Abstract
Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.
Highlights
Developable surface is a special ruled surface with vanishing Gaussian curvature
The main drawback of this method is the production of coupled equations that is very difficult for the designing of developable surfaces in a computer aided design (CAD) system
Based on the C-Bézier basis functions and cubic C-Bézier curve defined in the last section, we propose two methods to design the developable surface with three parameters due to the following Definition 3
Summary
Developable surface is a special ruled surface with vanishing Gaussian curvature. They can be unfolded or developed onto a plane without stretching and tearing. The main drawback of this method is the production of coupled equations that is very difficult for the designing of developable surfaces in a CAD system Another approach is projective geometry, which is proposed by Pottmann and Farin [4]. Zhu and Han [17] constructed four new cubic rational Bernstein-like basis functions with two parameters by using the blossom method. These basis functions can form a normalized B-basis. Zhou et al [19] constructed developable surfaces of C-Bézier basis functions with one shape parameter. In order to solve the shape handling problem of the developable surface, we introduce new basis functions with n parameters by integral. We do the examples on a 1.8 GHz PC by the software Maple, and the examples show the method is effective
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