Problem of Apollonius
Implement a solution to the 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).
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
There is an algebraic solution which is pretty straightforward.
The solutions to the example in the code are shown in the diagram (below and to the right).
The red circle is "internally tangent" to all three black circles, and the green circle is "externally tangent" to all three black circles.
Ada
apollonius.ads: <lang Ada>package Apollonius is
type Point is record
X, Y : Long_Float := 0.0;
end record;
type Circle is record
Center : Point;
Radius : Long_Float := 0.0;
end record;
type Tangentiality is (External, Internal);
function Solve_CCC
(Circle_1, Circle_2, Circle_3 : Circle;
T1, T2, T3 : Tangentiality := External)
return Circle;
end Apollonius;</lang> apollonius.adb: <lang Ada>with Ada.Numerics.Generic_Elementary_Functions;
package body Apollonius is
package Math is new Ada.Numerics.Generic_Elementary_Functions
(Long_Float);
function Solve_CCC
(Circle_1, Circle_2, Circle_3 : Circle;
T1, T2, T3 : Tangentiality := External)
return Circle
is
S1 : Long_Float := 1.0;
S2 : Long_Float := 1.0;
S3 : Long_Float := 1.0;
X1 : Long_Float renames Circle_1.Center.X;
Y1 : Long_Float renames Circle_1.Center.Y;
R1 : Long_Float renames Circle_1.Radius;
X2 : Long_Float renames Circle_2.Center.X;
Y2 : Long_Float renames Circle_2.Center.Y;
R2 : Long_Float renames Circle_2.Radius;
X3 : Long_Float renames Circle_3.Center.X;
Y3 : Long_Float renames Circle_3.Center.Y;
R3 : Long_Float renames Circle_3.Radius;
begin
if T1 = Internal then
S1 := -S1;
end if;
if T2 = Internal then
S2 := -S2;
end if;
if T3 = Internal then
S3 := -S3;
end if;
declare
V11 : constant Long_Float := 2.0 * X2 - 2.0 * X1;
V12 : constant Long_Float := 2.0 * Y2 - 2.0 * Y1;
V13 : constant Long_Float :=
X1 * X1 - X2 * X2 + Y1 * Y1 - Y2 * Y2 - R1 * R1 + R2 * R2;
V14 : constant Long_Float := 2.0 * S2 * R2 - 2.0 * S1 * R1;
V21 : constant Long_Float := 2.0 * X3 - 2.0 * X2;
V22 : constant Long_Float := 2.0 * Y3 - 2.0 * Y2;
V23 : constant Long_Float :=
X2 * X2 - X3 * X3 + Y2 * Y2 - Y3 * Y3 - R2 * R2 + R3 * R3;
V24 : constant Long_Float := 2.0 * S3 * R3 - 2.0 * S2 * R2;
W12 : constant Long_Float := V12 / V11;
W13 : constant Long_Float := V13 / V11;
W14 : constant Long_Float := V14 / V11;
W22 : constant Long_Float := V22 / V21 - W12;
W23 : constant Long_Float := V23 / V21 - W13;
W24 : constant Long_Float := V24 / V21 - W14;
P : constant Long_Float := -W23 / W22;
Q : constant Long_Float := W24 / W22;
M : constant Long_Float := -W12 * P - W13;
N : constant Long_Float := W14 - W12 * Q;
A : constant Long_Float := N * N + Q * Q - 1.0;
B : constant Long_Float :=
2.0 * M * N -
2.0 * N * X1 +
2.0 * P * Q -
2.0 * Q * Y1 +
2.0 * S1 * R1;
C : constant Long_Float :=
X1 * X1 +
M * M -
2.0 * M * X1 +
P * P +
Y1 * Y1 -
2.0 * P * Y1 -
R1 * R1;
D : constant Long_Float := B * B - 4.0 * A * C;
RS : constant Long_Float := (-B - Math.Sqrt (D)) / (2.0 * A);
begin
return (Center => (X => M + N * RS, Y => P + Q * RS), Radius => RS);
end;
end Solve_CCC;
end Apollonius;</lang>
example test_apollonius.adb: <lang Ada>with Ada.Text_IO; with Apollonius;
procedure Test_Apollonius is
use Apollonius; package Long_Float_IO is new Ada.Text_IO.Float_IO (Long_Float);
C1 : constant Circle := (Center => (X => 0.0, Y => 0.0), Radius => 1.0); C2 : constant Circle := (Center => (X => 4.0, Y => 0.0), Radius => 1.0); C3 : constant Circle := (Center => (X => 2.0, Y => 4.0), Radius => 2.0);
R1 : Circle := Solve_CCC (C1, C2, C3, External, External, External); R2 : Circle := Solve_CCC (C1, C2, C3, Internal, Internal, Internal);
begin
Ada.Text_IO.Put_Line ("R1:");
Long_Float_IO.Put (R1.Center.X, Aft => 3, Exp => 0);
Long_Float_IO.Put (R1.Center.Y, Aft => 3, Exp => 0);
Long_Float_IO.Put (R1.Radius, Aft => 3, Exp => 0);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line ("R2:");
Long_Float_IO.Put (R2.Center.X, Aft => 3, Exp => 0);
Long_Float_IO.Put (R2.Center.Y, Aft => 3, Exp => 0);
Long_Float_IO.Put (R2.Radius, Aft => 3, Exp => 0);
Ada.Text_IO.New_Line;
end Test_Apollonius;</lang>
output:
