public static Circle2d[] CircleCircleCircle(Circle2d ci1, Circle2d ci2, Circle2d ci3)
{
// see http://www.arcenciel.co.uk/geometry/ for explanation
List <Circle2d> result = null;
double r1, r2, r3, a, b, c, t, A, B, C;
double fRadius, xc, yc, distc1c2;
double[] roots;
// if all circles concentric, there are no solutions
distc1c2 = ci1.Center.Distance(ci2.Center);
if (MathUtil.IsZero(distc1c2) && MathUtil.IsZero(ci2.Center.Distance(ci3.Center)))
{
return(null);
}
// make sure first 2 circles are not concentric
// if so swap ci2,ci3
if (MathUtil.IsZero(distc1c2))
{
var tmp = ci2; ci2 = ci3; ci3 = ci2;
}
// transform input so that ci1 is at origo and ci2 is on xaxis
Transform2d trans = Transform2d.Translate(Point2d.Origo - ci1.Center) * Transform2d.Rotate(-ci1.Center.Angle(ci2.Center));
ci1 = new Circle2d(ci1);
ci1.Transform(trans);
ci2 = new Circle2d(ci2);
ci2.Transform(trans);
ci3 = new Circle2d(ci3);
ci3.Transform(trans);
// Negate the radii to get all combinations
for (int iCase = 0; iCase < 8; ++iCase)
{
r1 = ((iCase & 1) == 0) ? ci1.Radius : -ci1.Radius;
r2 = ((iCase & 2) == 0) ? ci2.Radius : -ci2.Radius;
r3 = ((iCase & 4) == 0) ? ci3.Radius : -ci3.Radius;
// special case where radii of first 2 circles are equal
if (MathUtil.Equals(r1, r2))
{
// Calculate x-cordinate of centre
xc = ci2.X / 2.0;
// if all radii are equal, there will be only one solution
if (MathUtil.Equals(r1, r3))
{
if (MathUtil.IsZero(ci3.Y))
{
continue;
}
// get y-coordinate of centre
yc = (ci3.X * ci3.X - 2.0 * xc * ci3.X + ci3.Y * ci3.Y) / (ci3.Y + ci3.Y);
// compute radius
A = 1;
B = 2 * r1;
C = r1 * r1 - xc * xc - yc * yc;
roots = RealPolynomial.SolveQuadric(A, B, C);
if (roots.Length > 0)
{
fRadius = roots[0];
if (fRadius <= 0.0)
{ //then try other root
if (roots.Length > 1)
{
fRadius = roots[1];
if (fRadius <= 0.0)
{
continue; //no posetive roots
}
}
}
AddResult(ref result, xc, yc, fRadius);
}
}
else
{
// compute constants
double k = r1 * r1 - r3 * r3 + ci3.X * ci3.X + ci3.Y * ci3.Y - 2 * xc * ci3.X;
A = 4 * ((r1 - r3) * (r1 - r3) - ci3.Y * ci3.Y);
B = 4 * (k * (r1 - r3) - 2 * ci3.Y * ci3.Y * r1);
C = 4 * xc * xc * ci3.Y * ci3.Y + k * k - 4 * ci3.Y * ci3.Y * r1 * r1;
if (!MathUtil.IsZero(A))
{
roots = RealPolynomial.SolveQuadric(A, B, C);
foreach (double radius in roots)
{
yc = (2 * radius * (r1 - r3) + k) / (2 * ci3.Y);
AddResult(ref result, xc, yc, radius);
}
}
}
continue;
} //end special case of r1==r2
// Get constants
a = 2 * (ci2.X * (r3 - r1) - ci3.X * (r2 - r1));
b = 2 * ci3.Y * (r1 - r2);
c = (r2 - r1) * (ci3.X * ci3.X + ci3.Y * ci3.Y - (r3 - r1) * (r3 - r1)) - (r3 - r1) * (ci2.X * ci2.X - (r2 - r1) * (r2 - r1));
t = (ci2.X * ci2.X + r1 * r1 - r2 * r2) / 2.0;
A = (r1 - r2) * (r1 - r2) * (a * a + b * b) - (ci2.X * ci2.X * b * b);
B = 2 * (t * (r1 - r2) * (a * a + b * b) + a * c * ci2.X * (r1 - r2) - (r1 * ci2.X * ci2.X * b * b));
C = t * t * (a * a + b * b) + (2 * a * c * ci2.X * t) + (c * c * ci2.X * ci2.X) - (r1 * r1 * ci2.X * ci2.X * b * b);
// Calculate radius
roots = RealPolynomial.SolveQuadric(A, B, C);
if (roots == null)
{
continue;
}
foreach (double radius in roots)
{
if (radius < minradius || radius > maxradius)
{
continue;
}
// get x coordinate of centre (x2 may not be zero)
xc = (radius * (r1 - r2) + t) / ci2.X;
// get y coordinate of centre. b should never be 0, as
// r1=r2 is special case and y3 may not be zero
yc = (-a * xc - c) / b;
AddResult(ref result, xc, yc, radius);
}
}
//convert back to original coordinate system by using the inverse
//of the original matrix
if (result != null)
{
trans = trans.Inversed;
