.center.y = 0.0 : c2.radius = 1.0
|
||||
c3.center.x = 2.0 : c3.center.y = 4.0 : c3.radius = 2.0
|
||||
|
||||
// External tangent
|
||||
c = fn SolveApollonius( c1, c2, c3, 1, 1, 1 )
|
||||
NSLog( @"%@", fn CircleToString( c ) )
|
||||
|
||||
// Internal tangent
|
||||
c = fn SolveApollonius( c1, c2, c3, -1, -1, -1 )
|
||||
NSLog( @"%@", fn CircleToString( c ) )
|
||||
|
||||
HandleEvents
|
||||
64
Task/Problem-of-Apollonius/Go/problem-of-apollonius.go
Normal file
64
Task/Problem-of-Apollonius/Go/problem-of-apollonius.go
Normal file
|
|
@ -0,0 +1,64 @@
|
|||
package main
|
||||
|
||||
import (
|
||||
"fmt"
|
||||
"math"
|
||||
)
|
||||
|
||||
type circle struct {
|
||||
x, y, r float64
|
||||
}
|
||||
|
||||
func main() {
|
||||
c1 := circle{0, 0, 1}
|
||||
c2 := circle{4, 0, 1}
|
||||
c3 := circle{2, 4, 2}
|
||||
fmt.Println(ap(c1, c2, c3, true))
|
||||
fmt.Println(ap(c1, c2, c3, false))
|
||||
}
|
||||
|
||||
func ap(c1, c2, c3 circle, s bool) circle {
|
||||
x1sq := c1.x * c1.x
|
||||
y1sq := c1.y * c1.y
|
||||
r1sq := c1.r * c1.r
|
||||
x2sq := c2.x * c2.x
|
||||
y2sq := c2.y * c2.y
|
||||
r2sq := c2.r * c2.r
|
||||
x3sq := c3.x * c3.x
|
||||
y3sq := c3.y * c3.y
|
||||
r3sq := c3.r * c3.r
|
||||
v11 := 2 * (c2.x - c1.x)
|
||||
v12 := 2 * (c2.y - c1.y)
|
||||
v13 := x1sq - x2sq + y1sq - y2sq - r1sq + r2sq
|
||||
v14 := 2 * (c2.r - c1.r)
|
||||
v21 := 2 * (c3.x - c2.x)
|
||||
v22 := 2 * (c3.y - c2.y)
|
||||
v23 := x2sq - x3sq + y2sq - y3sq - r2sq + r3sq
|
||||
v24 := 2 * (c3.r - c2.r)
|
||||
if s {
|
||||
v14 = -v14
|
||||
v24 = -v24
|
||||
}
|
||||
w12 := v12 / v11
|
||||
w13 := v13 / v11
|
||||
w14 := v14 / v11
|
||||
w22 := v22/v21 - w12
|
||||
w23 := v23/v21 - w13
|
||||
w24 := v24/v21 - w14
|
||||
p := -w23 / w22
|
||||
q := w24 / w22
|
||||
m := -w12*p - w13
|
||||
n := w14 - w12*q
|
||||
a := n*n + q*q - 1
|
||||
b := m*n - n*c1.x + p*q - q*c1.y
|
||||
if s {
|
||||
b -= c1.r
|
||||
} else {
|
||||
b += c1.r
|
||||
}
|
||||
b *= 2
|
||||
c := x1sq + m*m - 2*m*c1.x + p*p + y1sq - 2*p*c1.y - r1sq
|
||||
d := b*b - 4*a*c
|
||||
rs := (-b - math.Sqrt(d)) / (2 * a)
|
||||
return circle{m + n*rs, p + q*rs, rs}
|
||||
}
|
||||
75
Task/Problem-of-Apollonius/Haskell/problem-of-apollonius.hs
Normal file
75
Task/Problem-of-Apollonius/Haskell/problem-of-apollonius.hs
Normal file
|
|
@ -0,0 +1,75 @@
|
|||
data Circle = Circle { x, y, r :: Double } deriving (Show, Eq)
|
||||
data Tangent = Externally | Internally deriving Eq
|
||||
|
||||
{--
|
||||
Solves the Problem of Apollonius (finding a circle tangent to three
|
||||
other circles in the plane).
|
||||
|
||||
Params:
|
||||
c1 = First circle of the problem.
|
||||
c2 = Second circle of the problem.
|
||||
c3 = Third circle of the problem.
|
||||
t1 = How is the solution tangent (externally or internally) to c1.
|
||||
t2 = How is the solution tangent (externally or internally) to c2.
|
||||
t3 = How is the solution tangent (externally or internally) to c3.
|
||||
|
||||
Returns: The Circle that is tangent to c1, c2 and c3.
|
||||
--}
|
||||
solveApollonius :: Circle -> Circle -> Circle ->
|
||||
Tangent -> Tangent -> Tangent ->
|
||||
Circle
|
||||
solveApollonius c1 c2 c3 t1 t2 t3 =
|
||||
Circle (m + n * rs) (p + q * rs) rs
|
||||
where
|
||||
s1 = if t1 == Externally then 1.0 else -1.0
|
||||
s2 = if t2 == Externally then 1.0 else -1.0
|
||||
s3 = if t3 == Externally then 1.0 else -1.0
|
||||
|
||||
v11 = 2 * x c2 - 2 * x c1
|
||||
v12 = 2 * y c2 - 2 * y c1
|
||||
v13 = x c1 ^ 2 - x c2 ^ 2 +
|
||||
y c1 ^ 2 - y c2 ^ 2 -
|
||||
r c1 ^ 2 + r c2 ^ 2
|
||||
v14 = 2 * s2 * r c2 - 2 * s1 * r c1
|
||||
|
||||
v21 = 2 * x c3 - 2 * x c2
|
||||
v22 = 2 * y c3 - 2 * y c2
|
||||
v23 = x c2 ^ 2 - x c3 ^ 2 +
|
||||
y c2 ^ 2 - y c3 ^ 2 -
|
||||
r c2 ^ 2 + r c3 ^ 2;
|
||||
v24 = 2 * s3 * r c3 - 2 * s2 * r c2
|
||||
|
||||
w12 = v12 / v11
|
||||
w13 = v13 / v11
|
||||
w14 = v14 / v11
|
||||
|
||||
w22 = v22 / v21 - w12
|
||||
w23 = v23 / v21 - w13
|
||||
w24 = v24 / v21 - w14
|
||||
|
||||
p = -w23 / w22
|
||||
q = w24 / w22
|
||||
m = -w12 * p - w13
|
||||
n = w14 - w12 * q
|
||||
|
||||
a = n * n + q ^ 2 - 1
|
||||
b = 2 * m * n - 2 * n * x c1 +
|
||||
2 * p * q - 2 * q * y c1 +
|
||||
2 * s1 * r c1
|
||||
c = x c1 ^ 2 + m ^ 2 - 2 * m * x c1 +
|
||||
p ^ 2 + y c1 ^ 2 - 2 * p * y c1 - r c1 ^ 2
|
||||
|
||||
-- Find a root of a quadratic equation.
|
||||
-- This requires the circle centers not to be e.g. colinear.
|
||||
d = b ^ 2 - 4 * a * c
|
||||
rs = (-b - sqrt d) / (2 * a)
|
||||
|
||||
main = do
|
||||
let c1 = Circle 0.0 0.0 1.0
|
||||
let c2 = Circle 4.0 0.0 1.0
|
||||
let c3 = Circle 2.0 4.0 2.0
|
||||
let te = Externally
|
||||
print $ solveApollonius c1 c2 c3 te te te
|
||||
|
||||
let ti = Internally
|
||||
print $ solveApollonius c1 c2 c3 ti ti ti
|
||||
68
Task/Problem-of-Apollonius/Icon/problem-of-apollonius.icon
Normal file
68
Task/Problem-of-Apollonius/Icon/problem-of-apollonius.icon
Normal file
|
|
@ -0,0 +1,68 @@
|
|||
link graphics
|
||||
|
||||
record circle(x,y,r)
|
||||
global scale,xoffset,yoffset,yadjust
|
||||
|
||||
procedure main()
|
||||
|
||||
WOpen("size=400,400") | stop("Unable to open Window")
|
||||
scale := 28
|
||||
xoffset := WAttrib("width") / 2
|
||||
yoffset := ( yadjust := WAttrib("height")) / 2
|
||||
|
||||
|
||||
WC(c1 := circle(0,0,1),"black")
|
||||
WC(c2 := circle(4,0,1),"black")
|
||||
WC(c3 := circle(2,4,2),"black")
|
||||
WC(c4 := Apollonius(c1,c2,c3,1,1,1),"green") #/ Expects "Circle[x=2.00,y=2.10,r=3.90]" (green circle in image)
|
||||
WC(c5 := Apollonius(c1,c2,c3,-1,-1,-1),"red") #/ Expects "Circle[x=2.00,y=0.83,r=1.17]" (red circle in image)
|
||||
|
||||
|
||||
WAttrib("fg=blue")
|
||||
DrawLine( 0*scale+xoffset, yadjust-(-1*scale+yoffset), 0*scale+xoffset, yadjust-(4*scale+yoffset) )
|
||||
DrawLine( -1*scale+xoffset, yadjust-(0*scale+yoffset), 4*scale+xoffset, yadjust-(0*scale+yoffset) )
|
||||
WDone()
|
||||
end
|
||||
|
||||
procedure WC(c,fg) # write and plot circle
|
||||
WAttrib("fg="||fg)
|
||||
DrawCircle(c.x*scale+xoffset, yadjust-(c.y*scale+yoffset), c.r*scale)
|
||||
return write("Circle(x,y,r) := (",c.x,", ",c.y,", ",c.r,")")
|
||||
end
|
||||
|
||||
procedure Apollonius(c1,c2,c3,s1,s2,s3) # solve Apollonius
|
||||
|
||||
v11 := 2.*(c2.x - c1.x)
|
||||
v12 := 2.*(c2.y - c1.y)
|
||||
v13 := c1.x^2 - c2.x^2 + c1.y^2 - c2.y^2 - c1.r^2 + c2.r^2
|
||||
v14 := 2.*(s2*c2.r - s1*c1.r)
|
||||
|
||||
v21 := 2.*(c3.x - c2.x)
|
||||
v22 := 2.*(c3.y - c2.y)
|
||||
v23 := c2.x^2 - c3.x^2 + c2.y^2 - c3.y^2 - c2.r^2 + c3.r^2
|
||||
v24 := 2.*(s3*c3.r - s2*c2.r)
|
||||
|
||||
w12 := v12/v11
|
||||
w13 := v13/v11
|
||||
w14 := v14/v11
|
||||
|
||||
w22 := v22/v21-w12
|
||||
w23 := v23/v21-w13
|
||||
w24 := v24/v21-w14
|
||||
|
||||
P := -w23/w22
|
||||
Q := w24/w22
|
||||
M := -w12*P-w13
|
||||
N := w14 - w12*Q
|
||||
|
||||
a := N*N + Q*Q - 1
|
||||
b := 2*M*N - 2*N*c1.x + 2*P*Q - 2*Q*c1.y + 2*s1*c1.r
|
||||
c := c1.x*c1.x + M*M - 2*M*c1.x + P*P + c1.y*c1.y - 2*P*c1.y - c1.r*c1.r
|
||||
|
||||
#// Find a root of a quadratic equation. This requires the circle centers not to be e.g. colinear
|
||||
D := b*b-4*a*c
|
||||
rs := (-b-sqrt(D))/(2*a)
|
||||
xs := M + N * rs
|
||||
ys := P + Q * rs
|
||||
return circle(xs,ys,rs)
|
||||
end
|
||||
19
Task/Problem-of-Apollonius/J/problem-of-apollonius-1.j
Normal file
19
Task/Problem-of-Apollonius/J/problem-of-apollonius-1.j
Normal file
|
|
@ -0,0 +1,19 @@
|
|||
require 'math/misc/amoeba'
|
||||
|
||||
NB.*apollonius v solves Apollonius problems
|
||||
NB. y is Cx0 Cy0 R0, Cx1 Cy1 R1,: Cx2 Cy2 R2
|
||||
NB. x are radius scale factors to control which circles are included
|
||||
NB. in the common tangent circle. 1 to surround, _1 to exclude.
|
||||
NB. returns Cxs Cys Rs
|
||||
apollonius =: verb define"1 _
|
||||
1 apollonius y
|
||||
:
|
||||
centers=. 2{."1 y
|
||||
radii=. x * {:"1 y
|
||||
goal=. 1e_20 NB. goal simplex volume
|
||||
dist=. radii + [: +/"1&.:*: centers -"1 ] NB. distances to tangents
|
||||
'soln err'=. ([: +/@:*:@, -/~@dist) f. amoeba goal centers
|
||||
if. err > 10 * goal do. '' return. end. NB. no solution found
|
||||
avg=. +/ % #
|
||||
(, avg@dist) soln
|
||||
)
|
||||
7
Task/Problem-of-Apollonius/J/problem-of-apollonius-2.j
Normal file
7
Task/Problem-of-Apollonius/J/problem-of-apollonius-2.j
Normal file
|
|
@ -0,0 +1,7 @@
|
|||
]rctst=: 0 0 1,4 0 1,:2 4 2 NB. Task circles
|
||||
0 0 1
|
||||
4 0 1
|
||||
2 4 2
|
||||
(_1 _1 _1 ,: 1 1 1) apollonius rctst NB. internally & externally tangent solutions
|
||||
2 0.83333333 1.1666667
|
||||
2 2.1 3.9
|
||||
93
Task/Problem-of-Apollonius/Java/problem-of-apollonius.java
Normal file
93
Task/Problem-of-Apollonius/Java/problem-of-apollonius.java
Normal file
|
|
@ -0,0 +1,93 @@
|
|||
public class Circle
|
||||
{
|
||||
public double[] center;
|
||||
public double radius;
|
||||
public Circle(double[] center, double radius)
|
||||
{
|
||||
this.center = center;
|
||||
this.radius = radius;
|
||||
}
|
||||
public String toString()
|
||||
{
|
||||
return String.format("Circle[x=%.2f,y=%.2f,r=%.2f]",center[0],center[1],
|
||||
radius);
|
||||
}
|
||||
}
|
||||
|
||||
public class ApolloniusSolver
|
||||
{
|
||||
/** Solves the Problem of Apollonius (finding a circle tangent to three other
|
||||
* circles in the plane). The method uses approximately 68 heavy operations
|
||||
* (multiplication, division, square-roots).