R1: 2.000 2.100 3.900 R2: 2.000 0.833 1.167
AutoHotkey
<lang AutoHotkey>#NoEnv
- SingleInstance, Force
SetBatchLines, -1
- Uncomment if Gdip.ahk is not in your standard library
- Include, Gdip.ahk
- Start gdi+
If !pToken := Gdip_Startup() { MsgBox, 48, gdiplus error!, Gdiplus failed to start. Please ensure you have gdiplus on your system ExitApp } OnExit, Exit
- 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
If (Gdip_LibraryVersion() < 1.30) { MsgBox, 48, version error!, Please download the latest version of the gdi+ library ExitApp } 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 , Apollonius(x1,y1,r1,x2,y2,r2,x3,y3,r3,s1,s2,s3,xs,ys,rs) , Width:=max(x1+r1 "," x2+r2 "," x3+r3 "," xs+rs)*1.1 , Height:=max(y1+r1 "," y2+r2 "," y3+r3 "," ys+rs)*1.1
Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs Gui, Show hwnd1 := WinExist() , hbm := CreateDIBSection(Width, Height) , hdc := CreateCompatibleDC() , obm := SelectObject(hdc, hbm) , G := Gdip_GraphicsFromHDC(hdc) , Gdip_SetSmoothingMode(G, 4) , bWhite := Gdip_BrushCreateSolid(0xffffffff) , Gdip_FillRectangle(G, bWhite, 0, 0, Width, Height) , pRed := Gdip_CreatePen(0x88ff0000, 3) , pGreen := Gdip_CreatePen(0x8800ff00, 3) , pBlue := Gdip_CreatePen(0x880000ff, 3) , pBlack := Gdip_CreatePen(0x88000000, 3) , Gdip_DrawCircle(G, pRed, x1, y1, r1) , Gdip_DrawCircle(G, pGreen, x2, y2, r2) , Gdip_DrawCircle(G, pBlue, x3, y3, r3) , Gdip_DrawCircle(G, pBlack, xs, ys, rs) , Gdip_DeletePen(pRed) , Gdip_DeletePen(pGreen) , Gdip_DeletePen(pBlue) , Gdip_DeletePen(pBlack) , UpdateLayeredWindow(hwnd1, hdc, 0, 0, Width, Height) Return
GuiEscape: GuiClose: Exit: SelectObject(hdc, obm) , DeleteObject(hbm) , DeleteDC(hdc) , Gdip_DeleteGraphics(G) , Gdip_Shutdown(pToken) ExitApp
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) { 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 - d**0.5)/(2*a) xs := m + n*rs ys := p + q*rs }
max(list) { Loop Parse, list, `, x := x < A_LoopField ? A_LoopField : x Return x }
- Gdip helper function
Gdip_DrawCircle(G, pPen, x, y, r) { Return Gdip_DrawEllipse(G, pPen, x-r, y-r, r*2, r*2) }</lang>
BBC BASIC
Note use made of array arithmetic. <lang bbcbasic> 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</lang>
Output:
Internal: x = 2.000, y = 0.833, r = 1.167 External: x = 2.000, y = 2.100, r = 3.900
C
C99. 2D vectors are actually complex numbers. The method here is unothordox if not insane. I can't prove that it should work, though it does seem to give correct answers for test cases I tried. <lang c>#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); }</lang>
C#
This code finds all 8 possible circles touching the three given circles. <lang csharp> 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;
}
}
} </lang>
CoffeeScript
<lang coffeescript> 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)
</lang> output <lang> > 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 } </lang>
D
<lang d>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;
}</lang>
- Output:
immutable(Circle)(2, 2.1, 3.9) immutable(Circle)(2, 0.833333, 1.16667)
Elixir
<lang elixir>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)</lang>
- Output:
{2.0, 2.1, 3.9}
{2.0, 0.8333333333333333, 1.1666666666666667}
F#
<lang fsharp>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</lang>
- Output:
Circle(x=2.00, y=2.10, r=3.90) Circle(x=2.00, y=0.83, r=1.17)
Fortran
<lang fortran>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</lang> Output
External tangent: 2.00000000 2.10000000 3.90000000 Internal tangent: 2.00000000 0.83333333 1.16666667
FutureBasic
<lang futurebasic>Probleme of Apollonius include "NSLog.incl"
begin record Circle CGPoint center double radius CFStringRef locator end record
local fn CircleToString( c as Circle ) as CFStringRef '~'1 CFStringRef result
result = fn StringWithFormat( @"%@ Circle( x = %0.3f, y = %0.3f, radius = %0.3f )", c.locator, c.center.x, c.center.y, c.radius ) end fn = result
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 </lang> Output
External Tangent: Circle( x = 2.000, y = 2.100, radius = 3.900 ) Internal Tangent: Circle( x = 2.000, y = 0.833, radius = 1.167 )
