论文标题
点正常细分曲线和表面
Point-Normal Subdivision Curves and Surfaces
论文作者
论文摘要
本文提议将线性细分方案推广到非线性细分方案，以通过完善顶点位置以及在顶点处的单位控制正态的细化，以进行曲线和表面建模。对于每一轮细分，通过线性细分正态到单位圆或球的投影获得新的控制正态，而新的顶点位置是通过沿着新细分的正态方向更新线性细分的顶点，从而获得了新的顶点位置。特别是，每个线性细分顶点的新位置都是通过圆形或螺旋弧的终点的加权平均值计算得出的，这些平均位置或螺旋弧的末端点插入了一端的旧顶点的位置和正：在另一端的旧顶点，而新近细分的正常。 所提出的细分方案的主要特征是三倍： （1）点正常（PN）细分方案可以使用控制点和控制正常的圆圈，圆柱和球体繁殖； （2）从收敛线性细分方案中概括的PN细分方案收敛，并且可以具有与线性方案相同的平滑度阶； （3）PN $ C^2 $细分方案概括了$ C^2 $ subdivision表面具有非凡点的线性细分方案，可以在视觉上产生$ c^2 $ subdivision表面，并具有非flat非凡点。 已经给出了实验示例，以显示拟议技术在曲线和表面建模中的有效性。
This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.
