
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.




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}
}