Immersion and Embedding of Self-Crossing Loops

Abstract

The process of generating a 3D model from a set of 2D planar curves is complex due to the existence of many solutions. In this paper we consider a self-intersecting planar closed loop curve, and determine the 3D layered surface P with the curve as its boundary. Specifically, we are interested in a particular class of closed loop curvesin 2D with multiple self-crossings which bound a surface homeomorphic to a topological disk. Given such a selfcrossingclosed loop curve in 2D, we find the deformation of the topological disk whose boundary is the given loop. Further, we find the surface in 3D whose orthographic projection is the computed deformed disk, thus assigning 3D coordinates for the points in the self-crossing loop and its interior space.We also make theoretical observationsas to when, given a topological disk in 2D, the computed 3D surface will self-intersect.