Boolean Algebra and Analysis Using Approximate Step Functions

Implicit equations of curves and surfaces have been shown to be useful for constructing solutions for boundary value problems such that the boundary conditions are satisfied exactly. This application has generated interest in constructing solid models where the geometry is represented using implicit equations rather than parametric equations. In this paper we present a method for constructing step functions of solids that have a unit value within the solid and zero outside. A level set of this step function can then be defined as the boundary of the solid. This step function can be used not only to apply boundary conditions but also to compute volume integrals over the solid. Methods for combining step functions of solid primitives using ordinary and regularized Boolean operations to construct step functions of the Boolean result are also presented.