Developable Triangulations of a Strip.pdf

Computer-Aided Design Applications, Vol. 2, Nos. 1-4, 2005, pp 233-242 233 Developable Triangulations of a Strip Charlie C. L. Wang1 and Kai Tang2 1Chinese University of Hong Kong, cwang@.hk 2Hong Kong University of Science and Technology, mektang@ust.hk ABSTRACT This paper presents a modeling algorithm that helps automatically generate a “most” developable triangulated surface interpolating a given closed 3D boundary curve. The boundary curve is given in the form of a narrow strip which is often encountered in CAD applications like shoe and clothing wrinkle design. The triangulation geometrically simulates the folding process of a sheet as it would occur when rolled from one end of the strip to the other. The objective of maintaining the surface developability is realized by finding the “best rate” of the rolling along the boundary curve. To achieve this optimization, we convert the problem into a single-source shortest path problem on a special directed weighted graph, and utilize the famous Dijkstra’s algorithm to find a minimum- weight path. Examples are provided to demonstrate useful applications of the presented methodology in CAD. Keywords: strip triangulation, developability, bending energy, weighted graph. 1. INTRODUCTION Designing a surface of folding pattern over a closed 3D curve has many practical interests, especially in the computer- aided design (CAD) of clothes and footwear. Previous research in computer graphics studying folds focused mostly on cloth modeling or in animations, which are driven more by visual realism, but allow large elastic deformations and usually completely ignore or avoid the material issue. However, fold design in apparel and shoe industries has a critical request to consider – the designed surface must be able to be manufactured. A product in the apparel and shoe industries is usually produced by sewing several two-dimensional patterns together. In the majority of cases, the material