|
||||
* @param c1 One of the circles in the problem
|
||||
* @param c2 One of the circles in the problem
|
||||
* @param c3 One of the circles in the problem
|
||||
* @param s1 An indication if the solution should be externally or internally
|
||||
* tangent (+1/-1) to c1
|
||||
* @param s2 An indication if the solution should be externally or internally
|
||||
* tangent (+1/-1) to c2
|
||||
* @param s3 An indication if the solution should be externally or internally
|
||||
* tangent (+1/-1) to c3
|
||||
* @return The circle that is tangent to c1, c2 and c3.
|
||||
*/
|
||||
public static Circle solveApollonius(Circle c1, Circle c2, Circle c3, int s1,
|
||||
int s2, int s3)
|
||||
{
|
||||
float x1 = c1.center[0];
|
||||
float y1 = c1.center[1];
|
||||
float r1 = c1.radius;
|
||||
float x2 = c2.center[0];
|
||||
float y2 = c2.center[1];
|
||||
float r2 = c2.radius;
|
||||
float x3 = c3.center[0];
|
||||
float y3 = c3.center[1];
|
||||
float r3 = c3.radius;
|
||||
|
||||
//Currently optimized for fewest multiplications. Should be optimized for
|
||||
//readability
|
||||
float v11 = 2*x2 - 2*x1;
|
||||
float v12 = 2*y2 - 2*y1;
|
||||
float v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2;
|
||||
float v14 = 2*s2*r2 - 2*s1*r1;
|
||||
|
||||
float v21 = 2*x3 - 2*x2;
|
||||
float v22 = 2*y3 - 2*y2;
|
||||
float v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3;
|
||||
float v24 = 2*s3*r3 - 2*s2*r2;
|
||||
|
||||
float w12 = v12/v11;
|
||||
float w13 = v13/v11;
|
||||
float w14 = v14/v11;
|
||||
|
||||
float w22 = v22/v21-w12;
|
||||
float w23 = v23/v21-w13;
|
||||
float w24 = v24/v21-w14;
|
||||
|
||||
float P = -w23/w22;
|
||||
float Q = w24/w22;
|
||||
float M = -w12*P-w13;
|
||||
float N = w14 - w12*Q;
|
||||
|
||||
float a = N*N + Q*Q - 1;
|
||||
float b = 2*M*N - 2*N*x1 + 2*P*Q - 2*Q*y1 + 2*s1*r1;
|
||||
float c = x1*x1 + M*M - 2*M*x1 + P*P + y1*y1 - 2*P*y1 - r1*r1;
|
||||
|
||||
// Find a root of a quadratic equation. This requires the circle centers not
|
||||
// to be e.g. colinear
|
||||
float D = b*b-4*a*c;
|
||||
float rs = (-b-Math.sqrt(D))/(2*a);
|
||||
float xs = M + N * rs;
|
||||
float ys = P + Q * rs;
|
||||
return new Circle(new double[]{xs,ys}, rs);
|
||||
}
|
||||
public static void main(final String[] args)
|
||||
{
|
||||
Circle c1 = new Circle(new double[]{0,0}, 1);
|
||||
Circle c2 = new Circle(new double[]{4,0}, 1);
|
||||
Circle c3 = new Circle(new double[]{2,4}, 2);
|
||||
// Expects "Circle[x=2.00,y=2.10,r=3.90]" (green circle in image)
|
||||
System.out.println(solveApollonius(c1,c2,c3,1,1,1));
|
||||
// Expects "Circle[x=2.00,y=0.83,r=1.17]" (red circle in image)
|
||||
System.out.println(solveApollonius(c1,c2,c3,-1,-1,-1));
|
||||
}
|
||||
}
|
||||
51
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-1.jq
Normal file
51
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-1.jq
Normal file
|
|
@ -0,0 +1,51 @@
|
|||
def circle:
|
||||
{"x": .[0], "y": .[1], "r": .[2]};
|
||||
|
||||
# Find the interior or exterior Apollonius circle of three circles:
|
||||
# ap(circle, circle, circle, boolean)
|
||||
# Specify s as true for interior; false for exterior
|
||||
def ap(c1; c2; c3; s):
|
||||
def sign: if s then -. else . end;
|
||||
(c1.x * c1.x) as $x1sq
|
||||
| (c1.y * c1.y) as $y1sq
|
||||
| (c1.r * c1.r) as $r1sq
|
||||
| (c2.x * c2.x) as $x2sq
|
||||
| (c2.y * c2.y) as $y2sq
|
||||
| (c2.r * c2.r) as $r2sq
|
||||
| (c3.x * c3.x) as $x3sq
|
||||
| (c3.y * c3.y) as $y3sq
|
||||
| (c3.r * c3.r) as $r3sq
|
||||
|
||||
| (2 * (c2.x - c1.x)) as $v11
|
||||
| (2 * (c2.y - c1.y)) as $v12
|
||||
| ($x1sq - $x2sq + $y1sq - $y2sq - $r1sq + $r2sq) as $v13
|
||||
|
||||
| (2 * (c2.r - c1.r) | sign) as $v14
|
||||
| (2 * (c3.x - c2.x)) as $v21
|
||||
| (2 * (c3.y - c2.y)) as $v22
|
||||
|
||||
| ($x2sq - $x3sq + $y2sq - $y3sq - $r2sq + $r3sq) as $v23
|
||||
| ( 2 * c3.r - c2.r | sign) as $v24
|
||||
| ($v12 / $v11) as $w12
|
||||
| ($v13 / $v11) as $w13
|
||||
| ($v14 / $v11) as $w14
|
||||
|
||||
| (($v22 / $v21) - $w12) as $w22
|
||||
| (($v23 / $v21) - $w13) as $w23
|
||||
| (($v24 / $v21) - $w14) as $w24
|
||||
|
||||
| (-$w23 / $w22) as $p
|
||||
| ( $w24 / $w22) as $q
|
||||
| ((-$w12*$p) - $w13) as $m
|
||||
| ( $w14 - ($w12*$q)) as $n
|
||||
|
||||
| ( $n*$n + $q*$q - 1 ) as $a
|
||||
| (2 * (($m*$n - $n*c1.x + $p*$q - $q*c1.y) + (c1.r|sign))) as $b
|
||||
| ($x1sq + $m*$m - 2*$m*c1.x + $p*$p + $y1sq - 2*$p*c1.y - $r1sq) as $c
|
||||
|
||||
| ( $b*$b - 4*$a*$c ) as $d # discriminant
|
||||
| (( -$b - (($d|sqrt))) / (2 * $a)) as $rs # root
|
||||
|
||||
| [$m + ($n*$rs), $p + ($q*$rs), $rs]
|
||||
| circle
|
||||
;
|
||||
7
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-2.jq
Normal file
7
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-2.jq
Normal file
|
|
@ -0,0 +1,7 @@
|
|||
def task:
|
||||
([0, 0, 1] | circle) as $c1
|
||||
| ([4, 0, 1] | circle) as $c2
|
||||
| ([2, 4, 2] | circle) as $c3
|
||||
| ( ap($c1; $c2; $c3; true), # interior
|
||||
ap($c1; $c2; $c3; false) ) # exterior
|
||||
;
|
||||
3
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-3.jq
Normal file
3
Task/Problem-of-Apollonius/Jq/problem-of-apollonius-3.jq
Normal file
|
|
@ -0,0 +1,3 @@
|
|||
$ jq -n -c -f apollonius.jq
|
||||
{"x":2,"y":0.8333333333333333,"r":1.1666666666666667}
|
||||
{"x":2,"y":2.1,"r":3.9}
|
||||
|
|
@ -0,0 +1,51 @@
|
|||
module ApolloniusProblems
|
||||
|
||||
using Polynomials, LinearAlgebra, Printf
|
||||
export Circle
|
||||
|
||||
struct Point{T<:Real}
|
||||
x::T
|
||||
y::T
|
||||
end
|
||||
|
||||
xcoord(p::Point) = p.x
|
||||
ycoord(p::Point) = p.y
|
||||
|
||||
struct Circle{T<:Real}
|
||||
c::Point{T}
|
||||
r::T
|
||||
end
|
||||
Circle(x::T, y::T, r::T) where T<:Real = Circle(Point(x, y), r)
|
||||
|
||||
radius(c::Circle) = c.r
|
||||
center(c::Circle) = c.c
|
||||
xcenter(c::Circle) = xcoord(center(c))
|
||||
ycenter(c::Circle) = ycoord(center(c))
|
||||
|
||||
Base.show(io::IO, c::Circle) =
|
||||
@printf(io, "centered at (%0.4f, %0.4f) with radius %0.4f",
|
||||
xcenter(c), ycenter(c), radius(c))
|
||||
|
||||
function solve(ap::Vector{Circle{T}}, enc=()) where T<:Real
|
||||
length(ap) == 3 || error("This Apollonius problem needs 3 circles.")
|
||||
x = @. xcenter(ap)
|
||||
y = @. ycenter(ap)
|
||||
r = map(u -> ifelse(u ∈ enc, -1, 1), 1:3) .* radius.(ap)
|
||||
@views begin
|
||||
a = 2x[1] .- 2x[2:3]
|
||||
b = 2y[1] .- 2y[2:3]
|
||||
c = 2r[1] .- 2r[2:3]
|
||||
d = (x[1] ^ 2 + y[1] ^ 2 - r[1] ^ 2) .- (x[2:3] .^ 2 .+ y[2:3] .^ 2 .- r[2:3] .^ 2)
|
||||
end
|
||||
u = Polynomial([-det([b d]), det([b c])] ./ det([a b]))
|
||||
v = Polynomial([det([a d]), -det([a c])] ./ det([a b]))
|
||||
w = Polynomial([r[1], 1.0]) ^ 2
|
||||
s = (u - x[1]) ^ 2 + (v - y[1]) ^ 2 - w
|
||||
r = filter(x -> iszero(imag(x)) && x > zero(x), roots(s))
|
||||
length(r) < 2 || error("The solution is not unique.")
|
||||
length(r) == 1 || error("There is no solution.")
|
||||
r = r[1]
|
||||
return Circle(evalpoly(r, u), evalpoly(r, v), r)
|
||||
end
|
||||
|
||||
end # module ApolloniusProblem
|
||||
|
|
@ -0,0 +1,8 @@
|
|||
include("module.jl")
|
||||
using .ApolloniusProblems
|
||||
|
||||
let test = [Circle(0.0, 0.0, 1.0), Circle(4.0, 0.0, 1.0), Circle(2.0, 4.0, 2.0)]
|
||||
println("The defining circles are: \n - ", join(test, "\n - "))
|
||||
println("The internal circle is:\n\t", ApolloniusProblems.solve(test))
|
||||
println("The external circle is:\n\t", ApolloniusProblems.solve(test, 1:3))
|
||||
end
|
||||
|
|
@ -0,0 +1,53 @@
|
|||
// version 1.1.3
|
||||
|
||||
data class Circle(val x: Double, val y: Double, val r: Double)
|
||||
|
||||
val Double.sq get() = this * this
|
||||
|
||||
fun solveApollonius(c1: Circle, c2: Circle, c3: Circle,
|
||||
s1: Int, s2: Int, s3: Int): Circle {
|
||||
val (x1, y1, r1) = c1
|
||||
val (x2, y2, r2) = c2
|
||||
val (x3, y3, r3) = c3
|
||||
|
||||
val v11 = 2 * x2 - 2 * x1
|
||||
val v12 = 2 * y2 - 2 * y1
|
||||
val v13 = x1.sq - x2.sq + y1.sq - y2.sq - r1.sq + r2.sq
|
||||
val v14 = 2 * s2 * r2 - 2 * s1 * r1
|
||||
|
||||
val v21 = 2 * x3 - 2 * x2
|
||||
val v22 = 2 * y3 - 2 * y2
|
||||
val v23 = x2.sq - x3.sq + y2.sq - y3.sq - r2.sq + r3.sq
|
||||
val v24 = 2 * s3 * r3 - 2 * s2 * r2
|
||||
|
||||
val w12 = v12 / v11
|
||||
val w13 = v13 / v11
|
||||
val w14 = v14 / v11
|
||||
|
||||
val w22 = v22 / v21 - w12
|
||||
val w23 = v23 / v21 - w13
|
||||
val w24 = v24 / v21 - w14
|
||||
|
||||
val p = -w23 / w22
|
||||
val q = w24 / w22
|
||||
val m = -w12 * p - w13
|
||||
val n = w14 - w12 * q
|
||||
|
||||
val a = n.sq + q.sq - 1
|
||||
val b = 2 * m * n - 2 * n * x1 + 2 * p * q - 2 * q * y1 + 2 * s1 * r1
|
||||
val c = x1.sq + m.sq - 2 * m * x1 + p.sq + y1.sq - 2 * p * y1 - r1.sq
|
||||
|
||||
val d = b.sq - 4 * a * c
|
||||
val rs = (-b - Math.sqrt(d)) / (2 * a)
|
||||
val xs = m + n * rs
|
||||
val ys = p + q * rs
|
||||
return Circle(xs, ys, rs)
|
||||
}
|
||||
|
||||
fun main(args: Array<String>) {
|
||||
val c1 = Circle(0.0, 0.0, 1.0)
|
||||
val c2 = Circle(4.0, 0.0, 1.0)
|
||||
val c3 = Circle(2.0, 4.0, 2.0)
|
||||
println(solveApollonius(c1, c2, c3, 1, 1, 1))
|
||||
println(solveApollonius(c1, c2, c3,-1,-1,-1))
|
||||
}
|
||||
59
Task/Problem-of-Apollonius/Lasso/problem-of-apollonius.lasso
Normal file
59
Task/Problem-of-Apollonius/Lasso/problem-of-apollonius.lasso
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
define solveApollonius(c1, c2, c3, s1, s2, s3) => {
|
||||
local(
|
||||
x1 = decimal(#c1->get(1)),
|
||||
y1 = decimal(#c1->get(2)),
|
||||
r1 = decimal(#c1->get(3))
|
||||
)
|
||||
local(
|
||||
x2 = decimal(#c2->get(1)),
|
||||
y2 = decimal(#c2->get(2)),
|
||||
r2 = decimal(#c2->get(3))
|
||||
)
|
||||
local(
|
||||
x3 = decimal(#c3->get(1)),
|
||||
y3 = decimal(#c3->get(2)),
|
||||
r3 = decimal(#c3->get(3))
|
||||
)
|
||||
local(
|
||||
v11 = 2*#x2 - 2*#x1,
|
||||
v12 = 2*#y2 - 2*#y1,
|
||||
v13 = #x1*#x1 - #x2*#x2 + #y1*#y1 - #y2*#y2 - #r1*#r1 + #r2*#r2,
|
||||
v14 = 2*#s2*#r2 - 2*#s1*#r1,
|
||||
|
||||
v21 = 2*#x3 - 2*#x2,
|
||||
v22 = 2*#y3 - 2*#y2,
|
||||
v23 = #x2*#x2 - #x3*#x3 + #y2*#y2 - #y3*#y3 - #r2*#r2 + #r3*#r3,
|
||||
v24 = 2*#s3*#r3 - 2*#s2*#r2,
|
||||
|
||||
w12 = #v12/#v11,
|
||||
w13 = #v13/#v11,
|
||||
w14 = #v14/#v11,
|
||||
|
||||
w22 = #v22/#v21-#w12,
|
||||
w23 = #v23/#v21-#w13,
|
||||
w24 = #v24/#v21-#w14,
|
||||
|
||||
P = -#w23/#w22,
|
||||
Q = #w24/#w22,
|
||||
M = -#w12*#P-#w13,
|
||||
N = #w14 - #w12*#Q,
|
||||
|
||||
a = #N*#N + #Q*#Q - 1,
|
||||
b = 2*#M*#N - 2*#N*#x1 + 2*#P*#Q - 2*#Q*#y1 + 2*#s1*#r1,
|
||||
c = #x1*#x1 + #M*#M - 2*#M*#x1 + #P*#P + #y1*#y1 - 2*#P*#y1 - #r1*#r1
|
||||
|
