具有正交等参线的Bézier曲面

Bézier Surfaces with Orthogonal Iso-parameter Curves

Bézier曲面的表示形式在很大程度上决定了渲染和离散的结果质量.为了改进曲面等参线的正交性,给出了双线性Bézier曲面和双二次Bézier曲面满足曲面等参线正交性的约束条件,以及相应曲面的构造方法.首先提出了具备正交等参线的双线性曲面只能是矩形;对于双二次Bézier曲面,通过将正交约束多项式的系数设置为0,整理推导出控制顶点需要满足的约束条件,再对每一组约束条件给出满足此约束条件的曲面构造性方法,得到在渲染和离散中的应用结果.纹理映射的实验结果表明,该方法是有效的.

The representation of Bézier surfaces greatly affects the results of rendering and tessellation applications.To improve the orthogonality of surface iso-parameter curves, the constraints and their correspondingconstruction method of surfaces with orthogonal iso-parameter curves are given in this paper. Theonly rational bilinear surface with orthogonal iso-parameter curves is a rectangle. For Bézier surfaces of degree2 × 2, we derive the explicit orthogonality condition of their control points, by using which a scheme toconstruct surfaces with orthogonal iso-parameter curves is then presented. Also the possible extension of theconstruction scheme for Bézier surfaces of any degree is discussed. Examples are given to show the performanceof our algorithm for rendering and tessellation applications.