public static Point2d[] HyperbolaHyperbola(Hyperbola2d hyp1, Hyperbola2d hyp2)
{
Transform2d tr = hyp1.ToStandardPosition;
hyp2 = new Hyperbola2d(hyp2); //copy for modification
hyp2.Transform(tr);
Point2dSet res = StdHyperbolaConic(hyp1.Ratio, hyp2);
res.InverseTransform(tr);
return(res.ToArray());
}
/// <summary>
/// Intersects a general conic with the curve y=x^2, returning a point set (possibly empty).
/// This function is used as a helper function for parabola dominant intersection functions.
/// </summary>
private static Point2dSet StandardPosParabolaGeneralConic(GeneralConic2d con) //tested ok
//c*x^4+b*x^3+(e+a)*x^2+d*x+f
{
Point2dSet res = new Point2dSet();
double[] xs = GetRealRoots(con.C, con.B, con.E + con.A, con.D, con.F);
if (xs == null)
{
return(res); //no solutions
}
foreach (double x in xs)
{
double tx = x;
double ty = x * x;
res.Add(new Point2d(tx, ty));
}
return(res);
}
public static Point2d[] HyperbolaEllipse(Hyperbola2d hyp, Ellipse2d elp)
{
//TODO: this is probably more stable intersecting hyperbola with unitcircle. Rewrite.
Transform2d tr = hyp.ToStandardPosition;
hyp = new Hyperbola2d(hyp);
elp = new Ellipse2d(elp);
hyp.Transform(tr);
elp.Transform(tr);
GeneralConic2d hcon = new GeneralConic2d(1, 0.0, -1 / (hyp.B * hyp.B), 0.0, 0.0, -1);
Point2dSet pset = new Point2dSet();
pset.AddRange(ConicConic(hcon, elp.ToGeneralConic()));
pset.Transform(tr.Inversed);
return(pset.ToArray());
}
public static Point2d[] EllipseEllipse2(Ellipse2d elp1, Ellipse2d elp2)
{
//TODO: check if this is better than EllipseEllipse in stabillity and replace it or remove this function
Transform2d tr = elp1.ToStandardPosition;
elp2 = new Ellipse2d(elp2); //dont alter the original ellipse
elp2.Transform(tr);
elp1 = new Ellipse2d(elp1);
elp1.Transform(tr);
GeneralConic2d con1 = new GeneralConic2d(1.0, 0.0, 1 / (elp1.Ratio * elp1.Ratio), 0.0, 0.0, -1.0);
GeneralConic2d con2 = elp2.ToGeneralConic(); // GeneralConic2d.FromEllipse(elp2);
Point2dSet pset = new Point2dSet();
pset.AddRange(ConicConic(con1, con2));
pset.Transform(tr.Inversed);
return(pset.ToArray());
}
/// <summary>
/// Intersect a hyperbola in standard position (a=1.0) with a general conic.
/// Helper function for other hyperbola intersection functions.
/// </summary>
/// <param name="hyp1_b"></param>
/// <param name="hyp2"></param>
private static Point2dSet StdHyperbolaConic(double b1, Conic2d con)
{
double R = 1.0 / (b1 * b1);
Point2dSet res = new Point2dSet();
GeneralConic2d gc = con.ToGeneralConic();
double A = gc.A, B = gc.B, C = gc.C, D = gc.D, E = gc.E, F = gc.F;
double y4 = A * A * R * R + (2 * A * C - B * B) * R + C * C;
double y3 = (2 * A * E - 2 * B * D) * R + 2 * C * E;
double y2 = (-D * D + 2 * A * A + 2 * F * A) * R + E * E + (2 * A + 2 * F) * C - B * B;
double y1 = (2 * A + 2 * F) * E - 2 * B * D;
double y0 = A * A + 2 * F * A + F * F - D * D;
double[] ys = GetRealRoots(y4, y3, y2, y1, y0);
if (ys == null)
{
return(null);
}
foreach (double y in ys)
{
//two possible x for each y. Try to find the correct one:
double x = Math.Sqrt(y * y * R + 1);
double nx = -x;
double err = A * x * x + B * x * y + C * y * y + D * x + E * y + F;
double nerr = A * nx * nx + B * nx * y + C * y * y + D * nx + E * y + F;
if (Math.Abs(err) < Math.Abs(nerr))
{
res.Add(new Point2d(x, y));
}
else
{
res.Add(new Point2d(nx, y));
}
}
return(res);
}
public static Point2d[] ConicConic(GeneralConic2d tcon1, GeneralConic2d tcon2)
{
// uses the beautiful solution to find the multiplier λ, so that Conic11+λ*Conic2 is
// a degenerate conic, that is a conic of lines (the 'pencil'). The lines in this conic
// intersects each of the conics in their common intersection points, thus the problem
// has been reduced to a Conic-Line intersection problem.