||||
)
|
||||
|
||||
// Find a root of a quadratic equation. This requires the circle centers not to be e.g. colinear
|
||||
local(
|
||||
D = #b*#b-4*#a*#c,
|
||||
rs = (-#b - #D->sqrt)/(2*#a),
|
||||
|
||||
xs = #M+#N*#rs,
|
||||
ys = #P+#Q*#rs
|
||||
)
|
||||
return (:#xs, #ys, #rs)
|
||||
}
|
||||
// Tests:
|
||||
solveApollonius((:0, 0, 1), (:4, 0, 1), (:2, 4, 2), 1,1,1)
|
||||
solveApollonius((:0, 0, 1), (:4, 0, 1), (:2, 4, 2), -1,-1,-1)
|
||||
|
|
@ -0,0 +1,54 @@
|
|||
circle1$ =" 0.000, 0.000, 1.000"
|
||||
circle2$ =" 4.000, 0.000, 1.000"
|
||||
circle3$ =" 2.000, 4.000, 2.000"
|
||||
|
||||
print " x_pos y_pos radius"
|
||||
print circle1$
|
||||
print circle2$
|
||||
print circle3$
|
||||
print
|
||||
print ApolloniusSolver$( circle1$, circle2$, circle3$, 1, 1, 1)
|
||||
print ApolloniusSolver$( circle1$, circle2$, circle3$, -1, -1, -1)
|
||||
|
||||
end
|
||||
|
||||
function ApolloniusSolver$( c1$, c2$, c3$, s1, s2, s3)
|
||||
x1 =val( word$( c1$, 1, ",")): y1 =val( word$( c1$, 2, ",")): r1 =val( word$( c1$, 3, ","))
|
||||
x2 =val( word$( c2$, 1, ",")): y2 =val( word$( c2$, 2, ",")): r2 =val( word$( c2$, 3, ","))
|
||||
x3 =val( word$( c3$, 1, ",")): y3 =val( word$( c3$, 2, ",")): r3 =val( word$( c3$, 3, ","))
|
||||
|
||||
v11 = 2 *x2 -2 *x1
|
||||
v12 = 2 *y2 -2*y1
|
||||
v13 = x1 *x1 - x2 *x2 + y1 *y1 - y2 *y2 -r1 *r1 +r2 *r2
|
||||
v14 = 2 *s2 *r2 -2 *s1 *r1
|
||||
|
||||
v21 = 2 *x3 -2 *x2
|
||||
v22 = 2 *y3 -2*y2
|
||||
v23 = x2 *x2 -x3 *x3 + y2 *y2 -y3 *y3 -r2 *r2 +r3 *r3
|
||||
v24 = 2 *s3 *r3 - 2 *s2 *r2
|
||||
|
||||
w12 = v12 /v11
|
||||
w13 = v13 /v11
|
||||
w14 = v14 /v11
|
||||
|
||||
w22 = v22 /v21 -w12
|
||||
w23 = v23 /v21 -w13
|
||||
w24 = v24 /v21 -w14
|
||||
|
||||
P = 0 -w23 /w22
|
||||
Q = w24 /w22
|
||||
M = 0 -w12 *P -w13
|
||||
N = w14 -w12 *Q
|
||||
|
||||
a = N *N + Q *Q -1
|
||||
b = 2 *M *N -2 *N *x1 + 2 *P *Q -2 *Q *y1 +2 *s1 *r1
|
||||
c = x1 *x1 +M *M -2 *M *x1 +P *P +y1 *y1 -2 *P *y1 -r1 *r1
|
||||
|
||||
D = b *b -4 *a *c
|
||||
|
||||
Radius =( 0 -b -Sqr( D)) /( 2 *a)
|
||||
XPos =M +N *Radius
|
||||
YPos =P +Q *Radius
|
||||
|
||||
ApolloniusSolver$ =using( "###.###", XPos) +"," +using( "###.###", YPos) +using( "###.###", Radius)
|
||||
end function
|
||||
42
Task/Problem-of-Apollonius/MUMPS/problem-of-apollonius.mumps
Normal file
42
Task/Problem-of-Apollonius/MUMPS/problem-of-apollonius.mumps
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
APOLLONIUS(CIR1,CIR2,CIR3,S1,S2,S3)
|
||||
;Circles are passed in as strings with three parts with a "^" separator in the order x^y^r
|
||||
;The three circles are CIR1, CIR2, and CIR3
|
||||
;The S1, S2, and S3 parameters determine if the solution will be internally or externally
|
||||
;tangent to the circle. (+1 external, -1 internal)
|
||||
;CIRR is the circle returned in the same format as the input circles
|
||||
;
|
||||
;Xn, Yn, and Rn are the values for a circle n - following the precedents from the
|
||||
;other examples because doing $Pieces would make this confusing to read
|
||||
NEW X1,X2,X3,Y1,Y2,Y3,R1,R2,R3,RS,V11,V12,V13,V14,V21,V22,V23,V24,W12,W13,W14,W22,W23,W24,P,M,N,Q,A,B,C,D
|
||||
NEW CIRR
|
||||
SET X1=$PIECE(CIR1,"^",1),X2=$PIECE(CIR2,"^",1),X3=$PIECE(CIR3,"^",1)
|
||||
SET Y1=$PIECE(CIR1,"^",2),Y2=$PIECE(CIR2,"^",2),Y3=$PIECE(CIR3,"^",2)
|
||||
SET R1=$PIECE(CIR1,"^",3),R2=$PIECE(CIR2,"^",3),R3=$PIECE(CIR3,"^",3)
|
||||
SET V11=(2*X2)-(2*X1)
|
||||
SET V12=(2*Y2)-(2*Y1)
|
||||
SET V13=(X1*X1)-(X2*X2)+(Y1*Y1)-(Y2*Y2)-(R1*R1)+(R2*R2)
|
||||
SET V14=(2*S2*R2)-(2*S1*R1)
|
||||
SET V21=(2*X3)-(2*X2)
|
||||
SET V22=(2*Y3)-(2*Y2)
|
||||
SET V23=(X2*X2)-(X3*X3)+(Y2*Y2)-(Y3*Y3)-(R2*R2)+(R3*R3)
|
||||
SET V24=(2*S3*R3)-(2*S2*R2)
|
||||
SET W12=V12/V11
|
||||
SET W13=V13/V11
|
||||
SET W14=V14/V11
|
||||
SET W22=(V22/V21)-W12 ;Parentheses for insurance - MUMPS evaluates left to right
|
||||
SET W23=(V23/V21)-W13
|
||||
SET W24=(V24/V21)-W14
|
||||
SET P=-W23/W22
|
||||
SET Q=W24/W22
|
||||
SET M=-(W12*P)-W13
|
||||
SET N=W14-(W12*Q)
|
||||
SET A=(N*N)+(Q*Q)-1
|
||||
SET B=(2*M*N)-(2*N*X1)+(2*P*Q)-(2*Q*Y1)+(2*S1*R1)
|
||||
SET C=(X1*X1)+(M*M)+(2*M*X1)+(P*P)+(Y1*Y1)-(2*P*Y1)-(R1*R1)
|
||||
SET D=(B*B)-(4*A*C)
|
||||
SET RS=(-B-(D**.5))/(2*A)
|
||||
SET $PIECE(CIRR,"^",1)=M+(N*RS)
|
||||
SET $PIECE(CIRR,"^",2)=P+(Q*RS)
|
||||
SET $PIECE(CIRR,"^",3)=RS
|
||||
KILL X1,X2,X3,Y1,Y2,Y3,R1,R2,R3,RS,V11,V12,V13,V14,V21,V22,V23,V24,W12,W13,W14,W22,W23,W24,P,M,N,Q,A,B,C,D
|
||||
QUIT CIRR
|
||||
|
|
@ -0,0 +1,27 @@
|
|||
Apolonius[a1_,b1_,c1_,a2_,b2_,c2_,a3_,b3_,c3_,S1_,S2_ ,S3_ ]:=
|
||||
Module[{x1=a1,y1=b1,r1=c1,x2=a2,y2=b2,r2=c2,x3=a3,y3=b3,r3=c3,s1=S1,s2=S2,s3=S3},
|
||||
v11 = 2*x2 - 2*x1; v12 = 2*y2 - 2*y1;
|
||||
v13 = x1^2 - x2^2 + y1^2 - y2^2 - r1^2 + r2^2;
|
||||
v14 = 2*s2*r2 - 2*s1*r1;
|
||||
|
||||
v21 = 2*x3-2*x2 ; v22 = 2*y3 - 2*y2;
|
||||
v23 = x2^2 - x3^2 + y2^2 - y3^2 - r2^2 + r3^2;
|
||||
v24 = 2*s3*r3 - 2*s2*r2;
|
||||
|
||||
w12 = v12/v11; w13 = v13/v11; w14 = v14/v11;
|
||||
|
||||
w22 = v22/v21 - w12;
|
||||
w23 = v23/v21 - w13;
|
||||
w24 = v24/v21 - w14;
|
||||
|
||||
p = -w23/w22; q=w24/w22;
|
||||
m = -w12*p - w13; n=w14 - w12*q;
|
||||
|
||||
a = n^2 + q^2-1;
|
||||
b = 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1;
|
||||
c = x1^2+m^2 - 2*m*x1 + p^2+y1^2 - 2*p*y1 - r1^2;
|
||||
|
||||
d= b^2 - 4*a*c;
|
||||
rs = (-b -Sqrt[d])/(2*a);
|
||||
xs = m + n*rs; ys = p + q*rs;
|
||||
Map[N,{xs, ys, rs} ]]
|
||||
48
Task/Problem-of-Apollonius/Nim/problem-of-apollonius.nim
Normal file
48
Task/Problem-of-Apollonius/Nim/problem-of-apollonius.nim
Normal file
|
|
@ -0,0 +1,48 @@
|
|||
import math
|
||||
|
||||
type Circle = tuple[x, y, r: float]
|
||||
|
||||
proc solveApollonius(c1, c2, c3: Circle; s1, s2, s3: float): Circle =
|
||||
let
|
||||
v11 = 2*c2.x - 2*c1.x
|
||||
v12 = 2*c2.y - 2*c1.y
|
||||
v13 = c1.x*c1.x - c2.x*c2.x + c1.y*c1.y - c2.y*c2.y - c1.r*c1.r + c2.r*c2.r
|
||||
v14 = 2*s2*c2.r - 2*s1*c1.r
|
||||
|
||||
v21 = 2*c3.x - 2*c2.x
|
||||
v22 = 2*c3.y - 2*c2.y
|
||||
v23 = c2.x*c2.x - c3.x*c3.x + c2.y*c2.y - c3.y*c3.y - c2.r*c2.r + c3.r*c3.r
|
||||
v24 = 2*s3*c3.r - 2*s2*c2.r
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21-w12
|
||||
w23 = v23/v21-w13
|
||||
w24 = v24/v21-w14
|
||||
|
||||
p = -w23/w22
|
||||
q = w24/w22
|
||||
m = -w12*p-w13
|
||||
n = w14 - w12*q
|
||||
|
||||
a = n*n + q*q - 1
|
||||
b = 2*m*n - 2*n*c1.x + 2*p*q - 2*q*c1.y + 2*s1*c1.r
|
||||
c = c1.x*c1.x + m*m - 2*m*c1.x + p*p + c1.y*c1.y - 2*p*c1.y - c1.r*c1.r
|
||||
|
||||
d = b*b-4*a*c
|
||||
rs = (-b-sqrt(d))/(2*a)
|
||||
|
||||
xs = m+n*rs
|
||||
ys = p+q*rs
|
||||
|
||||
return (xs, ys, rs)
|
||||
|
||||
let
|
||||
c1: Circle = (0.0, 0.0, 1.0)
|
||||
c2: Circle = (4.0, 0.0, 1.0)
|
||||
c3: Circle = (2.0, 4.0, 2.0)
|
||||
|
||||
echo solveApollonius(c1, c2, c3, 1.0, 1.0, 1.0)
|
||||
echo solveApollonius(c1, c2, c3, -1.0, -1.0, -1.0)
|
||||
76
Task/Problem-of-Apollonius/OCaml/problem-of-apollonius.ocaml
Normal file
76
Task/Problem-of-Apollonius/OCaml/problem-of-apollonius.ocaml
Normal file
|
|
@ -0,0 +1,76 @@
|
|||
type point = { x:float; y:float }
|
||||
type circle = {
|
||||
center: point;
|
||||
radius: float;
|
||||
}
|
||||
|
||||
let new_circle ~x ~y ~r =
|
||||
{ center = { x=x; y=y };
|
||||
radius = r }
|
||||
|
||||
let print_circle ~c =
|
||||
Printf.printf "Circle(x=%.2f, y=%.2f, r=%.2f)\n"
|
||||
c.center.x c.center.y c.radius
|
||||
|
||||
let defxyr c =
|
||||
(c.center.x,
|
||||
c.center.y,
|
||||
c.radius)
|
||||
|
||||
let solve_apollonius ~c1 ~c2 ~c3
|
||||
~s1 ~s2 ~s3 =
|
||||
let ( * ) = ( *. ) in
|
||||
let ( / ) = ( /. ) in
|
||||
let ( + ) = ( +. ) in
|
||||
let ( - ) = ( -. ) in
|
||||
|
||||
let x1, y1, r1 = defxyr c1
|
||||
and x2, y2, r2 = defxyr c2
|
||||
and x3, y3, r3 = defxyr c3 in
|
||||
|
||||
let v11 = 2.0 * x2 - 2.0 * x1
|
||||
and v12 = 2.0 * y2 - 2.0 * y1
|
||||
and v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
and v14 = (2.0 * s2 * r2) - (2.0 * s1 * r1)
|
||||
|
||||
and v21 = 2.0 * x3 - 2.0 * x2
|
||||
and v22 = 2.0 * y3 - 2.0 * y2
|
||||
and v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
and v24 = (2.0 * s3 * r3) - (2.0 * s2 * r2) in
|
||||
|
||||
let w12 = v12 / v11
|
||||
and w13 = v13 / v11
|
||||
and w14 = v14 / v11 in
|
||||
|
||||
let w22 = v22 / v21 - w12
|
||||
and w23 = v23 / v21 - w13
|
||||
and w24 = v24 / v21 - w14 in
|
||||
|
||||
let p = -. w23 / w22
|
||||
and q = w24 / w22 in
|
||||
let m = -. w12 * p - w13
|
||||
and n = w14 - w12 * q in
|
||||
|
||||
let a = n*n + q*q - 1.0
|
||||
and b = 2.0*m*n - 2.0*n*x1 + 2.0*p*q - 2.0*q*y1 + 2.0*s1*r1
|
||||
and c = x1*x1 + m*m - 2.0*m*x1 + p*p + y1*y1 - 2.0*p*y1 - r1*r1 in
|
||||
|
||||
let d = b * b - 4.0 * a * c in
|
||||
let rs = (-. b - (sqrt d)) / (2.0 * a) in
|
||||
|
||||
let xs = m + n * rs
|
||||
and ys = p + q * rs in
|
||||
|
||||
(new_circle xs ys rs)
|
||||
|
||||
let () =
|
||||
let c1 = new_circle 0.0 0.0 1.0
|
||||
and c2 = new_circle 4.0 0.0 1.0
|
||||
and c3 = new_circle 2.0 4.0 2.0 in
|
||||
|
||||
let r1 = solve_apollonius c1 c2 c3 1.0 1.0 1.0 in
|
||||
print_circle r1;
|
||||
|
||||
let r2 = solve_apollonius c1 c2 c3 (-1.) (-1.) (-1.) in
|
||||
print_circle r2;
|
||||
;;
|
||||
69
Task/Problem-of-Apollonius/PL-I/problem-of-apollonius.pli
Normal file
69
Task/Problem-of-Apollonius/PL-I/problem-of-apollonius.pli
Normal file
|
|
@ -0,0 +1,69 @@
|
|||
Apollonius: procedure options (main); /* 29 October 2013 */
|
||||
|
||||
define structure
|
||||
1 circle,
|
||||
2 x float (15),
|
||||
2 y float (15),
|
||||
2 radius float (15);
|
||||
|
||||
declare (c1 , c2, c3, result) type (circle);
|
||||
|
||||
c1.x = 0; c1.y = 0; c1.radius = 1;
|
||||
c2.x = 4; c2.y = 0; c2.radius = 1;
|
||||
c3.x = 2; c3.y = 4; c3.radius = 2;
|
||||
|
||||
result = Solve_Apollonius(c1, c2, c3, 1, 1, 1);
|
||||
put skip edit ('External tangent:', result.x, result.y, result.radius) (a, 3 f(12,8));
|
||||
|
||||
result = Solve_Apollonius(c1, c2, c3, -1, -1, -1);
|
||||
put skip edit ('Internal tangent:', result.x, result.y, result.radius) (a, 3 f(12,8));
|
||||
|
||||
|
||||
|
||||
Solve_Apollonius: procedure (c1, c2, c3, s1, s2, s3) returns(type(circle));
|
||||
declare (c1, c2, c3) type(circle);
|
||||
declare res type (circle);
|
||||
declare (s1, s2, s3) fixed binary;
|
||||
|
||||
declare (
|
||||
v11, v12, v13, v14,
|
||||
v21, v22, v23, v24,
|
||||
w12, w13, w14,
|
||||
w22, w23, w24,
|
||||
p, q, m, n, a, b, c, det) float (15);
|
||||
|
||||
v11 = 2*c2.x - 2*c1.x;
|
||||
v12 = 2*c2.y - 2*c1.y;
|
||||
v13 = c1.x**2 - c2.x**2 + c1.y**2 - c2.y**2 - c1.radius**2 + c2.radius**2;
|
||||
