BOUNDARY TRIANGULATIONS APPROXIMATING DEVELOPABLE SURFACES THAT INTERPOLATE A CLOSED SPACE CURVE

Abstract

A common operation in fabricating many products is bending a thin, flat sheet of material over a three-dimensional boundary curve that outlines the desired shape. Typically the shape of the closed boundary curve is given, and the geometric problem of interest is to compute the developable surface that forms when the material is bent to conform to this curve. A classical method of solving this problem on the drawing board is by triangulation development: approximating the boundary curve by line segments and the surface by triangles whose vertices are the boundary points. Choosing the proper boundary triangulation was the job of the draftsman. Automating this process on a computer requires identifying triangulations approximating developable surfaces. Moreover, since there are usually two or more developable surfaces that interpolate a given space curve, there must be a means of selecting the most appropriate solution for the application at hand. This paper describes a computer-based method that generates a boundary triangulation by geometrically simulating the bending of the sheet as it would occur during closing of the blankholder in a sheet-metal forming problem.