Go
Simplified to produce only the fully interior and fully exterior solutions. <lang go>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}
}</lang> Output:
{2 0.8333333333333333 1.1666666666666667}
{2 2.1 3.9}
Haskell
<lang haskell>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</lang>
- Output:
Circle {x = 2.0, y = 2.1, r = 3.9}
Circle {x = 2.0, y = 0.8333333333333333, r = 1.1666666666666667}
Icon and Unicon
This is a translation of the Java version.
<lang Icon>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</lang>
Output:
Circle(x,y,r) := (0, 0, 1) Circle(x,y,r) := (4, 0, 1) Circle(x,y,r) := (2, 4, 2) Circle(x,y,r) := (2.0, 2.1, 3.9) Circle(x,y,r) := (2.0, 0.8333333333333333, 1.166666666666667)
J
Solution <lang j>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
)</lang>
Usage <lang j> ]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 </lang>
Java
<lang Java>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));
}
}</lang>
jq
<lang jq>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
- </lang>
The task: <lang jq>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; $c3; false) ) # exterior
- </lang>
- Output:
<lang sh>$ jq -n -c -f apollonius.jq {"x":2,"y":0.8333333333333333,"r":1.1666666666666667} {"x":2,"y":2.1,"r":3.9}</lang>
Julia
This solution follows the algebraic solution from Weisstein, Eric W. "Apollonius' Problem." From MathWorld--A Wolfram Web Resource. The Polynomials package is used to solve the quadratic equation for the radius (equation 1 in the reference) rather than hard coding it.
The enc array passed to the apollonius function, specifies which of the three defining circles are to be enclosed in the solution. For this task only the "internal" (enc=[]) and "external" (enc=[1:3]) are called for.
Module: <lang julia>using Printf
module ApolloniusProblems
using Polynomials 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 = Poly([-det([b d]), det([b c])] ./ det([a b]))
v = Poly([det([a d]), -det([a c])] ./ det([a b]))
w = Poly([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(polyval(u, r), polyval(v, r), r)
end
end # module ApolloniusProblem</lang>
Main: <lang julia>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</lang>
- Output:
The defining circles are:
- centered at (0.0000, 0.0000) with radius 1.0000
- centered at (4.0000, 0.0000) with radius 1.0000
- centered at (2.0000, 4.0000) with radius 2.0000
The internal circle is:
centered at (2.0000, 0.8333) with radius 1.1667
The external circle is:
centered at (2.0000, 2.1000) with radius 3.9000
Kotlin
<lang scala>// 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))
}</lang>
- Output:
Circle(x=2.0, y=2.1, r=3.9) Circle(x=2.0, y=0.8333333333333333, r=1.1666666666666667)
Lasso
<lang Lasso>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) </lang>
- Output:
staticarray(2.000000, 2.100000, 3.900000) staticarray(2.000000, 0.833333, 1.166667)
Liberty BASIC
Uses the string Circle$ to hold "xPos, yPos, radius" as csv data. A GUI representation is very easily added. <lang lb>
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 </lang>
x_pos y_pos radius 0.000, 0.000, 1.000 4.000, 0.000, 1.000 2.000, 4.000, 2.000
2.000, 2.100, 3.900 2.000, 0.833, 1.167
Mathematica
<lang Mathematica>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} ]]</lang>
Apolonius[0,0,1,2,4,2,4,0,1,1,1,1]
->{2.,2.1,3.9}
Apolonius[0,0,1,2,4,2,4,0,1,-1,-1,-1]
->{2.,0.833333,1.16667}
MUMPS
<lang MUMPS>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</lang>
In use:
USER>WRITE C1 0^0^1 USER>WRITE C2 4^0^1 USER>WRITE C3 2^4^2 USER>WRITE $$APOLLONIUS^ROSETTA(C1,C2,C3,1,1,1) 2^2.1^3.9 USER>WRITE $$APOLLONIUS^ROSETTA(C1,C2,C3,-1,-1,-1) 2^.833333333333333333^1.166666666666666667
Nim
<lang nim>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)</lang> Output:
(x: 2.0, y: 2.1, r: 3.9) (x: 2.0, y: 0.8333333333333333, r: 1.166666666666667)
OCaml
<lang ocaml>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;
- </lang>
Perl
Using the module Math::Cartesian::Product to generate the values to allow iteration through all solutions.