v14 = 2*s2*c2.radius - 2*s1*c1.radius;
|
||||
|
||||
v21 = 2*c3.x - 2*c2.x;
|
||||
v22 = 2*c3.y - 2*c2.y;
|
||||
v23 = c2.x**2 - c3.x**2 + c2.y**2 - c3.y**2 - c2.radius**2 + c3.radius**2;
|
||||
v24 = 2*s3*c3.radius - 2*s2*c2.radius;
|
||||
|
||||
w12 = v12/v11;
|
||||
w13 = v13/v11;
|
||||
w14 = v14/v11;
|
||||
|
||||
w22 = v22/v21-w12;
|
||||
w23 = v23/v21-w13;
|
||||
w24 = v24/v21-w14;
|
||||
|
||||
p = -w23/w22;
|
||||
q = w24/w22;
|
||||
m = -w12*P - w13;
|
||||
n = w14 - w12*q;
|
||||
|
||||
a = n*n + q*q - 1;
|
||||
b = 2*m*n - 2*n*c1.x + 2*p*q - 2*q*c1.y + 2*s1*c1.radius;
|
||||
c = c1.x**2 + m*m - 2*m*c1.x + p*p + c1.y**2 - 2*p*c1.y - c1.radius**2;
|
||||
|
||||
det = b*b - 4*a*c;
|
||||
res.radius = (-b-sqrt(det)) / (2*a);
|
||||
res.x = m + n*res.radius;
|
||||
res.y = p + q*res.radius;
|
||||
|
||||
return (res);
|
||||
end Solve_Apollonius;
|
||||
end Apollonius;
|
||||
73
Task/Problem-of-Apollonius/Perl/problem-of-apollonius.pl
Normal file
73
Task/Problem-of-Apollonius/Perl/problem-of-apollonius.pl
Normal file
|
|
@ -0,0 +1,73 @@
|
|||
use utf8;
|
||||
use Math::Cartesian::Product;
|
||||
|
||||
package Circle;
|
||||
|
||||
sub new {
|
||||
my ($class, $args) = @_;
|
||||
my $self = {
|
||||
x => $args->{x},
|
||||
y => $args->{y},
|
||||
r => $args->{r},
|
||||
};
|
||||
bless $self, $class;
|
||||
}
|
||||
|
||||
sub show {
|
||||
my ($self, $args) = @_;
|
||||
sprintf "x =%7.3f y =%7.3f r =%7.3f\n", $args->{x}, $args->{y}, $args->{r};
|
||||
}
|
||||
|
||||
package main;
|
||||
|
||||
sub circle {
|
||||
my($x,$y,$r) = @_;
|
||||
Circle->new({ x => $x, y=> $y, r => $r });
|
||||
}
|
||||
|
||||
sub solve_Apollonius {
|
||||
my($c1, $c2, $c3, $s1, $s2, $s3) = @_;
|
||||
|
||||
my $𝑣11 = 2 * $c2->{x} - 2 * $c1->{x};
|
||||
my $𝑣12 = 2 * $c2->{y} - 2 * $c1->{y};
|
||||
my $𝑣13 = $c1->{x}**2 - $c2->{x}**2 + $c1->{y}**2 - $c2->{y}**2 - $c1->{r}**2 + $c2->{r}**2;
|
||||
my $𝑣14 = 2 * $s2 * $c2->{r} - 2 * $s1 * $c1->{r};
|
||||
|
||||
my $𝑣21 = 2 * $c3->{x} - 2 * $c2->{x};
|
||||
my $𝑣22 = 2 * $c3->{y} - 2 * $c2->{y};
|
||||
my $𝑣23 = $c2->{x}**2 - $c3->{x}**2 + $c2->{y}**2 - $c3->{y}**2 - $c2->{r}**2 + $c3->{r}**2;
|
||||
my $𝑣24 = 2 * $s3 * $c3->{r} - 2 * $s2 * $c2->{r};
|
||||
|
||||
my $𝑤12 = $𝑣12 / $𝑣11;
|
||||
my $𝑤13 = $𝑣13 / $𝑣11;
|
||||
my $𝑤14 = $𝑣14 / $𝑣11;
|
||||
|
||||
my $𝑤22 = $𝑣22 / $𝑣21 - $𝑤12;
|
||||
my $𝑤23 = $𝑣23 / $𝑣21 - $𝑤13;
|
||||
my $𝑤24 = $𝑣24 / $𝑣21 - $𝑤14;
|
||||
|
||||
my $𝑃 = -$𝑤23 / $𝑤22;
|
||||
my $𝑄 = $𝑤24 / $𝑤22;
|
||||
my $𝑀 = -$𝑤12 * $𝑃 - $𝑤13;
|
||||
my $𝑁 = $𝑤14 - $𝑤12 * $𝑄;
|
||||
|
||||
my $𝑎 = $𝑁**2 + $𝑄**2 - 1;
|
||||
my $𝑏 = 2 * $𝑀 * $𝑁 - 2 * $𝑁 * $c1->{x} + 2 * $𝑃 * $𝑄 - 2 * $𝑄 * $c1->{y} + 2 * $s1 * $c1->{r};
|
||||
my $𝑐 = $c1->{x}**2 + $𝑀**2 - 2 * $𝑀 * $c1->{x} + $𝑃**2 + $c1->{y}**2 - 2 * $𝑃 * $c1->{y} - $c1->{r}**2;
|
||||
|
||||
my $𝐷 = $𝑏**2 - 4 * $𝑎 * $𝑐;
|
||||
my $rs = (-$𝑏 - sqrt $𝐷) / (2 * $𝑎);
|
||||
|
||||
my $xs = $𝑀 + $𝑁 * $rs;
|
||||
my $ys = $𝑃 + $𝑄 * $rs;
|
||||
|
||||
circle($xs, $ys, $rs);
|
||||
}
|
||||
|
||||
$c1 = circle(0, 0, 1);
|
||||
$c2 = circle(4, 0, 1);
|
||||
$c3 = circle(2, 4, 2);
|
||||
|
||||
for (cartesian {@_} ([-1,1])x3) {
|
||||
print Circle->show( solve_Apollonius $c1, $c2, $c3, @$_);
|
||||
}
|
||||
65
Task/Problem-of-Apollonius/Phix/problem-of-apollonius.phix
Normal file
65
Task/Problem-of-Apollonius/Phix/problem-of-apollonius.phix
Normal file
|
|
@ -0,0 +1,65 @@
|
|||
(phixonline)-->
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">Apollonius</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">calc</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">circles</span><span style="color: #0000FF;">)</span>
|
||||
|
||||
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">s1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">calc</span>
|
||||
|
||||
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r1</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">circles</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">circles</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">],</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">x3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">circles</span><span style="color: #0000FF;">[</span><span style="color: #000000;">3</span><span style="color: #0000FF;">],</span>
|
||||
|
||||
<span style="color: #000000;">v11</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v12</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y1</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v13</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">x2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">y1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">y2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">r1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">r2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v14</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r1</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">v21</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x3</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x2</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v22</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y3</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y2</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v23</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">x3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x3</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">y2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y2</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">y3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y3</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">r2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">r3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r3</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">v24</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r3</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">w12</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v12</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v11</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">w13</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v13</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v11</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">w14</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v14</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v11</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">w22</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v22</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v21</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">w12</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">w23</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v23</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v21</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">w13</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">w24</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v24</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">v21</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">w14</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">P</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">w23</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">w22</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">Q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">w24</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">w22</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">M</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">w12</span><span style="color: #0000FF;">*</span><span style="color: #000000;">P</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">w13</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">N</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">w14</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">w12</span><span style="color: #0000FF;">*</span><span style="color: #000000;">Q</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">N</span><span style="color: #0000FF;">*</span><span style="color: #000000;">N</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">Q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">Q</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">M</span><span style="color: #0000FF;">*</span><span style="color: #000000;">N</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">N</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">P</span><span style="color: #0000FF;">*</span><span style="color: #000000;">Q</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">Q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r1</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">M</span><span style="color: #0000FF;">*</span><span style="color: #000000;">M</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">M</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">P</span><span style="color: #0000FF;">*</span><span style="color: #000000;">P</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">y1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">P</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">r1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">r1</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a</span><span style="color: #0000FF;">*</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span>
|
||||
|
||||
<span style="color: #000000;">rs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(-</span><span style="color: #000000;">b</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">))</span> <span style="color: #0000FF;">/</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
|
||||
|
||||
<span style="color: #000000;">xs</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">M</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">N</span><span style="color: #0000FF;">*</span><span style="color: #000000;">rs</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">ys</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">P</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">Q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">rs</span>
|
||||
|
||||
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">xs</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ys</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rs</span><span style="color: #0000FF;">}</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">circles</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">}}</span>
|
||||
|
||||
