Conformal Surface Parameterization for Texture Mapping

摘要

In this paper, we give an explicit method for mapping any simply connected surface onto the sphere in a manner which preserves angles. This technique relies on certain conformal mappings from differential geometry. Our method provides a new way to automatically assign texture coordinates to complex undulating surfaces. We demonstrate a finite element method that can be used to apply our mapping technique to a triangulated geometric description of a surface. The technique of texture mapping is based on mapping an image either synthesized or digitized onto a given surface. The computer graphics literature has many such works on this topic, e.g., see 12 and the references therein. We will not review all the literature on this subject here, but just some of the key works in order to contrast our approach. Parameterization of polygonal geometric surfaces has direct use for what Gomes et al. refer to as image mapping 13, a category of techniques which includes texture mapping 6 14 15, reflection or environment mapping 4, bump mapping 5, and light mapping 20. As its name implies, image mapping associates a coordinate in image space with every location on the surface of a geometric object. The discrete samples of an image raster addressed by the coordinates in image space are applied to the geometric object. The mathematical basis of this mapping determines whether the image provides the object with the appearance of surface texture, a specular reflection, or some other effect. In this paper, we will focus only on mappings such as texture mapping that are a function of surface location and not those, for example, based on surface orientation.