private static Double2[] AdjustCircles(ref Double2 c1, ref double r1, Double2 n1, ref Double2 c2, ref double r2, Double2 n2)
{
Double2 pt1, pt2;
// Negate if necessary, so we are working with positive values of the radius
bool neg1 = r1 < 0, neg2 = r2 < 0;
if (neg1)
{
r1 = -r1; n1 = -n1;
}
if (neg2)
{
r2 = -r2; n2 = -n2;
}
// There are two cases: if the circles are outside (their radius sum is less than their distance)
// and if the circles are inside (their radius difference is less than their distance)
var rs = r1 + r2;
// The first case
if (c1.DistanceSquaredTo(c2) > rs * rs)
{
// Compute the coefficients of the polynomials
var p = n1.DistanceSquaredTo(n2);
var q = 2 * (c1 - c2).Dot(n1 - n2);
var a = c1.DistanceSquaredTo(c2);
// There are two polyomials that, solved, give the necessary displacement for this adjustment
// One of them is pλ² + qλ + a² - (r1-r2)² and the other is (p-4)λ² + (q - 4(r1+r2))λ + a - (r1+r2)²
// Experimentally, though, only the latter one gives a "useful" value
// We still have an edge case, though, as there might be no positive root to this (this happens if
// the circles have to grow at opposite direction). On this case, return an empty array
var roots = Equations.SolveQuadratic(p - 4, q - 4 * rs, a - rs * rs);
if (!roots.Any(r => r >= 0))
{
pt1 = pt2 = new Double2(double.NaN, double.NaN);
}
else
{
var l = roots.Where(r => r >= 0).Min();
// Correctly adjust the centers and radii
c1 += n1 * l;
c2 += n2 * l;
r1 += l;
r2 += l;
// The single point as a double root
var dir = (c2 - c1).Normalized;
pt1 = c1 + dir * r1;
pt2 = c2 - dir * r2;
}
}
// The second case
else
{
// Swap the circles if the first one is smaller
bool swap = r1 < r2;
if (swap)
{
Swap(ref c1, ref c2);
Swap(ref r1, ref r2);
Swap(ref n1, ref n2);
}
// Compute the coefficients of the polynomials
var p = (n1 + n2).LengthSquared;
var q = 2 * (c1 - c2).Dot(n1 + n2);
var a = c1.DistanceSquaredTo(c2);
// Again, here, there are two polynomials: pλ² - qλ + a² - (r1+r2)² and (p-4)λ² + (-q + 4(r1-r2))λ + a - (r1-r2)²
// And again, experimentally only the second one will give the "correct" result
var rd = r1 - r2;
var roots = Equations.SolveQuadratic(p - 4, -q + 4 * rd, a - rd * rd);
var l = roots.Where(r => r >= 0).Min();
// Correctly adjust the centers and radii
c1 -= n1 * l;
c2 += n2 * l;
r1 -= l;
r2 += l;
// Pick the single point intersection
var dir = (c2 - c1).Normalized;
pt1 = c1 + dir * r1;
pt2 = c2 + dir * r2;
// Finally, swap them again if we had to swap earlier
if (swap)
{
Swap(ref c1, ref c2);
Swap(ref r1, ref r2);
}
}
// Return back their signs
if (neg1)
{
r1 = -r1;
}
if (neg2)
{
r2 = -r2;
}
if (double.IsNaN(pt1.X))
{
return(new Double2[0]);
}
return(new[] { (pt1 + pt2) / 2, (pt1 + pt2) / 2 });
}