<span style="color: #000080;font-style:italic;">-- +1: externally tangental, -1: internally tangental</span>
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">calcs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span>
|
||||
<span style="color: #0000FF;">{+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}}</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">8</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">xs</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ys</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rs</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">Apollonius</span><span style="color: #0000FF;">(</span><span style="color: #000000;">calcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">circles</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">th</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #008000;">"st (external)"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"nd (internal)"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"rd"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"th"</span><span style="color: #0000FF;">}[</span><span style="color: #7060A8;">min</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)]</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%d%s solution: x=%+f, y=%+f, r=%f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">th</span><span style="color: #0000FF;">,</span><span style="color: #000000;">xs</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ys</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rs</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,69 @@
|
|||
function Measure-Apollonius
|
||||
{
|
||||
[CmdletBinding()]
|
||||
[OutputType([PSCustomObject])]
|
||||
Param
|
||||
(
|
||||
[int]$Counter,
|
||||
[double]$x1,
|
||||
[double]$y1,
|
||||
[double]$r1,
|
||||
[double]$x2,
|
||||
[double]$y2,
|
||||
[double]$r2,
|
||||
[double]$x3,
|
||||
[double]$y3,
|
||||
[double]$r3
|
||||
)
|
||||
|
||||
switch ($Counter)
|
||||
{
|
||||
{$_ -eq 2} {$s1 = -1; $s2 = -1; $s3 = -1; break}
|
||||
{$_ -eq 3} {$s1 = 1; $s2 = -1; $s3 = -1; break}
|
||||
{$_ -eq 4} {$s1 = -1; $s2 = 1; $s3 = -1; break}
|
||||
{$_ -eq 5} {$s1 = -1; $s2 = -1; $s3 = 1; break}
|
||||
{$_ -eq 6} {$s1 = 1; $s2 = 1; $s3 = -1; break}
|
||||
{$_ -eq 7} {$s1 = -1; $s2 = 1; $s3 = 1; break}
|
||||
{$_ -eq 8} {$s1 = 1; $s2 = -1; $s3 = 1; break}
|
||||
Default {$s1 = 1; $s2 = 1; $s3 = 1; break}
|
||||
}
|
||||
|
||||
[double]$v11 = 2 * $x2 - 2 * $x1
|
||||
[double]$v12 = 2 * $y2 - 2 * $y1
|
||||
[double]$v13 = $x1 * $x1 - $x2 * $x2 + $y1 * $y1 - $y2 * $y2 - $r1 * $r1 + $r2 * $r2
|
||||
[double]$v14 = 2 * $s2 * $r2 - 2 * $s1 * $r1
|
||||
|
||||
[double]$v21 = 2 * $x3 - 2 * $x2
|
||||
[double]$v22 = 2 * $y3 - 2 * $y2
|
||||
[double]$v23 = $x2 * $x2 - $x3 * $x3 + $y2 * $y2 - $y3 * $y3 - $r2 * $r2 + $r3 * $r3
|
||||
[double]$v24 = 2 * $s3 * $r3 - 2 * $s2 * $r2
|
||||
|
||||
[double]$w12 = $v12 / $v11
|
||||
[double]$w13 = $v13 / $v11
|
||||
[double]$w14 = $v14 / $v11
|
||||
|
||||
[double]$w22 = $v22 / $v21 - $w12
|
||||
[double]$w23 = $v23 / $v21 - $w13
|
||||
[double]$w24 = $v24 / $v21 - $w14
|
||||
|
||||
[double]$P = -$w23 / $w22
|
||||
[double]$Q = $w24 / $w22
|
||||
[double]$M = -$w12 * $P - $w13
|
||||
[double]$N = $w14 - $w12 * $Q
|
||||
|
||||
[double]$a = $N * $N + $Q * $Q - 1
|
||||
[double]$b = 2 * $M * $N - 2 * $N * $x1 + 2 * $P * $Q - 2 * $Q * $y1 + 2 * $s1 * $r1
|
||||
[double]$c = $x1 * $x1 + $M * $M - 2 * $M * $x1 + $P * $P + $y1 * $y1 - 2 * $P * $y1 - $r1 * $r1
|
||||
|
||||
[double]$D = $b * $b - 4 * $a * $c
|
||||
|
||||
[double]$rs = (-$b - [Double]::Parse([Math]::Sqrt($D).ToString())) / (2 * [Double]::Parse($a.ToString()))
|
||||
[double]$xs = $M + $N * $rs
|
||||
[double]$ys = $P + $Q * $rs
|
||||
|
||||
[PSCustomObject]@{
|
||||
X = $xs
|
||||
Y = $ys
|
||||
Radius = $rs
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,4 @@
|
|||
for ($i = 1; $i -le 8; $i++)
|
||||
{
|
||||
Measure-Apollonius -Counter $i -x1 0 -y1 0 -r1 1 -x2 4 -y2 0 -r2 1 -x3 2 -y3 4 -r3 2
|
||||
}
|
||||
|
|
@ -0,0 +1,74 @@
|
|||
Structure Circle
|
||||
XPos.f
|
||||
YPos.f
|
||||
Radius.f
|
||||
EndStructure
|
||||
|
||||
Procedure ApolloniusSolver(*c1.Circle,*c2.Circle,*c3.Circle, s1, s2, s3)
|
||||
Define.f ; This tells the compiler that all non-specified new variables
|
||||
; should be of float type (.f).
|
||||
x1=*c1\XPos: y1=*c1\YPos: r1=*c1\Radius
|
||||
x2=*c2\XPos: y2=*c2\YPos: r2=*c2\Radius
|
||||
x3=*c3\XPos: y3=*c3\YPos: r3=*c3\Radius
|
||||
|
||||
v11 = 2*x2 - 2*x1
|
||||
v12 = 2*y2 - 2*y1
|
||||
v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
v21 = 2*x3 - 2*x2
|
||||
v22 = 2*y3 - 2*y2
|
||||
v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21-w12
|
||||
w23 = v23/v21-w13
|
||||
w24 = v24/v21-w14
|
||||
|
||||
P = -w23/w22
|
||||
Q = w24/w22
|
||||
M = -w12*P-w13
|
||||
N = w14-w12*Q
|
||||
|
||||
a = N*N + Q*Q - 1
|
||||
b = 2*M*N - 2*N*x1 + 2*P*Q - 2*Q*y1 + 2*s1*r1
|
||||
c = x1*x1 + M*M - 2*M*x1 + P*P + y1*y1 - 2*P*y1 - r1*r1
|
||||
|
||||
D= b*b - 4*a*c
|
||||
|
||||
Define *result.Circle=AllocateMemory(SizeOf(Circle))
|
||||
; Allocate memory for a returned Structure of type Circle.
|
||||
; This memory should be freed later but if not, PureBasic’s
|
||||
; internal framework will do so when the program shuts down.
|
||||
If *result
|
||||
*result\Radius=(-b-Sqr(D))/(2*a)
|
||||
*result\XPos =M+N * *result\Radius
|
||||
*result\YPos =P+Q * *result\Radius
|
||||
EndIf
|
||||
ProcedureReturn *result ; Sending back a pointer
|
||||
EndProcedure
|
||||
|
||||
If OpenConsole()
|
||||
Define.Circle c1, c2, c3
|
||||
Define *c.Circle ; '*c' is defined as a pointer to a circle-structure.
|
||||
c1\Radius=1
|
||||
c2\XPos=4: c2\Radius=1
|
||||
c3\XPos=2: c3\YPos=4: c3\Radius=2
|
||||
|
||||
*c=ApolloniusSolver(@c1, @c2, @c3, 1, 1, 1)
|
||||
If *c ; Verify that *c got allocated
|
||||
PrintN("Circle [x="+StrF(*c\XPos,2)+", y="+StrF(*c\YPos,2)+", r="+StrF(*c\Radius,2)+"]")
|
||||
FreeMemory(*c) ; We are done with *c for the first calculation
|
||||
EndIf
|
||||
|
||||
*c=ApolloniusSolver(@c1, @c2, @c3,-1,-1,-1)
|
||||
If *c
|
||||
PrintN("Circle [x="+StrF(*c\XPos,2)+", y="+StrF(*c\YPos,2)+", r="+StrF(*c\Radius,2)+"]")
|
||||
FreeMemory(*c)
|
||||
EndIf
|
||||
Print("Press ENTER to exit"): Input()
|
||||
EndIf
|
||||
56
Task/Problem-of-Apollonius/Python/problem-of-apollonius.py
Normal file
56
Task/Problem-of-Apollonius/Python/problem-of-apollonius.py
Normal file
|
|
@ -0,0 +1,56 @@
|
|||
from collections import namedtuple
|
||||
import math
|
||||
|
||||
Circle = namedtuple('Circle', 'x, y, r')
|
||||
|
||||
def solveApollonius(c1, c2, c3, s1, s2, s3):
|
||||
'''
|
||||
>>> solveApollonius((0, 0, 1), (4, 0, 1), (2, 4, 2), 1,1,1)
|
||||
Circle(x=2.0, y=2.1, r=3.9)
|
||||
>>> solveApollonius((0, 0, 1), (4, 0, 1), (2, 4, 2), -1,-1,-1)
|
||||
Circle(x=2.0, y=0.8333333333333333, r=1.1666666666666667)
|
||||
'''
|
||||
x1, y1, r1 = c1
|
||||
x2, y2, r2 = c2
|
||||
x3, y3, r3 = c3
|
||||
|
||||
v11 = 2*x2 - 2*x1
|
||||
v12 = 2*y2 - 2*y1
|
||||
v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
v21 = 2*x3 - 2*x2
|
||||
v22 = 2*y3 - 2*y2
|
||||
v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21-w12
|
||||
w23 = v23/v21-w13
|
||||
w24 = v24/v21-w14
|
||||
|
||||
P = -w23/w22
|
||||
Q = w24/w22
|
||||
M = -w12*P-w13
|
||||
N = w14 - w12*Q
|
||||
|
||||
a = N*N + Q*Q - 1
|
||||
b = 2*M*N - 2*N*x1 + 2*P*Q - 2*Q*y1 + 2*s1*r1
|
||||
c = x1*x1 + M*M - 2*M*x1 + P*P + y1*y1 - 2*P*y1 - r1*r1
|
||||
|
||||
# Find a root of a quadratic equation. This requires the circle centers not to be e.g. colinear
|
||||
D = b*b-4*a*c
|
||||
rs = (-b-math.sqrt(D))/(2*a)
|
||||
|
||||
xs = M+N*rs
|
||||
ys = P+Q*rs
|
||||
|
||||
return Circle(xs, ys, rs)
|
||||
|
||||
if __name__ == '__main__':
|
||||
c1, c2, c3 = Circle(0, 0, 1), Circle(4, 0, 1), Circle(2, 4, 2)
|
||||
print(solveApollonius(c1, c2, c3, 1, 1, 1)) #Expects "Circle[x=2.00,y=2.10,r=3.90]" (green circle in image)
|
||||
print(solveApollonius(c1, c2, c3, -1, -1, -1)) #Expects "Circle[x=2.00,y=0.83,r=1.17]" (red circle in image)
|
||||
30
Task/Problem-of-Apollonius/REXX/problem-of-apollonius.rexx
Normal file
30
Task/Problem-of-Apollonius/REXX/problem-of-apollonius.rexx
Normal file
|
|
@ -0,0 +1,30 @@
|
|||
/*REXX program solves the problem of Apollonius, named after the Greek Apollonius of */
|
||||
/*────────────────────────────────────── Perga [Pergæus] (circa 262 BCE ──► 190 BCE). */
|
||||
numeric digits 15; x1= 0; y1= 0; r1= 1
|
||||
x2= 4; y2= 0; r2= 1
|
||||
x3= 2; y3= 4; r3= 2
|
||||
call tell 'external tangent: ', Apollonius( 1, 1, 1)
|
||||
call tell 'internal tangent: ', Apollonius(-1, -1, -1)
|
||||
exit /*stick a fork in it, we're all done. */
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
Apollonius: parse arg s1,s2,s3 /*could be internal or external tangent*/
|
||||
numeric digits digits() * 3 /*reduce rounding with thrice digits. */
|
||||
va= x2*2 - x1*2; vb= y2*2 - y1*2
|
||||
vc= x1**2 - x2**2 + y1**2 - y2**2 - r1**2 + r2**2
|
||||
vd= s2*r2*2 - s1*r1*2; ve= x3*2 - x2*2; vf= y3*2 - y2*2
|
||||
vg= x2**2 - x3**2 + y2**2 - y3**2 - r2**2 + r3**2; vh= s3*r3*2 - s2*r2*2
|
||||
vj= vb/va; vk= vc/va; vm= vd/va; vn= vf/ve - vj
|
||||
vp= vg/ve - vk; vr= vh/ve - vm; p = -vp/vn; q = vr/vn
|
||||
m = -vj*p - vk; n = vm - vj*q
|
||||
a = n**2 + q**2 - 1
|
||||
b = (m*n - n*x1 + p*q - q*y1 + s1*r1) * 2
|
||||
c = x1**2 + y1**2 + m**2 - r1**2 + p**2 - (m*x1 + p*y1) * 2
|
||||
$r= (-b - sqrt(b**2 - a*c*4) ) / (a+a)
|
||||
return (m + n*$r) (p + q*$r) ($r) /*return 3 arguments.*/
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); h=d+6; numeric digits
|
||||
m.=9; numeric form; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_%2
|
||||
do j=0 while h>9; m.j=h; h= h % 2 + 1; end /*j*/
|
||||
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g) * .5; end /*k*/; return g
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
tell: parse arg _,a b c; w=digits()+4; say _ left(a/1,w%2) left(b/1,w) left(c/1,w); return
|
||||
84
Task/Problem-of-Apollonius/Racket/problem-of-apollonius.rkt
Normal file
84
Task/Problem-of-Apollonius/Racket/problem-of-apollonius.rkt
Normal file
|
|
@ -0,0 +1,84 @@
|
|||
#lang slideshow
|
||||
|
||||
(struct circle (x y r) #:prefab)
|
||||
|
||||
(define (apollonius c1 c2 c3 s1 s2 s3)
|
||||
(define x1 (circle-x c1))
|
||||
(define y1 (circle-y c1))
|
||||
(define r1 (circle-r c1))
|
||||
(define x2 (circle-x c2))
|
||||
(define y2 (circle-y c2))
|
||||
(define r2 (circle-r c2))
|
||||
(define x3 (circle-x c3))
|
||||
(define y3 (circle-y c3))
|
||||
(define r3 (circle-r c3))
