Interactive Construction of Smoothly Blended Star Solids

We introduce a computationally efficient method for interactive construction of implicitly represented star solids. These solids smoothly approximate control shapes that are defined by exact union and intersections over half-spaces containing the origin. Based on our algorithm, computation of a new solid shape when a new half-space is added or when the position of an existing half-space is changed can be performed in constant time and in space linear in the number of half-spaces. Our implicit shape construction is based on a family of non-polynomials called ray-linears [Akl93]. Computation of an implicitly represented shape is a root finding process and in general can be extremely difficult. However since ray-linear implicit representations can easily be parameterized, the computation of any ray-linearly represented shape simplifies to evaluation of a parametric equation instead of root finding. But the related parametric equations are non-polynomials and their complexity increases as the number of building blocks (in this case halfspaces) increases. Our algorithm makes the computation of this parametric equation independent of the number of half-spaces. We develop an interactive platform based on our algorithm with which we are able to construct star solids that resemble human faces. CR