private double EllipticArc_NearestPointTo(Double2 v)
{
// First, center and un-rotate the point
var p = (v - Center).RotScale(ComplexRot.NegateY);
var k = p.X * p.X / (Radii.X * Radii.X) + p.Y * p.Y / (Radii.Y * Radii.Y) - 1;
// If it is a circle, just calculate its angle
double t;
if (RoughlyEquals(Radii.X, Radii.Y))
{
t = p.Angle;
}
// If the point is over the ellipse, calculate its angle the "dumb" form
else if (RoughlyZero(k))
{
t = Math.Atan2(Radii.X * p.Y, Radii.Y * p.X);
}
else
{
// Make the point on the first quadrant
bool nx = p.X < 0, ny = p.Y < 0;
if (nx)
{
p.X = -p.X;
}
if (ny)
{
p.Y = -p.Y;
}
// Radii difference
var rx = Radii.X;
var ry = Radii.Y;
var rr = ry * ry - rx * rx;
// Function to test for roots
double f(double tx) => 0.5 * rr * Math.Sin(2 * tx) + rx * p.X * Math.Sin(tx) - ry * p.Y * Math.Cos(tx);
// There will always be a single root on this equation
t = Equations.Bisection(f, 0, Pi_2);
// Now, update the guess for the correct quadrants
if (nx && ny)
{
t += Pi;
}
else if (nx)
{
t = Pi - t;
}
else
{
t = -t;
}
}
// If the guess is inside the interval
var tu = AngleToParameter(t);
if (!double.IsInfinity(tu))
{
return(tu);
}
// Else, pick the nearest of 0 or 1
return(EllipticArc_At(0).DistanceSquaredTo(v) < EllipticArc_At(1).DistanceSquaredTo(v) ? 0 : 1);
}