|
||||
|
||||
(define v11 (- (* 2 x2) (* 2 x1)))
|
||||
(define v12 (- (* 2 y2) (* 2 y1)))
|
||||
(define v13 (+ (- (* x1 x1) (* x2 x2))
|
||||
(- (* y1 y1) (* y2 y2))
|
||||
(- (* r2 r2) (* r1 r1))))
|
||||
(define v14 (- (* 2 s2 r2) (* 2 s1 r1)))
|
||||
|
||||
(define v21 (- (* 2 x3) (* 2 x2)))
|
||||
(define v22 (- (* 2 y3) (* 2 y2)))
|
||||
(define v23 (+ (- (* x2 x2) (* x3 x3))
|
||||
(- (* y2 y2) (* y3 y3))
|
||||
(- (* r3 r3) (* r2 r2))))
|
||||
(define v24 (- (* 2 s3 r3) (* 2 s2 r2)))
|
||||
|
||||
(define w12 (/ v12 v11))
|
||||
(define w13 (/ v13 v11))
|
||||
(define w14 (/ v14 v11))
|
||||
|
||||
(define w22 (- (/ v22 v21) w12))
|
||||
(define w23 (- (/ v23 v21) w13))
|
||||
(define w24 (- (/ v24 v21) w14))
|
||||
|
||||
(define P (- (/ w23 w22)))
|
||||
(define Q (/ w24 w22))
|
||||
(define M (- (+ (* w12 P) w13)))
|
||||
(define N (- w14 (* w12 Q)))
|
||||
|
||||
(define a (+ (* N N) (* Q Q) -1))
|
||||
(define b (+ (- (* 2 M N) (* 2 N x1))
|
||||
(- (* 2 P Q) (* 2 Q y1))
|
||||
(* 2 s1 r1)))
|
||||
(define c (- (+ (* x1 x1) (* M M) (* P P) (* y1 y1))
|
||||
(+ (* 2 M x1) (* 2 P y1) (* r1 r1))))
|
||||
|
||||
(define D (- (* b b) (* 4 a c)))
|
||||
(define rs (/ (- (+ b (sqrt D))) (* 2 a)))
|
||||
(define xs (+ M (* N rs)))
|
||||
(define ys (+ P (* Q rs)))
|
||||
(circle xs ys rs))
|
||||
|
||||
(define c1 (circle 0.0 0.0 1.0))
|
||||
(define c2 (circle 4.0 0.0 1.0))
|
||||
(define c3 (circle 2.0 4.0 2.0))
|
||||
|
||||
;; print solutions
|
||||
(apollonius c1 c2 c3 1.0 1.0 1.0)
|
||||
(apollonius c1 c2 c3 -1.0 -1.0 -1.0)
|
||||
|
||||
;; visualize solutions
|
||||
(require racket/gui/base)
|
||||
(define (show-circles . circles+colors)
|
||||
(define f (new frame% [label "Apollonius"] [width 300] [height 300]))
|
||||
(define c
|
||||
(new canvas% [parent f]
|
||||
[paint-callback
|
||||
(lambda (canvas dc)
|
||||
(send* dc (set-origin 100 100)
|
||||
(set-scale 20 20)
|
||||
(set-pen "black" 1/10 'solid)
|
||||
(set-brush "white" 'transparent))
|
||||
(for ([x circles+colors])
|
||||
(if (string? x)
|
||||
(send dc set-pen x 1/5 'solid)
|
||||
(let ([x (circle-x x)] [y (circle-y x)] [r (circle-r x)])
|
||||
(send dc draw-ellipse (- x r) (- y r) (* 2 r) (* 2 r))))))]))
|
||||
(send f show #t))
|
||||
(show-circles "black" c1 c2 c3
|
||||
"green" (apollonius c1 c2 c3 1.0 1.0 1.0)
|
||||
"red" (apollonius c1 c2 c3 -1.0 -1.0 -1.0))
|
||||
50
Task/Problem-of-Apollonius/Raku/problem-of-apollonius.raku
Normal file
50
Task/Problem-of-Apollonius/Raku/problem-of-apollonius.raku
Normal file
|
|
@ -0,0 +1,50 @@
|
|||
class Circle {
|
||||
has $.x;
|
||||
has $.y;
|
||||
has $.r;
|
||||
method gist { sprintf "%s =%7.3f " xx 3, (:$!x,:$!y,:$!r)».kv }
|
||||
}
|
||||
|
||||
sub circle($x,$y,$r) { Circle.new: :$x, :$y, :$r }
|
||||
|
||||
sub solve-Apollonius([\c1, \c2, \c3], [\s1, \s2, \s3]) {
|
||||
my \𝑣11 = 2 * c2.x - 2 * c1.x;
|
||||
my \𝑣12 = 2 * c2.y - 2 * c1.y;
|
||||
my \𝑣13 = c1.x² - c2.x² + c1.y² - c2.y² - c1.r² + c2.r²;
|
||||
my \𝑣14 = 2 * s2 * c2.r - 2 * s1 * c1.r;
|
||||
|
||||
my \𝑣21 = 2 * c3.x - 2 * c2.x;
|
||||
my \𝑣22 = 2 * c3.y - 2 * c2.y;
|
||||
my \𝑣23 = c2.x² - c3.x² + c2.y² - c3.y² - c2.r² + c3.r²;
|
||||
my \𝑣24 = 2 * s3 * c3.r - 2 * s2 * c2.r;
|
||||
|
||||
my \𝑤12 = 𝑣12 / 𝑣11;
|
||||
my \𝑤13 = 𝑣13 / 𝑣11;
|
||||
my \𝑤14 = 𝑣14 / 𝑣11;
|
||||
|
||||
my \𝑤22 = 𝑣22 / 𝑣21 - 𝑤12;
|
||||
my \𝑤23 = 𝑣23 / 𝑣21 - 𝑤13;
|
||||
my \𝑤24 = 𝑣24 / 𝑣21 - 𝑤14;
|
||||
|
||||
my \𝑃 = -𝑤23 / 𝑤22;
|
||||
my \𝑄 = 𝑤24 / 𝑤22;
|
||||
my \𝑀 = -𝑤12 * 𝑃 - 𝑤13;
|
||||
my \𝑁 = 𝑤14 - 𝑤12 * 𝑄;
|
||||
|
||||
my \𝑎 = 𝑁² + 𝑄² - 1;
|
||||
my \𝑏 = 2 * 𝑀 * 𝑁 - 2 * 𝑁 * c1.x + 2 * 𝑃 * 𝑄 - 2 * 𝑄 * c1.y + 2 * s1 * c1.r;
|
||||
my \𝑐 = c1.x² + 𝑀² - 2 * 𝑀 * c1.x + 𝑃² + c1.y² - 2 * 𝑃 * c1.y - c1.r²;
|
||||
|
||||
my \𝐷 = 𝑏² - 4 * 𝑎 * 𝑐;
|
||||
my \rs = (-𝑏 - sqrt 𝐷) / (2 * 𝑎);
|
||||
|
||||
my \xs = 𝑀 + 𝑁 * rs;
|
||||
my \ys = 𝑃 + 𝑄 * rs;
|
||||
|
||||
circle(xs, ys, rs);
|
||||
}
|
||||
|
||||
my @c = circle(0, 0, 1), circle(4, 0, 1), circle(2, 4, 2);
|
||||
for ([X] [-1,1] xx 3) -> @i {
|
||||
say (solve-Apollonius @c, @i).gist;
|
||||
}
|
||||
57
Task/Problem-of-Apollonius/Ruby/problem-of-apollonius.rb
Normal file
57
Task/Problem-of-Apollonius/Ruby/problem-of-apollonius.rb
Normal file
|
|
@ -0,0 +1,57 @@
|
|||
class Circle
|
||||
def initialize(x, y, r)
|
||||
@x, @y, @r = [x, y, r].map(&:to_f)
|
||||
end
|
||||
attr_reader :x, :y, :r
|
||||
|
||||
def self.apollonius(c1, c2, c3, s1=1, s2=1, s3=1)
|
||||
x1, y1, r1 = c1.x, c1.y, c1.r
|
||||
x2, y2, r2 = c2.x, c2.y, c2.r
|
||||
x3, y3, r3 = c3.x, c3.y, c3.r
|
||||
|
||||
v11 = 2*x2 - 2*x1
|
||||
v12 = 2*y2 - 2*y1
|
||||
v13 = x1**2 - x2**2 + y1**2 - y2**2 - r1**2 + r2**2
|
||||
v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
v21 = 2*x3 - 2*x2
|
||||
v22 = 2*y3 - 2*y2
|
||||
v23 = x2**2 - x3**2 + y2**2 - y3**2 - r2**2 + r3**2
|
||||
v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
w12 = v12/v11
|
||||
w13 = v13/v11
|
||||
w14 = v14/v11
|
||||
|
||||
w22 = v22/v21 - w12
|
||||
w23 = v23/v21 - w13
|
||||
w24 = v24/v21 - w14
|
||||
|
||||
p = -w23/w22
|
||||
q = w24/w22
|
||||
m = -w12*p - w13
|
||||
n = w14 - w12*q
|
||||
|
||||
a = n**2 + q**2 - 1
|
||||
b = 2*m*n - 2*n*x1 + 2*p*q - 2*q*y1 + 2*s1*r1
|
||||
c = x1**2 + m**2 - 2*m*x1 + p**2 + y1**2 - 2*p*y1 - r1**2
|
||||
|
||||
d = b**2 - 4*a*c
|
||||
rs = (-b - Math.sqrt(d)) / (2*a)
|
||||
xs = m + n*rs
|
||||
ys = p + q*rs
|
||||
|
||||
self.new(xs, ys, rs)
|
||||
end
|
||||
|
||||
def to_s
|
||||
"Circle: x=#{@x}, y=#{@y}, r=#{@r}"
|
||||
end
|
||||
end
|
||||
|
||||
puts c1 = Circle.new(0, 0, 1)
|
||||
puts c2 = Circle.new(2, 4, 2)
|
||||
puts c3 = Circle.new(4, 0, 1)
|
||||
|
||||
puts Circle.apollonius(c1, c2, c3)
|
||||
puts Circle.apollonius(c1, c2, c3, -1, -1, -1)
|
||||
81
Task/Problem-of-Apollonius/Scala/problem-of-apollonius.scala
Normal file
81
Task/Problem-of-Apollonius/Scala/problem-of-apollonius.scala
Normal file
|
|
@ -0,0 +1,81 @@
|
|||
object ApolloniusSolver extends App {
|
||||
case class Circle(x: Double, y: Double, r: Double)
|
||||
object Tangent extends Enumeration {
|
||||
type Tangent = Value
|
||||
val intern = Value(-1)
|
||||
val extern = Value(1)
|
||||
}
|
||||
|
||||
import Tangent._
|
||||
import scala.Math._
|
||||
|
||||
val solveApollonius: (Circle, Circle, Circle, Triple[Tangent, Tangent, Tangent]) => Circle = (c1, c2, c3, tangents) => {
|
||||
val fv: (Circle, Circle, Int, Int) => Tuple4[Double, Double, Double, Double] = (c1, c2, s1, s2) => {
|
||||
val v11 = 2 * c2.x - 2 * c1.x
|
||||
val v12 = 2 * c2.y - 2 * c1.y
|
||||
val v13 = pow(c1.x, 2) - pow(c2.x, 2) + pow(c1.y, 2) - pow(c2.y, 2) - pow(c1.r, 2) + pow(c2.r, 2)
|
||||
val v14 = 2 * s2 * c2.r - 2 * s1 * c1.r
|
||||
Tuple4(v11, v12, v13, v14)
|
||||
}
|
||||
val (s1, s2, s3) = (tangents._1.id, tangents._2.id, tangents._3.id)
|
||||
|
||||
val (v11, v12, v13, v14) = fv(c1, c2, s1, s2)
|
||||
val (v21, v22, v23, v24) = fv(c2, c3, s2, s3)
|
||||
|
||||
val w12 = v12 / v11
|
||||
val w13 = v13 / v11
|
||||
val w14 = v14 / v11
|
||||
|
||||
val w22 = v22 / v21 - w12
|
||||
val w23 = v23 / v21 - w13
|
||||
val w24 = v24 / v21 - w14
|
||||
|
||||
val P = -w23 / w22
|
||||
val Q = w24 / w22
|
||||
val M = -w12 * P - w13
|
||||
val N = w14 - w12 * Q
|
||||
|
||||
val a = N*N + Q*Q - 1
|
||||
val b = 2*M*N - 2*N*c1.x +
|
||||
2*P*Q - 2*Q*c1.y +
|
||||
2*s1*c1.r
|
||||
val c = pow(c1.x, 2) + M*M - 2*M*c1.x +
|
||||
P*P + pow(c1.y, 2) - 2*P*c1.y - pow(c1.r, 2)
|
||||
|
||||
// Find a root of a quadratic equation. This requires the circle centers not to be e.g. colinear
|
||||
val D = b*b - 4*a*c
|
||||
val rs = (-b - sqrt(D)) / (2*a)
|
||||
|
||||
Circle(x=M + N*rs, y=P + Q*rs, r=rs)
|
||||
}
|
||||
|
||||
val c1 = Circle(x=0.0, y=0.0, r=1.0)
|
||||
val c2 = Circle(x=4.0, y=0.0, r=1.0)
|
||||
val c3 = Circle(x=2.0, y=4.0, r=2.0)
|
||||
|
||||
println("c1: "+c1)
|
||||
println("c2: "+c2)
|
||||
println("c3: "+c3)
|
||||
|
||||
println{
|
||||
val tangents = Triple(intern, intern, intern)
|
||||
"red circle: tangents="+tangents+" cs=" + solveApollonius(c1, c2, c3, tangents)
|
||||
}
|
||||
println{
|
||||
val tangents = Triple(extern, extern, extern)
|
||||
"green circle: tangents="+tangents+" cs=" + solveApollonius(c1, c2, c3, tangents)
|
||||
}
|
||||
|
||||
println("all combinations:")
|
||||
for ( ti <- Tangent.values)
|
||||
for ( tj <- Tangent.values)
|
||||
for ( tk <- Tangent.values) {
|
||||
println{
|
||||
val format: Circle => String = c => {
|
||||
"Circle(x=%8.5f, y=%8.5f, r=%8.5f)".format(c.x, c.y, c.r)
|
||||
}
|
||||
val tangents = Triple(ti, tj, tk)
|
||||
"tangents: " + tangents + " -> cs=" + format(solveApollonius(c1, c2, c3, tangents))
|
||||
}
|
||||
}
|
||||
}
|
||||
48
Task/Problem-of-Apollonius/Sidef/problem-of-apollonius.sidef
Normal file
48
Task/Problem-of-Apollonius/Sidef/problem-of-apollonius.sidef
Normal file
|
|
@ -0,0 +1,48 @@
|
|||
class Circle(x,y,r) {
|
||||
method to_s { "Circle(#{x}, #{y}, #{r})" }
|
||||
}
|
||||
|
||||
func solve_apollonius(c, s) {
|
||||
|
||||
var(c1, c2, c3) = c...;
|
||||
var(s1, s2, s3) = s...;
|
||||
|
||||
var 𝑣11 = (2*c2.x - 2*c1.x);
|
||||
var 𝑣12 = (2*c2.y - 2*c1.y);
|
||||
var 𝑣13 = (c1.x**2 - c2.x**2 + c1.y**2 - c2.y**2 - c1.r**2 + c2.r**2);
|
||||
var 𝑣14 = (2*s2*c2.r - 2*s1*c1.r);
|
||||
|
||||
var 𝑣21 = (2*c3.x - 2*c2.x);
|
||||
var 𝑣22 = (2*c3.y - 2*c2.y);
|
||||
var 𝑣23 = (c2.x**2 - c3.x**2 + c2.y**2 - c3.y**2 - c2.r**2 + c3.r**2);
|
||||
var 𝑣24 = (2*s3*c3.r - 2*s2*c2.r);
|
||||
|
||||
var 𝑤12 = (𝑣12 / 𝑣11);
|
||||
var 𝑤13 = (𝑣13 / 𝑣11);
|
||||
var 𝑤14 = (𝑣14 / 𝑣11);
|
||||
|
||||
var 𝑤22 = (𝑣22/𝑣21 - 𝑤12);
|
||||
var 𝑤23 = (𝑣23/𝑣21 - 𝑤13);
|
||||
var 𝑤24 = (𝑣24/𝑣21 - 𝑤14);
|
||||
|
||||
var 𝑃 = (-𝑤23 / 𝑤22);
|
||||
var 𝑄 = (𝑤24 / 𝑤22);
|
||||
var 𝑀 = (-𝑤12*𝑃 - 𝑤13);
|
||||
var 𝑁 = (𝑤14 - 𝑤12*𝑄);
|
||||
|
||||
var 𝑎 = (𝑁**2 + 𝑄**2 - 1);
|
||||
var 𝑏 = (2*𝑀*𝑁 - 2*𝑁*c1.x + 2*𝑃*𝑄 - 2*𝑄*c1.y + 2*s1*c1.r);
|
||||
var 𝑐 = (c1.x**2 + 𝑀**2 - 2*𝑀*c1.x + 𝑃**2 + c1.y**2 - 2*𝑃*c1.y - c1.r**2);
|
||||
|
||||
var 𝐷 = (𝑏**2 - 4*𝑎*𝑐);
|
||||
var rs = ((-𝑏 - 𝐷.sqrt) / 2*𝑎);
|
||||
|
||||
var xs = (𝑀 + 𝑁*rs);
|
||||
var ys = (𝑃 + 𝑄*rs);
|
||||
|
||||
Circle(xs, ys, rs);
|
||||
}
|
||||
|
||||
var c = [Circle(0, 0, 1), Circle(4, 0, 1), Circle(2, 4, 2)];
|
||||
say solve_apollonius(c, %n<1 1 1>);
|
||||
say solve_apollonius(c, %n<-1 -1 -1>);
|
||||
72
Task/Problem-of-Apollonius/Swift/problem-of-apollonius.swift
Normal file
72
Task/Problem-of-Apollonius/Swift/problem-of-apollonius.swift
Normal file
|
|
@ -0,0 +1,72 @@
|
|||
import Foundation
|
||||
|
||||
struct Circle {
|
||||
let center:[Double]!
