(3,4)次可展Bézier曲面及其性质

A New Method for Constructing Developable (3,4)-Bézier Surfaces

在两个平行平面上分别选取 3次和 4次 Bézier曲线以生成直纹面 ,得到了此直纹面为可展曲面时 ,两个控制多边形应该满足的相对几何位置关系 ,给出了设计曲线所需满足的充分必要条件以及关于这种可展曲面上平行截曲线的凸性、拐点及奇异性 (尖点 )。最后又给出了两条边界插值于指定型值点列的 G1组合可展 Bézier曲面的构造方法。

Developable surfaces are frequently employed in computer aided geometric design and manufacturing of products such as aerofoil, car bodies and pipes. G. Aumann [4] gave a method for constructing a developable surface, but his method requires that the projection points of the four end points of the design curve and adjoint curve on a perpendicular plane form a rectangle. In section 1, we first present a new method to construct developable Bézier surfaces of degrees (3,4). Since we use Bernstein form to present the matching function, our method only requires that the design curve and adjoint curve are in two parallel planes; so it is very convenient for the designers. Section 1 also gives the geometrical relationship of control polygons and centers of gravity between design curve and adjoint curve (eq.(5)). We then discuss how to use control points to facilitate the design of developable Bézier surfaces and study the singular and convex properties of the curves on developable surfaces. Finally, in section 2 we give 5 steps for constructing G 1-continuous developable Bézier surfaces which are valuable for shape design.