for (int l = 0; l < result.Count; l++)
{
result[l].Transform(trans);
}
return(result.ToArray());
}
return(null);
}
public static Circle2d[] CircleCircleLine(Circle2d ci1, Circle2d ci2, Line2d li)
{
// see http://www.arcenciel.co.uk/geometry/ for explanation
List <Circle2d> result = null;
double a1, b1, c1, t, r2, r3, a, b, c, u, s;
double A, B, C, xc, yc;
//transform input so that c1 is at origo and c2 is on xaxis
Transform2d trans = Transform2d.Translate(Point2d.Origo - ci1.Center) * Transform2d.Rotate(-ci1.Center.Angle(ci2.Center));
ci1 = new Circle2d(ci1);
ci1.Transform(trans);
ci2 = new Circle2d(ci2);
ci2.Transform(trans);
li = new Line2d(li);
li.Transform(trans);
if (!li.ToEquation(out a1, out b1, out c1))
{
return(null); //degenerate line
}
for (int signcase = 0; signcase < 8; ++signcase)
{
t = ((signcase & 1) == 0) ? 1 : -1;
r2 = ((signcase & 2) == 0) ? ci1.Radius : -ci1.Radius;
r3 = ((signcase & 4) == 0) ? ci2.Radius : -ci2.Radius;
// Get constants
a = 2 * (a1 * (r2 - r3) - ci2.X * t);
b = 2 * b1 * (r2 - r3);
c = 2 * c1 * (r2 - r3) + t * (r2 * r2 - r3 * r3 + ci2.X * ci2.X);
if (!MathUtil.IsZero(b))
{
u = b1 * c - b * c1;
s = a1 * b - a * b1;
A = t * t * b * b * (a * a + b * b) - b * b * s * s;
B = 2 * (u * t * b * (a * a + b * b) + a * c * s * t * b - b * b * s * s * r2);
C = u * u * (a * a + b * b) + 2 * a * c * s * u + c * c * s * s - b * b * s * s * r2 * r2;
}
else
{
u = a1 * c - a * c1;
s = a * b1;
A = a * a * (t * t * a * a - s * s);
B = 2 * a * a * (u * t * a - s * s * r2);
C = u * u * a * a + c * c * s * s - a * a * s * s * r2 * r2;
}
// Calculate radius
double[] roots = RealPolynomial.SolveQuadric(A, B, C);
if (roots != null)
{
foreach (double radius in roots)
{
if (radius < minradius || radius > maxradius)
{
continue;
}
// compute x coordinates of centers
List <double> xsols = new List <double>();
if (!MathUtil.IsZero(ci2.X)) //circles are not concentric
{
xc = ((r2 + radius) * (r2 + radius) - (r3 + radius) * (r3 + radius) + ci2.X * ci2.X) / (2 * ci2.X);
xsols.Add(xc);
}
else // If circles are concentric there can be 2 solutions for x
{
A = (a1 * a1 + b1 * b1);
B = -2 * a1 * (radius * t - c1);
C = (radius * t - c1) * (radius * t - c1) - b1 * b1 * (r2 + radius) * (r2 + radius);
double[] roots2 = RealPolynomial.SolveQuadric(A, B, C);
if (roots2 != null)
{
foreach (double x in roots2)
{
xsols.Add(x);
}
}
}
// now compute y coordinates from the calculated x:es
// and input the final solution
foreach (double x in xsols)
{
if (!MathUtil.IsZero(b1))
{
yc = (-a1 * x - c1 + radius * t) / b1;
}
else
{
double ycSquare = (r2 + radius) * (r2 + radius) - (x * x);
if (ycSquare < 0.0)
{
continue;
}
yc = Math.Sqrt(ycSquare);
}
AddResult(ref result, x, yc, radius);
if (MathUtil.IsZero(b1))
{
AddResult(ref result, x, -yc, radius);
}
}
}
}
}
//convert back to original coordinate system by using the inverse
//of the original matrix
if (result != null)
{
trans = trans.Inversed;
for (int l = 0; l < result.Count; l++)
{
result[l].Transform(trans);
}
return(result.ToArray());
}
return(null);
}
public static Ellipse2d FromCircle(Circle2d c)
{
return(new Ellipse2d(c.Center, c.Radius, c.Radius));
}
public static Circle2d[] CircleLineLine(Circle2d ci, Line2d l1, Line2d l2)
{
// see http://www.arcenciel.co.uk/geometry/ for explanation
List <Circle2d> result = null;
//translate everyting so circle center at origo
double dx = ci.X, dy = ci.Y;
ci = new Circle2d(0, 0, ci.Radius);
l1 = new Line2d(l1.X1 - dx, l1.Y1 - dy, l1.X2 - dx, l1.Y2 - dy);
l2 = new Line2d(l2.X1 - dx, l2.Y1 - dy, l2.X2 - dx, l2.Y2 - dy);
//if first line vertical, swap lines...