|
||||
let radius:Double!
|
||||
|
||||
init(center:[Double], radius:Double) {
|
||||
self.center = center
|
||||
self.radius = radius
|
||||
}
|
||||
|
||||
func toString() -> String {
|
||||
return "Circle[x=\(center[0]),y=\(center[1]),r=\(radius)]"
|
||||
}
|
||||
}
|
||||
|
||||
func solveApollonius(c1:Circle, c2:Circle, c3:Circle,
|
||||
s1:Double, s2:Double, s3:Double) -> Circle {
|
||||
|
||||
let x1 = c1.center[0]
|
||||
let y1 = c1.center[1]
|
||||
let r1 = c1.radius
|
||||
let x2 = c2.center[0]
|
||||
let y2 = c2.center[1]
|
||||
let r2 = c2.radius
|
||||
let x3 = c3.center[0]
|
||||
let y3 = c3.center[1]
|
||||
let r3 = c3.radius
|
||||
|
||||
let v11 = 2*x2 - 2*x1
|
||||
let v12 = 2*y2 - 2*y1
|
||||
let v13 = x1*x1 - x2*x2 + y1*y1 - y2*y2 - r1*r1 + r2*r2
|
||||
let v14 = 2*s2*r2 - 2*s1*r1
|
||||
|
||||
let v21 = 2*x3 - 2*x2
|
||||
let v22 = 2*y3 - 2*y2
|
||||
let v23 = x2*x2 - x3*x3 + y2*y2 - y3*y3 - r2*r2 + r3*r3
|
||||
let v24 = 2*s3*r3 - 2*s2*r2
|
||||
|
||||
let w12 = v12/v11
|
||||
let w13 = v13/v11
|
||||
let w14 = v14/v11
|
||||
|
||||
let w22 = v22/v21-w12
|
||||
let w23 = v23/v21-w13
|
||||
let w24 = v24/v21-w14
|
||||
|
||||
let P = -w23/w22
|
||||
let Q = w24/w22
|
||||
let M = -w12*P-w13
|
||||
let N = w14 - w12*Q
|
||||
|
||||
let a = N*N + Q*Q - 1
|
||||
let b = 2*M*N - 2*N*x1 + 2*P*Q - 2*Q*y1 + 2*s1*r1
|
||||
let c = x1*x1 + M*M - 2*M*x1 + P*P + y1*y1 - 2*P*y1 - r1*r1
|
||||
|
||||
let D = b*b-4*a*c
|
||||
|
||||
let rs = (-b - sqrt(D)) / (2*a)
|
||||
let xs = M + N * rs
|
||||
let ys = P + Q * rs
|
||||
|
||||
return Circle(center: [xs,ys], radius: rs)
|
||||
|
||||
}
|
||||
|
||||
let c1 = Circle(center: [0,0], radius: 1)
|
||||
let c2 = Circle(center: [4,0], radius: 1)
|
||||
let c3 = Circle(center: [2,4], radius: 2)
|
||||
|
||||
println(solveApollonius(c1,c2,c3,1,1,1).toString())
|
||||
println(solveApollonius(c1,c2,c3,-1,-1,-1).toString())
|
||||
56
Task/Problem-of-Apollonius/Tcl/problem-of-apollonius-1.tcl
Normal file
56
Task/Problem-of-Apollonius/Tcl/problem-of-apollonius-1.tcl
Normal file
|
|
@ -0,0 +1,56 @@
|
|||
package require TclOO; # Just so we can make a circle class
|
||||
|
||||
oo::class create circle {
|
||||
variable X Y Radius
|
||||
constructor {x y radius} {
|
||||
namespace import ::tcl::mathfunc::double
|
||||
set X [double $x]; set Y [double $y]; set Radius [double $radius]
|
||||
}
|
||||
method values {} {list $X $Y $Radius}
|
||||
method format {} {
|
||||
format "Circle\[o=(%.2f,%.2f),r=%.2f\]" $X $Y $Radius
|
||||
}
|
||||
}
|
||||
|
||||
proc solveApollonius {c1 c2 c3 {s1 1} {s2 1} {s3 1}} {
|
||||
if {abs($s1)!=1||abs($s2)!=1||abs($s3)!=1} {
|
||||
error "wrong sign; must be 1 or -1"
|
||||
}
|
||||
|
||||
lassign [$c1 values] x1 y1 r1
|
||||
lassign [$c2 values] x2 y2 r2
|
||||
lassign [$c3 values] x3 y3 r3
|
||||
|
||||
set v11 [expr {2*($x2 - $x1)}]
|
||||
set v12 [expr {2*($y2 - $y1)}]
|
||||
set v13 [expr {$x1**2 - $x2**2 + $y1**2 - $y2**2 - $r1**2 + $r2**2}]
|
||||
set v14 [expr {2*($s2*$r2 - $s1*$r1)}]
|
||||
|
||||
set v21 [expr {2*($x3 - $x2)}]
|
||||
set v22 [expr {2*($y3 - $y2)}]
|
||||
set v23 [expr {$x2**2 - $x3**2 + $y2**2 - $y3**2 - $r2**2 + $r3**2}]
|
||||
set v24 [expr {2*($s3*$r3 - $s2*$r2)}]
|
||||
|
||||
set w12 [expr {$v12 / $v11}]
|
||||
set w13 [expr {$v13 / $v11}]
|
||||
set w14 [expr {$v14 / $v11}]
|
||||
|
||||
set w22 [expr {$v22 / $v21 - $w12}]
|
||||
set w23 [expr {$v23 / $v21 - $w13}]
|
||||
set w24 [expr {$v24 / $v21 - $w14}]
|
||||
|
||||
set P [expr {-$w23 / $w22}]
|
||||
set Q [expr {$w24 / $w22}]
|
||||
set M [expr {-$w12 * $P - $w13}]
|
||||
set N [expr {$w14 - $w12 * $Q}]
|
||||
|
||||
set a [expr {$N**2 + $Q**2 - 1}]
|
||||
set b [expr {2*($M*$N - $N*$x1 + $P*$Q - $Q*$y1 + $s1*$r1)}]
|
||||
set c [expr {($x1-$M)**2 + ($y1-$P)**2 - $r1**2}]
|
||||
|
||||
set rs [expr {(-$b - sqrt($b**2 - 4*$a*$c)) / (2*$a)}]
|
||||
set xs [expr {$M + $N*$rs}]
|
||||
set ys [expr {$P + $Q*$rs}]
|
||||
|
||||
return [circle new $xs $ys $rs]
|
||||
}
|
||||
|
|
@ -0,0 +1,7 @@
|
|||
set c1 [circle new 0 0 1]
|
||||
set c2 [circle new 4 0 1]
|
||||
set c3 [circle new 2 4 2]
|
||||
set sA [solveApollonius $c1 $c2 $c3]
|
||||
set sB [solveApollonius $c1 $c2 $c3 -1 -1 -1]
|
||||
puts [$sA format]
|
||||
puts [$sB format]
|
||||
196
Task/Problem-of-Apollonius/VBA/problem-of-apollonius.vba
Normal file
196
Task/Problem-of-Apollonius/VBA/problem-of-apollonius.vba
Normal file
|
|
@ -0,0 +1,196 @@
|
|||
Option Explicit
|
||||
Option Base 0
|
||||
|
||||
Private Const intBase As Integer = 0
|
||||
|
||||
Private Type tPoint
|
||||
X As Double
|
||||
Y As Double
|
||||
End Type
|
||||
Private Type tCircle
|
||||
Centre As tPoint
|
||||
Radius As Double
|
||||
End Type
|
||||
|
||||
Private Sub sApollonius()
|
||||
Dim Circle1 As tCircle
|
||||
Dim Circle2 As tCircle
|
||||
Dim Circle3 As tCircle
|
||||
Dim CTanTanTan(intBase + 0 to intBase + 7) As tCircle
|
||||
|
||||
With Circle1
|
||||
With .Centre
|
||||
.X = 0
|
||||
.Y = 0
|
||||
End With
|
||||
.Radius = 1
|
||||
End With
|
||||
|
||||
With Circle2
|
||||
With .Centre
|
||||
.X = 4
|
||||
.Y = 0
|
||||
End With
|
||||
.Radius = 1
|
||||
End With
|
||||
|
||||
With Circle3
|
||||
With .Centre
|
||||
.X = 2
|
||||
.Y = 4
|
||||
End With
|
||||
.Radius = 2
|
||||
End With
|
||||
|
||||
Call fApollonius(Circle1,Circle2,Circle3,CTanTanTan()))
|
||||
|
||||
End Sub
|
||||
|
||||
Public Function fApollonius(ByRef C1 As tCircle, _
|
||||
ByRef C2 As tCircle, _
|
||||
ByRef C3 As tCircle, _
|
||||
ByRef CTanTanTan() As tCircle) As Boolean
|
||||
' Solves the Problem of Apollonius (finding a circle tangent to three other circles in the plane)
|
||||
' (x_s - x_1)^2 + (y_s - y_1)^2 = (r_s - Tan_1 * r_1)^2
|
||||
' (x_s - x_2)^2 + (y_s - y_2)^2 = (r_s - Tan_2 * r_2)^2
|
||||
' (x_s - x_3)^2 + (y_s - y_3)^2 = (r_s - Tan_3 * r_3)^2
|
||||
' x_s = M + N * r_s
|
||||
' y_s = P + Q * r_s
|
||||
|
||||
' Parameters:
|
||||
' C1, C2, C3 (circles in the problem)
|
||||
' Tan1 := An indication if the solution should be externally or internally tangent (+1/-1) to Circle1 (C1)
|
||||
' Tan2 := An indication if the solution should be externally or internally tangent (+1/-1) to Circle2 (C2)
|
||||
' Tan3 := An indication if the solution should be externally or internally tangent (+1/-1) to Circle3 (C3)
|
||||
|
||||
Dim Tangent(intBase + 0 To intBase + 7, intBase + 0 To intBase + 2) As Integer
|
||||
Dim lgTangent As Long
|
||||
Dim Tan1 As Integer
|
||||
Dim Tan2 As Integer
|
||||
Dim Tan3 As Integer
|
||||
|
||||
Dim v11 As Double
|
||||
Dim v12 As Double
|
||||
Dim v13 As Double
|
||||
Dim v14 As Double
|
||||
Dim v21 As Double
|
||||
Dim v22 As Double
|
||||
Dim v23 As Double
|
||||
Dim v24 As Double
|
||||
Dim w12 As Double
|
||||
Dim w13 As Double
|
||||
Dim w14 As Double
|
||||
Dim w22 As Double
|
||||
Dim w23 As Double
|
||||
Dim w24 As Double
|
||||
|
||||
Dim p As Double
|
||||
Dim Q As Double
|
||||
Dim M As Double
|
||||
Dim N As Double
|
||||
|
||||
Dim A As Double
|
||||
Dim b As Double
|
||||
Dim c As Double
|
||||
Dim D As Double
|
||||
|
||||
'Check if circle centers are colinear
|
||||
If fColinearPoints(C1.Centre, C2.Centre, C3.Centre) Then
|
||||
fApollonius = False
|
||||
Exit Function
|
||||
End If
|
||||
|
||||
Tangent(intBase + 0, intBase + 0) = -1
|
||||
Tangent(intBase + 0, intBase + 1) = -1
|
||||
Tangent(intBase + 0, intBase + 2) = -1
|
||||
|
||||
Tangent(intBase + 1, intBase + 0) = -1
|
||||
Tangent(intBase + 1, intBase + 1) = -1
|
||||
Tangent(intBase + 1, intBase + 2) = 1
|
||||
|
||||
Tangent(intBase + 2, intBase + 0) = -1
|
||||
Tangent(intBase + 2, intBase + 1) = 1
|
||||
Tangent(intBase + 2, intBase + 2) = -1
|
||||
|
||||
Tangent(intBase + 3, intBase + 0) = -1
|
||||
Tangent(intBase + 3, intBase + 1) = 1
|
||||
Tangent(intBase + 3, intBase + 2) = 1
|
||||
|
||||
Tangent(intBase + 4, intBase + 0) = 1
|
||||
Tangent(intBase + 4, intBase + 1) = -1
|
||||
Tangent(intBase + 4, intBase + 2) = -1
|
||||
|
||||
Tangent(intBase + 5, intBase + 0) = 1
|
||||
Tangent(intBase + 5, intBase + 1) = -1