<lang perl>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, $c3, @$_);
}</lang>
- Output:
x = 2.000 y = 0.833 r = 1.167 x = 2.000 y = 3.214 r = 2.786 x = 3.002 y = 0.123 r = 2.005 x = 4.127 y = 3.252 r = 4.255 x = 0.998 y = 0.123 r = 2.005 x = -0.127 y = 3.252 r = 4.255 x = 2.000 y = -1.500 r = 3.500 x = 2.000 y = 2.100 r = 3.900
Phix
<lang Phix>function Apollonius(sequence calc, circles)
integer {s1,s2,s3} = calc
atom {x1,y1,r1} = circles[1],
{x2,y2,r2} = circles[2],
{x3,y3,r3} = circles[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 = -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-sqrt(d)) / (2*a),
xs = M + N*rs,
ys = P + Q*rs
return {xs,ys,rs}
end function
constant circles = {{0,0,1},
{4,0,1},
{2,4,2}}
-- +1: externally tangental, -1: internally tangental constant calcs = {{+1,+1,+1},
{-1,-1,-1},
{+1,-1,-1},
{-1,+1,-1},
{-1,-1,+1},
{+1,+1,-1},
{-1,+1,+1},
{+1,-1,+1}}
for i=1 to 8 do
atom {xs,ys,rs} = Apollonius(calcs[i],circles)
string th = {"st (external)","nd (internal)","rd","th"}[min(i,4)]
printf(1,"%d%s solution: x=%+f, y=%+f, r=%f\n",{i,th,xs,ys,rs})
end for</lang>
- Output:
1st (external) solution: x=+2.000000, y=+2.100000, r=3.900000 2nd (internal) solution: x=+2.000000, y=+0.833333, r=1.166667 3rd solution: x=+0.997502, y=+0.122502, r=2.004996 4th solution: x=+3.002498, y=+0.122502, r=2.004996 5th solution: x=+2.000000, y=+3.214286, r=2.785714 6th solution: x=+2.000000, y=-1.500000, r=3.500000 7th solution: x=+4.127498, y=+3.252498, r=4.254996 8th solution: x=-0.127498, y=+3.252498, r=4.254996
PL/I
<lang PL/I>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;</lang> Results:
External tangent: 2.00000000 2.10000000 3.90000000 Internal tangent: 2.00000000 0.83333333 1.16666667
PowerShell
<lang PowerShell> 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 = $rs
}
} </lang> <lang PowerShell> for ($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
} </lang>
- Output:
X Y Radius
- - ------
2 2.1 3.9
2 0.833333333333333 1.16666666666667
0.997501996806385 0.122501996806385 2.00499600638723
3.00249800319362 0.122501996806385 2.00499600638723
2 3.21428571428571 2.78571428571429
2 -1.5 3.5
4.12749800319362 3.25249800319362 4.25499600638723
-0.127498003193615 3.25249800319362 4.25499600638723
PureBasic
<lang PureBasic>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</lang>
Circle [x=2.00, y=2.10, r=3.90] Circle [x=2.00, y=0.83, r=1.17] Press ENTER to exit
Python
. Although a Circle class is defined, the solveApollonius function is defined in such a way that any three valued tuple or list could be used instead of c1, c2, and c3. The function calls near the end use instances of the Circle class, whereas the docstring shows how the same can be achieved using simple tuples. (And also serves as a simple doctest)
<lang python> 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)</lang>
Sample Output
Circle(x=2.0, y=2.1, r=3.9) Circle(x=2.0, y=0.8333333333333333, r=1.1666666666666667)
Racket
<lang Racket>
- 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))
</lang>
Raku
(formerly Perl 6) This program is written mostly in the "sigilless" style for several reasons. First, sigils tend to imply variables, and these sigilless symbols are not variables, but readonly bindings to values that are calculated only once, so leaving off the sigil emphasizes the fact that they are not variables, but merely named intermediate results.