// This technique is described in book Graphics Gems V, of which we are inspired although
// this code differs a somewhat from the one in the book.
// work in standard space for conic 1, gives a more stable computation and speeds up
// the multiple line-conic intersections later on.
GeneralConic2d con1 = new GeneralConic2d(tcon1);
GeneralConic2d con2 = new GeneralConic2d(tcon2);
//TODO: does not work properly, probably because line extractor does not work correctly in some cases
//convert conic coefficients to their matrix form
double a = con1.A, b = con1.B * 0.5, c = con1.C, d = con1.D * 0.5, e = con1.E * 0.5, f = con1.F;
double A = con2.A, B = con2.B * 0.5, C = con2.C, D = con2.D * 0.5, E = con2.E * 0.5, F = con2.F;
//TODO: since conic 1 is in standard position, thoose can be simplified: b,d,e terms are always zero
double c3 = (A * C - B * B) * F - A * E * E + 2 * B * D * E - C * D * D;
double c2 = (a * C - 2 * b * B + c * A) * F - a * E * E + (2 * b * D + 2 * d * B - 2 * e * A) * E - c * D * D + (2 * e * B - 2 * d * C) * D + f * A * C - f * B * B;
double c1 = (a * c - b * b) * F + (2 * b * d - 2 * a * e) * E + (2 * b * e - 2 * c * d) * D + (a * f - d * d) * C + (2 * d * e - 2 * b * f) * B + (c * f - e * e) * A;
double c0 = (a * c - b * b) * f - a * e * e + 2 * b * d * e - c * d * d;
double[] lambdas2 = RealPolynomial.SolveCubic2(c3, c2, c1, c0); //up to three coefficients that will turn conic sum to degenerate lines
double[] lambdas = GetRealRoots(c3, c2, c1, c0);
if (lambdas == null)
{
return(null); //this can never happen on a 3d degree equation but we check it anyway
}
Point2dSet res = new Point2dSet();
foreach (double lambda in lambdas)
{
GeneralConic2d pencil = new GeneralConic2d(
a + lambda * A,
(b + lambda * B) * 2,
c + lambda * C,
(d + lambda * D) * 2,
(e + lambda * E) * 2,
f + lambda * F);
Line2d[] lines = pencil.ToLines();
if (lines != null)
{
foreach (Line2d lin in lines)
{ //max 2 lines
Point2d[] intpts = ConicLine(con1, lin);
if (intpts == null)
{
continue;
}
//validate each point satisfying the conic equations (they can be out of range for finite conics such as ellipses)
foreach (Point2d pt in intpts)
{
double x = pt.X;
double y = pt.Y;
double err1 = con1.A * x * x + con1.B * x * y + con1.C * y * y + con1.D * x + con1.E * y + con1.F;
if (MathUtil.IsZero(err1, 5.0))
{
double err2 = con2.A * x * x + con2.B * x * y + con2.C * y * y + con2.D * x + con2.E * y + con2.F;
if (MathUtil.IsZero(err2, 5.0))
{
res.Add(pt);
}
}
}
}
}
}
//res.Transform(tr.Inversed);
return(res.ToArray());
}
public static Point2d[] EllipseCircle(Ellipse2d el, Circle2d ci)
{
Transform2d tr = el.ToStandardPosition;
ci = new Circle2d(ci); //dont modify original circle, but this copy
ci.Transform(tr);
double b = el.Ratio, b2 = b * b, b4 = b2 * b2;
double i = ci.Center.X, i2 = i * i, i4 = i2 * i2;
double j = ci.Center.Y, j2 = j * j, j4 = j2 * j2;
