We present an efficient and robust algorithm for computing the self-intersection of a freeform surface, based on a special representation of miter points, using sufficiently small quadrangles in the parameter domain. A self-intersecting surface changes its normal direction quite dramatically around miter points, located at the open endpoints of the self-intersection curve. This undesirable behavior causes serious problems in the stability of geometric algorithms on the surface. To facilitate a stable detection of miter points, we employ osculating toroidal patches and their intersections, and consider a gradual change to degenerate intersections as a signal for the detection of miter points. The exact location of each miter point is bounded by a tiny ball in the Euclidean space and is also represented as a small quadrangle in the parameter space. The surface self-intersection curve is then constructed, using a hybrid Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches and miter quadrangles. We demonstrate the effectiveness of our approach by using test examples of computing the self-intersection of freeform surfaces.

基于斜接点的特殊表示，我们提出了一种高效且鲁棒的算法，用于计算自由曲面的自相交，在参数域中使用足够小的四边形。自相交的曲面在位于自相交曲线的开放端点处的斜接点附近极大地改变了其法线方向。这种不希望的行为在表面上的几何算法的稳定性方面引起严重的问题。为了便于稳定地检测斜接点，我们采用了闭合的环形斑块及其相交点，并考虑了逐渐变化的退化退化相交点作为检测斜接点的信号。每个斜接点的确切位置都由欧氏空间中的一个小球界定，并且在参数空间中也表示为一个小四边形。然后，使用混合边界体积层次（BVH）构造表面自相交曲线，叶节点包含闭合的环形面片和斜切四边形。通过使用计算自由曲面的自相交的测试示例，我们证明了该方法的有效性。