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

Or you can just find the formula of both:
line: (x-x0)/a=(y-y0)/b=(z-z0)/c, which are symmetric equations of the line segment between the points you can find.
sphere: (x-xc)^2+(y-yc)^2+(z-zc)^2 = R^2.

Use the symmetric equation to find relationship between x and y, and x and z.

Then plug in y and z in terms of x into the equation of the sphere.
Then find x, and then you can find y and z.

If x gives you an imaginary result, that means the line and the sphere doesn't intersect.

Answer 2

Don't have enough reputation to comment on M Katz answer, but his answer assumes that the line can go on infinitely in each direction. If you need only the line SEGMENT's intersection points, you need t1 and t2 to be less than one (based on the definition of a parameterized equation). Please see my answer in C# below:

        public static Point3D[] FindLineSphereIntersections(Point3D linePoint0, Point3D linePoint1, Point3D circleCenter, double circleRadius)
    {

        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;

        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);

        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);

        if (D < 0 || t1 > 1 || t2 >1)
        {
            return new Point3D[0];
        }
        else if (D == 0)
        {
            return new [] { solution1 };
        }
        else
        {
            return new [] { solution1, solution2 };
        }
    }