ja72 / so.ellipsecontact

Sample code for answer on SO which asks for finding a tangent circle to an ellipse, if the circle is constrained to be on an arc.

EllipseContact

Sample code for answer on SO which asks for finding a tangent circle to an ellipse, if the circle is constrained to be on an arc.

My solution involves two numerical methods, a bisection of the location of the tangent circle as an outer loop, and a single point iteration for each circe location to find the closest point to the ellipse.

Search for circle-ellipse tangency

The code snipper below used a bisection method to find the circle's location that is tangent to the ellipse.

  Ellipse ellipse = new Ellipse(7, 1);
  Point pivot = new Point(10, -4);
  Circle arc = new Circle(pivot, 8);

  double RootFun(double u)
  {
      Circle cen = new Circle(arc.GetPoint(u), 1);
      Point trgt = ellipse.GetClosestPoint(cen, NumericalMethods.LooseTolerance);
      return trgt.DistanceTo(cen);
  }

  double u_sol = NumericalMethods.BisectionRoot(RootFun, 0, PI/2, 1e-6);
  Circle circle = new Circle(arc.GetPoint(u_sol), 1);
  Point closest = ellipse.GetClosestPoint(circle, NumericalMethods.LooseTolerance);
  double distance = closest.DistanceTo(circle);

Closest Point on ellipse to outside point

The code snippet below shows the single-point iteration algorithm used for the ellipse-circle distance

public Point GetClosestPoint(Point point, double tol)
{
    double cx = point.X - Center.X, cy = point.Y - Center.Y;
    double Q = MajorAxis*MajorAxis-MinorAxis*MinorAxis;
    double A = 2*cx*MajorAxis/Q;
    double B = 2*cy*MinorAxis/Q;

    double IterFun(double z)
    {
        return 1/A*(B + 2*z/Sqrt(1+z*z));
    }

    double z_sol = NumericalMethods.GaussPointIteration(IterFun, 0, tol);

    double t = Atan(z_sol);

    return GetPoint(t);
}

With the sample output below:

Arc     = Circle(x=10, y=-4, r=8)
Target  = Ellipse(x=0, y=0, rx=7, ry=1)
Circle  = Circle(x=4.46565766050686, y=1.77676854905003, r=1)
Contact = Point(x=4.35294539705547, y=0.783136894362217)
distance= 3.95975714484464E-06

Closest Point on ellipse to infinite line

The code snippet below find the closest point on an ellipse to a line in the plane.

public Point2 GetClosestPoint(Line2 line, double tol)
{
	var (a, b, c)= line.Coords;
	var (cx, cy) = Center.Coords;
	// transform line coordinates to being relative to ellipse center
	c -= -a*cx - b*cy;
	double rx = MajorAxis, ry = MinorAxis;

	double IterFun(double z)
	{
		return (b*ry-a*rx*z)*(a*rx+b*ry*z)/(a*c*rx*Sqrt(1+z*z))+(b*ry)/(a*rx);
	}

	double z_sol = NumericalMethods.GaussPointIteration(IterFun, 0, tol);

	double t = Atan(z_sol);

	return GetPoint(t);

}

With the sample output below

Ellipse = Ellipse(x=1, y=1, rx=7, ry=1)
Line    = 4*x + 19*y + -131 = 0
Contact = Point(x=6.79233084048069, y=1.56150132997128)
Target  = Point(x=7.57919716799924, y=5.29911638568437)

Related

A closed-form solution to the distance problem is claimed to be found here:

• https://yehar.com/blog/?p=2926