Modeling Developable Surfaces using Quintic Bézier and Hermite Curves

Abstract

The nature of the developable surfaces has similarities to the industrial materials that are not amenable to stretching. Regarding the benefit, the developable surfaces are widely used to model plat-metal-based industry products such as automobiles, ship hulls, and ducts. For this reason, we introduce a new approach for designing developable surfaces limited by two space curves. The method consists of these steps. First, we define a generalized cone and a cylinder surface by posing restrictions: a fixed summit point of the cone has to be outside a plane; a static nonzero constant vector is unparallel to the plane; and two quintic Bézier curves are placed on the different sides of the plane. Second, computing the control points on the plane is determined by the intersection between the control lines of the cone/cylinder surface and the plane. Third, using these obtained control points, we evaluate the required boundary curves profile and the shape of the developable Bézier surfaces, that are limited by these quintic Bézier curves. Finally, we also apply this method to design the developable Hermite surfaces. As a result, this introduced method can provide the equations and procedures for modeling developable surfaces with boundary curves in space. Also, it is useable to design these surfaces in many arches and shapes. Moreover, this method is effective for modifying and adjusting the desired boundary curves profile of the surfaces.

Keywords

• boundary curves
• developable surfaces
• generalized cone/cylinder surfaces
• quintic bézier/hermite curves