Using the CW-complex to represent the topological structure of implicit surfaces and solids

Abstract

We investigate the CW-complex as a data structure for visualizing and controlling the topology of implicit surfaces. Previous methods for contolling the blending of implicit surfaces redefined the contribution of a metaball or unioned blended components. Morse theory provides new insight into the topology of the surface a function implicitly defines by studying the critical points of the function. These critical points are organized by a separatrix structure into a CW-complex. This CW-complex forms a topological skeleton of the object, indicating connectedness and the possibility of connectedness at various locations in the surface model. Definitions, algorithms and applications for the CW-complex of an implicit surface and the solid it bounds are given as a preliminary step toward direct control of the topology of an implicit surface.