A second order geometric method for ray/parametric surface intersection

• A novel second order method is proposed to compute the intersection between a ray and a parametric surface using the second order approximation.
• It is a geometric iteration scheme which is less sensitive to initial conditions than Newton–Raphson and Halley methods.
• It handles cases that Newton–Raphson or Halley method fails to do: inappropriate initial condition, tangential and multiple intersections.

In this paper, a new method for computing intersection between a ray and a parametric surface is proposed, which finds many applications in computer graphics, robotics and geometric modeling. The method uses the second order derivative of the surface, which can handle inherent problems that Newton–Raphson and Halley methods have such as instability caused by inappropriate initial conditions and tangential intersection. Case examples are presented to demonstrate the capability of the proposed method.