Approximation of Transitional Developable Surfaces between Plane Curve and Polygon

Abstract

The paper describes and explains the computational geometry method that simulates the physical model of transitional pipeline parts whose cross sections are plane curve and polygon and are made of materials that cannot be wrinkled or stretched. This geometrical concept is taken from the classical line geometry and can be directly applied to the creation of computer algorithm that generates the transitional surfaces between plane curve and polygon lying in two planes. Mathematical concept of the solution for this problem is based on the approximation of the boundary surface triangulation and is presented in MATLAB program. The solutions are given in the form of three-dimensional surface models from which the necessary technical documentation can quickly be made using standard CAD/CAM tools thus shortening the time necessary to prepare the construction and production. In particular, the paper presents the basic computer graphics theory of ruled and developable surfaces, the concept of a geometrical solution for the problem, the algorithm for generation of transitional surface between plane curve and polygon, and MATLAB program which is useful for many engineering purposes.