Second, it looks more like the original mathematical formulas to do it this way.
Third, together with the use of Unicode, we are emphasizing the social contract between the writer and the reader, which has a clause in it that indicates code is read much more often than it is written, therefore the writer is obligated to undergo vicarious suffering on behalf of the reader to make things clear. If the reader doesn't understand, it's the writer's fault, in other words. Or in other other words, figure out how to type those Unicode characters, even if it's hard. And you should type them whenever it makes things clearer to the reader.
Finally, writing in an SSA style tends to help the optimizer.
<lang perl6>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;
}</lang>
- Output:
x = 2.000 y = 0.833 r = 1.167 x = 2.000 y = 3.214 r = 2.786 x = 3.002 y = 0.123 r = 2.005 x = 4.127 y = 3.252 r = 4.255 x = 0.998 y = 0.123 r = 2.005 x = -0.127 y = 3.252 r = 4.255 x = 2.000 y = -1.500 r = 3.500 x = 2.000 y = 2.100 r = 3.900
REXX
Programming note: REXX has no sqrt (square root) function, so a RYO version is included here. <lang rexx>/*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</lang> Programming note: in REXX, dividing by unity normalizes the number.
- output when using the default input:
external tangent: 2 2.1 3.9 internal tangent: 2 0.833333333333333 1.16666666666667
Ruby
<lang ruby>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)</lang>
- Output:
Circle: x=0.0, y=0.0, r=1.0 Circle: x=2.0, y=4.0, r=2.0 Circle: x=4.0, y=0.0, r=1.0 Circle: x=2.0, y=2.1, r=3.9 Circle: x=2.0, y=0.8333333333333333, r=1.1666666666666667
Scala
<lang scala>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))
}
}
}</lang> Output:
c1: Circle(0.0,0.0,1.0) c2: Circle(4.0,0.0,1.0) c3: Circle(2.0,4.0,2.0) red circle: tangents=(intern,intern,intern) cs=Circle(2.0,0.8333333333333333,1.1666666666666667) green circle: tangents=(extern,extern,extern) cs=Circle(2.0,2.1,3.9) all combinations: tangents: (intern,intern,intern) -> cs=Circle(x= 2,00000, y= 0,83333, r= 1,16667) tangents: (intern,intern,extern) -> cs=Circle(x= 2,00000, y= 3,21429, r= 2,78571) tangents: (intern,extern,intern) -> cs=Circle(x= 3,00250, y= 0,12250, r= 2,00500) tangents: (intern,extern,extern) -> cs=Circle(x= 4,12750, y= 3,25250, r= 4,25500) tangents: (extern,intern,intern) -> cs=Circle(x= 0,99750, y= 0,12250, r= 2,00500) tangents: (extern,intern,extern) -> cs=Circle(x=-0,12750, y= 3,25250, r= 4,25500) tangents: (extern,extern,intern) -> cs=Circle(x= 2,00000, y=-1,50000, r= 3,50000) tangents: (extern,extern,extern) -> cs=Circle(x= 2,00000, y= 2,10000, r= 3,90000)
Sidef
<lang ruby>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>);</lang>
- Output:
Circle(2, 2.1, 3.9) Circle(2, 0.83333333333333333333333333333333333333325, 1.166666666666666666666666666666666666667)
Swift
<lang Swift>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())</lang>
- Output:
Circle[x=2.0,y=2.1,r=3.9] Circle[x=2.0,y=0.833333333333333,r=1.16666666666667]
Tcl
or
<lang tcl>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]
}</lang> Demonstration code: <lang tcl>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]</lang> Output:
Circle[o=(2.00,2.10),r=3.90] Circle[o=(2.00,0.83),r=1.17]
Note that the Tcl code uses the ** (exponentiation) operator to shorten and simplify some operations, and that the circle class is forcing the interpretation of every circle's coordinates as double-precision floating-point numbers.
VBA
<lang VBA/VBasic 6.0> 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
</lang>
Wren
<lang ecmascript>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))")</lang>
- Output:
Circle(2, 2.1, 3.9) Circle(2, 0.83333333333333, 1.1666666666667)
zkl
<lang zkl>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); }
}</lang> <lang zkl>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");</lang>
- Output:
Circle(2.000000,2.100000,3.900000) Outside Circle(2.000000,0.833333,1.166667) Inside