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());
}
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());
}
