CSG Operations of Arbitrary Primitives with Interval Arithmetic and Real-Time Ray Casting

Abstract

We apply Knoll et al.'s algorithm [Knoll et al., "Fast ray tracing of arbitrary implicit surfaces with interval and affine arithmetic.",  Comput. Graph. Forum, 28(1):26–40, 2009] to interactively ray-cast constructive solid geometry (CSG) objects of arbitrary primitives represented as implicit functions. Whereas modeling globally with implicit surfaces suffers from a lack of control, implicits are well-suited for arbitrary primitives and can be combined through various operations. The conventional way to represent union and intersection with interval arithmetic (IA) is simply using min and max but other operations such as the product of two forms can be useful in modeling joints between multiple objects. 

Typical primitives are objects of simple shape, e.g. cubes, cylinders, spheres, etc. Our method handles arbitrary primitives, e.g. superquadrics or non-algebraic implicits. Subdivision and interval arithmetic guarantee robustness whereas GPU ray casting allows for fast and aesthetic rendering. Indeed, ray casting parallelizes efficiently and trivially and thus takes advantage of the continuous increasing computational power of hardware (CPUs and GPUs); moreover it lends itself to multi-bounce effects, such as shadows and transparency, which help for the visualization of complicated objects. With our system, we are able to render multi-material CSG trees of implicits robustly, in interactive time and with good visual quality.