distance from given point to given ellipse

Question

I have an ellipse, defined by Center Point, radiusX and radiusY, and I have a Point. I want to find the point on the ellipse that is closest to the given point. In the illustrat

Answer 1

Here is the code translated to C# implemented from this paper to solve for the ellipse: http://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf

Note that this code is untested - if you find any errors let me know.

    //Pseudocode for robustly computing the closest ellipse point and distance to a query point. It
    //is required that e0 >= e1 > 0, y0 >= 0, and y1 >= 0.
    //e0,e1 = ellipse dimension 0 and 1, where 0 is greater and both are positive.
    //y0,y1 = initial point on ellipse axis (center of ellipse is 0,0)
    //x0,x1 = intersection point

    double GetRoot ( double r0 , double z0 , double z1 , double g )
    {
        double n0 = r0*z0;
        double s0 = z1 - 1; 
        double s1 = ( g < 0 ? 0 : Math.Sqrt(n0*n0+z1*z1) - 1 ) ;
        double s = 0;
        for ( int i = 0; i < maxIter; ++i ){
            s = ( s0 + s1 ) / 2 ;
            if ( s == s0 || s == s1 ) {break; }
            double ratio0 = n0 /( s + r0 );
            double ratio1 = z1 /( s + 1 );
            g = ratio0*ratio0 + ratio1*ratio1 - 1 ;
            if (g > 0) {s0 = s;} else if (g < 0) {s1 = s ;} else {break ;}
        }
        return s;
    }
    double DistancePointEllipse( double e0 , double e1 , double y0 , double y1 , out double x0 , out double x1)
    {
        double distance;
        if ( y1 > 0){
            if ( y0 > 0){
                double z0 = y0 / e0; 
                double z1 = y1 / e1; 
                double g = z0*z0+z1*z1 - 1;
                if ( g != 0){
                    double r0 = (e0/e1)*(e0/e1);
                    double sbar = GetRoot(r0 , z0 , z1 , g);
                    x0 = r0 * y0 /( sbar + r0 );
                    x1 = y1 /( sbar + 1 );
                    distance = Math.Sqrt( (x0-y0)*(x0-y0) + (x1-y1)*(x1-y1) );
                    }else{
                        x0 = y0; 
                        x1 = y1;
                        distance = 0;
                    }
                }
                else // y0 == 0
                    x0 = 0 ; x1 = e1 ; distance = Math.Abs( y1 - e1 );
        }else{ // y1 == 0
            double numer0 = e0*y0 , denom0 = e0*e0 - e1*e1;
            if ( numer0 < denom0 ){
                    double xde0 = numer0/denom0;
                    x0 = e0*xde0 ; x1 = e1*Math.Sqrt(1 - xde0*xde0 );
                    distance = Math.Sqrt( (x0-y0)*(x0-y0) + x1*x1 );
                }else{
                    x0 = e0; 
                    x1 = 0; 
                    distance = Math.Abs( y0 - e0 );
            }
        }
        return distance;
    }