Sphere Tracing: Simple Robust Antialiased Rendering of Distance-Based Implicit Surfaces

Sphere tracing is a new numerical method for nd-ing the intersection of a ray with an implicit surface. Unlike Newton's method or regula falsi, it is robust, converging if and only if the ray intersects the implicit surface, and it converges on the rst intersection | the intersection closest to the ray origin. It is signiicantly simpler to implement than other robust intersection techniques, such as interval analysis or LG-surfaces. Sphere tracing beneets from implicit surfaces that return distance, but requires only the imposition of the Lipschitz condition on implicit functions, and so has a wide application base. Sphere tracing also reduces to cone tracing, producing antialiased renderings though casting only one ray through each pixel. Appropriate distance functions are derived for solid modeling primitives, CSG operations, two blends, and several deformations. Finally , resulting sphere traced images of blended and deformed implicit surfaces are exhibited.