Intersection between a line and a sphere

Question

I\'m trying to find the point of intersection between a sphere and a line but honestly, I don\'t have any idea of how to do so. Could anyone help me on this one ?

Answer 1

I believe there is an inaccuracy in the solution by Markus Jarderot. Not sure what the problem is, but I'm pretty sure I translated it faithfully to code, and when I tried to find the intersection of a line segment known to cross into a sphere, I got a negative discriminant (no solutions).

I found this: http://www.codeproject.com/Articles/19799/Simple-Ray-Tracing-in-C-Part-II-Triangles-Intersec, which gives a similar but slightly different derivation.

I turned that into the following C# code and it works for me:

    public static Point3D[] FindLineSphereIntersections( Point3D linePoint0, Point3D linePoint1, Point3D circleCenter, double circleRadius )
    {
        // http://www.codeproject.com/Articles/19799/Simple-Ray-Tracing-in-C-Part-II-Triangles-Intersec

        double cx = circleCenter.X;
        double cy = circleCenter.Y;
        double cz = circleCenter.Z;

        double px = linePoint0.X;
        double py = linePoint0.Y;
        double pz = linePoint0.Z;

        double vx = linePoint1.X - px;
        double vy = linePoint1.Y - py;
        double vz = linePoint1.Z - pz;

        double A = vx * vx + vy * vy + vz * vz;
        double B = 2.0 * (px * vx + py * vy + pz * vz - vx * cx - vy * cy - vz * cz);
        double C = px * px - 2 * px * cx + cx * cx + py * py - 2 * py * cy + cy * cy +
                   pz * pz - 2 * pz * cz + cz * cz - circleRadius * circleRadius;

        // discriminant
        double D = B * B - 4 * A * C;

        if ( D < 0 )
        {
            return new Point3D[ 0 ];
        }

        double t1 = ( -B - Math.Sqrt ( D ) ) / ( 2.0 * A );

        Point3D solution1 = new Point3D( linePoint0.X * ( 1 - t1 ) + t1 * linePoint1.X,
                                         linePoint0.Y * ( 1 - t1 ) + t1 * linePoint1.Y,
                                         linePoint0.Z * ( 1 - t1 ) + t1 * linePoint1.Z );
        if ( D == 0 )
        {
            return new Point3D[] { solution1 };
        }

        double t2 = ( -B + Math.Sqrt( D ) ) / ( 2.0 * A );
        Point3D solution2 = new Point3D( linePoint0.X * ( 1 - t2 ) + t2 * linePoint1.X,
                                         linePoint0.Y * ( 1 - t2 ) + t2 * linePoint1.Y,
                                         linePoint0.Z * ( 1 - t2 ) + t2 * linePoint1.Z );

        // prefer a solution that's on the line segment itself

        if ( Math.Abs( t1 - 0.5 ) < Math.Abs( t2 - 0.5 ) )
        {
            return new Point3D[] { solution1, solution2 };
        }

        return new Point3D[] { solution2, solution1 };
    }