
Task

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).

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.




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));
 }
}