|
||||
Tangent(intBase + 5, intBase + 2) = 1
|
||||
|
||||
Tangent(intBase + 6, intBase + 0) = 1
|
||||
Tangent(intBase + 6, intBase + 1) = 1
|
||||
Tangent(intBase + 6, intBase + 2) = -1
|
||||
|
||||
Tangent(intBase + 7, intBase + 0) = 1
|
||||
Tangent(intBase + 7, intBase + 1) = 1
|
||||
Tangent(intBase + 7, intBase + 2) = 1
|
||||
|
||||
For lgTangent = LBound(Tangent) To UBound(Tangent)
|
||||
Tan1 = Tangent(lgTangent, intBase + 0)
|
||||
Tan2 = Tangent(lgTangent, intBase + 1)
|
||||
Tan3 = Tangent(lgTangent, intBase + 2)
|
||||
|
||||
v11 = 2 * (C2.Centre.X - C1.Centre.X)
|
||||
v12 = 2 * (C2.Centre.Y - C1.Centre.Y)
|
||||
v13 = (C1.Centre.X * C1.Centre.X) _
|
||||
- (C2.Centre.X * C2.Centre.X) _
|
||||
+ (C1.Centre.Y * C1.Centre.Y) _
|
||||
- (C2.Centre.Y * C2.Centre.Y) _
|
||||
- (C1.Radius * C1.Radius) _
|
||||
+ (C2.Radius * C2.Radius)
|
||||
v14 = 2 * (Tan2 * C2.Radius - Tan1 * C1.Radius)
|
||||
|
||||
v21 = 2 * (C3.Centre.X - C2.Centre.X)
|
||||
v22 = 2 * (C3.Centre.Y - C2.Centre.Y)
|
||||
v23 = (C2.Centre.X * C2.Centre.X) _
|
||||
- (C3.Centre.X * C3.Centre.X) _
|
||||
+ (C2.Centre.Y * C2.Centre.Y) _
|
||||
- (C3.Centre.Y * C3.Centre.Y) _
|
||||
- (C2.Radius * C2.Radius) _
|
||||
+ (C3.Radius * C3.Radius)
|
||||
v24 = 2 * ((Tan3 * C3.Radius) - (Tan2 * C2.Radius))
|
||||
|
||||
w12 = v12 / v11
|
||||
w13 = v13 / v11
|
||||
w14 = v14 / v11
|
||||
|
||||
w22 = (v22 / v21) - w12
|
||||
w23 = (v23 / v21) - w13
|
||||
w24 = (v24 / v21) - w14
|
||||
|
||||
p = -w23 / w22
|
||||
Q = w24 / w22
|
||||
M = -(w12 * p) - w13
|
||||
N = w14 - (w12 * Q)
|
||||
|
||||
A = (N * N) + (Q * Q) - 1
|
||||
b = 2 * ((M * N) - (N * C1.Centre.X) + (p * Q) - (Q * C1.Centre.Y) + (Tan1 * C1.Radius))
|
||||
c = (C1.Centre.X * C1.Centre.X) _
|
||||
+ (M * M) _
|
||||
- (2 * M * C1.Centre.X) _
|
||||
+ (p * p) _
|
||||
+ (C1.Centre.Y * C1.Centre.Y) _
|
||||
- (2 * p * C1.Centre.Y) _
|
||||
- (C1.Radius * C1.Radius)
|
||||
|
||||
'Find a root of a quadratic equation (requires the circle centers not to be e.g. colinear)
|
||||
D = (b * b) - (4 * A * c)
|
||||
|
||||
With CTanTanTan(lgTangent)
|
||||
.Radius = (-b - VBA.Sqr(D)) / (2 * A)
|
||||
.Centre.X = M + (N * .Radius)
|
||||
.Centre.Y = p + (Q * .Radius)
|
||||
End With
|
||||
|
||||
Next lgTangent
|
||||
|
||||
fApollonius = True
|
||||
|
||||
End Function
|
||||
56
Task/Problem-of-Apollonius/Wren/problem-of-apollonius.wren
Normal file
56
Task/Problem-of-Apollonius/Wren/problem-of-apollonius.wren
Normal file
|
|
@ -0,0 +1,56 @@
|
|||
import "/dynamic" for Tuple
|
||||
|
||||
var Circle = Tuple.create("Circle", ["x", "y", "r"])
|
||||
|
||||
var solveApollonius = Fn.new { |c1, c2, c3, s1, s2, s3|
|
||||
var x1 = c1.x
|
||||
var y1 = c1.y
|
||||
var r1 = c1.r
|
||||
|
||||
var x2 = c2.x
|
||||
var y2 = c2.y
|
||||
var r2 = c2.r
|
||||
|
||||
var x3 = c3.x
|
||||
var y3 = c3.y
|
||||
var r3 = c3.r
|
||||
|
||||
var v11 = 2 * x2 - 2 * x1
|
||||
var v12 = 2 * y2 - 2 * y1
|
||||
var v13 = x1 * x1 - x2 * x2 + y1 * y1 - y2 * y2 - r1 * r1 + r2 * r2
|
||||
var v14 = 2 * s2 * r2 - 2 * s1 * r1
|
||||
|
||||
var v21 = 2 * x3 - 2 * x2
|
||||
var v22 = 2 * y3 - 2 * y2
|
||||
var v23 = x2 * x2 - x3 * x3 + y2 * y2 - y3 * y3 - r2 * r2 + r3 * r3
|
||||
var v24 = 2 * s3 * r3 - 2 * s2 * r2
|
||||
|
||||
var w12 = v12 / v11
|
||||
var w13 = v13 / v11
|
||||
var w14 = v14 / v11
|
||||
|
||||
var w22 = v22 / v21 - w12
|
||||
var w23 = v23 / v21 - w13
|
||||
var w24 = v24 / v21 - w14
|
||||
|
||||
var p = -w23 / w22
|
||||
var q = w24 / w22
|
||||
var m = -w12 * p - w13
|
||||
var n = w14 - w12 * q
|
||||
|
||||
var a = n * n + q * q - 1
|
||||
var b = 2 * m * n - 2 * n * x1 + 2 * p * q - 2 * q * y1 + 2 * s1 * r1
|
||||
var c = x1 * x1 + m * m - 2 * m * x1 + p * p + y1 * y1 - 2 * p * y1 - r1 * r1
|
||||
|
||||
var d = b * b - 4 * a * c
|
||||
var rs = (-b - d.sqrt) / (2 * a)
|
||||
var xs = m + n * rs
|
||||
var ys = p + q * rs
|
||||
return Circle.new(xs, ys, rs)
|
||||
}
|
||||
|
||||
var c1 = Circle.new(0, 0, 1)
|
||||
var c2 = Circle.new(4, 0, 1)
|
||||
var c3 = Circle.new(2, 4, 2)
|
||||
System.print("Circle%(solveApollonius.call(c1, c2, c3, 1, 1, 1))")
|
||||
System.print("Circle%(solveApollonius.call(c1, c2, c3, -1, -1, -1))")
|
||||
65
Task/Problem-of-Apollonius/XPL0/problem-of-apollonius.xpl0
Normal file
65
Task/Problem-of-Apollonius/XPL0/problem-of-apollonius.xpl0
Normal file
|
|
@ -0,0 +1,65 @@
|
|||
real Circle(3);
|
||||
def \Circle\ X, Y, R;
|
||||
|
||||
proc real AP(C1, C2, C3, S);
|
||||
real C1, C2, C3; int S;
|
||||
real X1sq, Y1sq, R1sq,
|
||||
X2sq, Y2sq, R2sq,
|
||||
X3sq, Y3sq, R3sq,
|
||||
V11, V12, V13, V14,
|
||||
V21, V22, V23, V24,
|
||||
W12, W13, W14,
|
||||
W22, W23, W24,
|
||||
P, Q, M, N, A, B, C, D, RS;
|
||||
[
|
||||
X1sq:= C1(X) * C1(X);
|
||||
Y1sq:= C1(Y) * C1(Y);
|
||||
R1sq:= C1(R) * C1(R);
|
||||
X2sq:= C2(X) * C2(X);
|
||||
Y2sq:= C2(Y) * C2(Y);
|
||||
R2sq:= C2(R) * C2(R);
|
||||
X3sq:= C3(X) * C3(X);
|
||||
Y3sq:= C3(Y) * C3(Y);
|
||||
R3sq:= C3(R) * C3(R);
|
||||
V11:= 2. * (C2(X) - C1(X));
|
||||
V12:= 2. * (C2(Y) - C1(Y));
|
||||
V13:= X1sq - X2sq + Y1sq - Y2sq - R1sq + R2sq;
|
||||
V14:= 2. * (C2(R) - C1(R));
|
||||
V21:= 2. * (C3(X) - C2(X));
|
||||
V22:= 2. * (C3(Y) - C2(Y));
|
||||
V23:= X2sq - X3sq + Y2sq - Y3sq - R2sq + R3sq;
|
||||
V24:= 2. * (C3(R) - C2(R));
|
||||
if S then [V14:= -V14; V24:= -V24];
|
||||
W12:= V12 / V11;
|
||||
W13:= V13 / V11;
|
||||
W14:= V14 / V11;
|
||||
W22:= V22/V21 - W12;
|
||||
W23:= V23/V21 - W13;
|
||||
W24:= V24/V21 - W14;
|
||||
P:= -W23 / W22;
|
||||
Q:= W24 / W22;
|
||||
M:= -W12*P - W13;
|
||||
N:= W14 - W12*Q;
|
||||
A:= N*N + Q*Q - 1.;
|
||||
B:= M*N - N*C1(X) + P*Q - Q*C1(Y);
|
||||
if S then B:= B - C1(R)
|
||||
else B:= B + C1(R);
|
||||
B:= B * 2.;
|
||||
C:= X1sq + M*M - 2.*M*C1(X) + P*P + Y1sq - 2.*P*C1(Y) - R1sq;
|
||||
D:= B*B - 4.*A*C;
|
||||
RS:= (-B - sqrt(D)) / (2.*A);
|
||||
Circle(X):= M + N*RS;
|
||||
Circle(Y):= P + Q*RS;
|
||||
Circle(R):= RS;
|
||||
];
|
||||
|
||||
real C1, C2, C3;
|
||||
[
|
||||
C1:= [0., 0., 1.];
|
||||
C2:= [4., 0., 1.];
|
||||
C3:= [2., 4., 2.];
|
||||
AP(C1, C2, C3, true);
|
||||
RlOut(0, Circle(X)); RlOut(0, Circle(Y)); RlOut(0, Circle(R)); CrLf(0);
|
||||
AP(C1, C2, C3, false);
|
||||
RlOut(0, Circle(X)); RlOut(0, Circle(Y)); RlOut(0, Circle(R)); CrLf(0);
|
||||
]
|
||||
42
Task/Problem-of-Apollonius/Zkl/problem-of-apollonius-1.zkl
Normal file
42
Task/Problem-of-Apollonius/Zkl/problem-of-apollonius-1.zkl
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
class Circle{
|
||||
fcn init(xpos,ypos,radius){
|
||||
var [const] x=xpos.toFloat(), y=ypos.toFloat(),r=radius.toFloat();
|
||||
}
|
||||
fcn toString{ "Circle(%f,%f,%f)".fmt(x,y,r) }
|
||||
fcn apollonius(c2,c3,outside=True){
|
||||
s1:=s2:=s3:=outside and 1 or -1;
|
||||
|
||||
v11:=2.0*(c2.x - x);
|
||||
v12:=2.0*(c2.y - y);
|
||||
v13:=x.pow(2) - c2.x.pow(2) +
|
||||
y.pow(2) - c2.y.pow(2) -
|
||||
r.pow(2) + c2.r.pow(2);
|
||||
v14:=2.0*(s2*c2.r - s1*r);
|
||||
|
||||
v21:=2.0*(c3.x - c2.x);
|
||||
v22:=2.0*(c3.y - c2.y);
|
||||
v23:=c2.x.pow(2) - c3.x.pow(2) +
|
||||
c2.y.pow(2) - c3.y.pow(2) -
|
||||
c2.r.pow(2) + c3.r.pow(2);
|
||||
v24:=2.0*(s3*c3.r - s2*c2.r);
|
||||
|
||||
w12,w13,w14:=v12/v11, v13/v11, v14/v11;
|
||||
w22,w23,w24:=v22/v21 - w12, v23/v21 - w13, v24/v21 - w14;
|
||||
|
||||
P:=-w23/w22;
|
||||
Q:= w24/w22;
|
||||
M:=-w12*P - w13;
|
||||
N:= w14 - w12*Q;
|
||||
|
||||
a:=N*N + Q*Q - 1;
|
||||
b:=2.0*(M*N - N*x + P*Q - Q*y + s1*r);
|
||||
c:=x*x + M*M - 2.0*M*x + P*P + y*y - 2.0*P*y - r*r;
|
||||
|
||||
// find a root of a quadratic equation.
|
||||
// This requires the circle centers not to be e.g. colinear
|
||||
D:=b*b - 4.0*a*c;
|
||||
rs:=(-b - D.sqrt())/(2.0*a);
|
||||
|
||||
Circle(M + N*rs, P + Q*rs, rs);
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,3 @@
|
|||
a,b,c:=Circle(0,0,1), Circle(4,0,1), Circle(2,4,2);
|
||||
a.apollonius(b,c).println(" Outside");
|
||||
a.apollonius(b,c,False).println(" Inside");
|
||||
Loading…
Add table
Add a link
Reference in a new issue