if (MathUtil.Equals(l1.X1, l1.X2))
{
var tmp = l1; l1 = l2; l2 = tmp;
}
//if first line still vertical, special case:
if (MathUtil.Equals(l1.X1, l1.X2))
{
double rad = (l1.X1 - l2.X1) / 2.0;
double xcenter = (l1.X1 + l2.X1) / 2.0;
double yc = Math.Sqrt((rad + ci.Radius) * (rad + ci.Radius) - xcenter * xcenter);
AddResult(ref result, xcenter, ci.Y + yc, rad);
AddResult(ref result, xcenter, ci.Y - yc, rad);
}
else
{
//now we know that first line is not vertical, and circle is centered at origo
double a1, b1, c1, a2, b2, c2, u, w, s, a, b, c, xcenter, ycenter, t1, t2, r3;
if (!l1.ToEquation(out a1, out b1, out c1))
{
return(null);
}
if (!l2.ToEquation(out a2, out b2, out c2))
{
return(null);
}
for (int signcase = 0; signcase < 8; signcase++)
{
t1 = ((signcase & 1) == 0) ? -1 : 1;
t2 = ((signcase & 2) == 0) ? -1 : 1;
r3 = ((signcase & 4) == 0) ? -ci.Radius : ci.Radius;
u = (t1 * b2) - (t2 * b1);
w = (b1 * c2) - (b2 * c1);
s = (a1 * b2) - (a2 * b1);
a = (u * u) - (2 * a1 * s * u * t1) + (t1 * t1 * s * s) - (b1 * b1 * s * s);
b = 2.0 * ((u * w) + (c1 * a1 * s * u) - (a1 * s * t1 * w) - (c1 * t1 * s * s) - (r3 * b1 * b1 * s * s));
c = (w * w) + (2 * a1 * s * c1 * w) + (c1 * c1 * s * s) - (b1 * b1 * r3 * r3 * s * s);
double[] roots = RealPolynomial.SolveQuadric(a, b, c);
if (roots != null)
{
foreach (double radius in roots)
{
if (radius < minradius || radius > maxradius)
{
continue;
}
if (!MathUtil.IsZero(s))
{ //non parallel lines, one center per root
xcenter = (radius * u + w) / s;
ycenter = ((-a1 * xcenter) - c1 + (radius * t1)) / b1;
AddResult(ref result, xcenter, ycenter, radius);
}
else //parallel lines, two centers per root
{
a = t1 * t1;
b = 2.0 * a1 * (c1 - (radius * t1));
c = ((radius * t1) - c1) * ((radius * t1) - c1) - (b1 * b2 * (r3 + radius) * (r3 + radius));
double[] roots2 = RealPolynomial.SolveQuadric(a, b, c);
if (roots2 != null)
{
foreach (double x in roots2)
{
ycenter = (-a1 * x - c1 + radius * t1) / b1;
AddResult(ref result, x, ycenter, radius);
}
}
}
}
}
}
}
//translate results back to original position
if (result != null)
{
foreach (Circle2d c in result)
{
c.X += dx;
c.Y += dy;
}
return(result.ToArray());
}
return(null);
}
public Circle2d(Circle2d tocopy)
{
center = tocopy.center;
radius = tocopy.radius;
}