double r = ci.Radius, r2 = r * r, r4 = r2 * r2;
double x4 = b4 - 2 * b2 + 1;
double x3 = 4 * b2 * i - 4 * i;
double x2 = b2 * (2 * r2 + 2 * j2 - 2 * i2 + 2) - 2 * r2 + 2 * j2 + 6 * i2 - 2 * b4;
double x1 = 4 * i * r2 - 4 * i * j2 - 4 * i * i * i - 4 * b2 * i;
double x0 = r4 + (-2 * j2 - 2 * i2) * r2 + b2 * (-2 * r2 - 2 * j2 + 2 * i2) + j4 + 2 * i2 * j2 + i4 + b4;
//double[] xs = RealPolynomial.SolveQuartic2(x4, x3, x2, x1, x0, 1e-30);
RealPolynomial rp = new RealPolynomial(x4, x3, x2, x1, x0);
double[] xs = rp.FindRoots(true);
if (xs == null)
{
return(null); //no intersections
}
Point2dSet resultset = new Point2dSet();
foreach (double x in xs)
{
//test the two possible y:s to be solutions for this x
double y = (1 - x * x) * b2;
if (y < 0.0)
{
continue;
}
y = Math.Sqrt(y);
for (int t = 0; t < 2; t++) //test booth y solutions...
{
double err = x * x + y * y / b2 - 1.0; //on ellipse
double err2 = MathUtil.Square(x - i) + MathUtil.Square(y - j) - r2; //on circle
if (MathUtil.IsZero(err, 1e-7) && MathUtil.IsZero(err2, MathUtil.Epsilon))
{
resultset.Add(new Point2d(x, y));
}
y = -y; // ...by inverting y in second turn
}
}
if (resultset.Count == 0)
{
return(null);
}
resultset.Transform(tr.Inversed); //back to original position
return(resultset.ToArray());
}
/// <summary>
/// Returns all the perpendicular points on the ellipse from a given point 'from'
/// </summary>
public Point2d[] Perpendicular(Point2d from)
{
// Solved by Robert.P. in december 2012
// Note on solutions:
// Quartic coefficients gotten from applying lagrange multiplier to minimize (x-i)^2+(y-j)^2
// with x^2/a^2+y^2/b^2-1=0 as constraint (a=1 because we work in standard position).
// This gives a system of three equations F_x,F_y,F_lambda, which were solved with
// resultant theory using 'eliminate' in maxima
//work in standard position, retranslate solutions last
Transform2d tostd = ToStandardPosition;
from = from.GetTransformed(tostd);
double b = sigratio, b2 = b * b, b4 = b2 * b2;
double i = from.X, i2 = i * i;
double j = from.Y, j2 = j * j;
double x4 = b4 - 2 * b2 + 1;
double x3 = 2 * b2 * i - 2 * i;
double x2 = b2 * j2 + i2 - b4 + 2 * b2 - 1;
double x1 = 2 * i - 2 * b2 * i;
double x0 = -i2;
double[] sols = RealPolynomial.SolveQuartic2(x4, x3, x2, x1, x0, 1e-16);
if (sols == null)
{
return(null);
}
Point2dSet respts = new Point2dSet();
foreach (double x in sols)
{
double y = (1 - x * x) * b2;
if (y < 0.0)
{
continue;
}
y = Math.Sqrt(y);
for (int l = 0; l < 2; l++)
{
//both posetive and negative y:s can be solutions. Check with each possible
//point that its perpendicular to ellipse (subtracting the inverse ellipse slope (=normal slope) with the slope from 'from' point)
double err;
err = y * (from.X - x) - x * b2 * (from.Y - y);
if (Math.Abs(err) < 1e-6)
{
respts.Add(new Point2d(x, y));
}
y = -y; //test negative solution as well
}
}
respts.Transform(tostd.Inversed);
return(respts.ToArray());
}
