11.3.2 Singularities of offset surfaces

Similar to the offset curve case, there are two types of singularities on offset surfaces, namely irregular points and self-intersections. It is apparent from (11.21) that offset surfaces become singular at points which satisfy

The vector-valued mapping of a curve in the -parametric space, which satisfies or , into three-dimensional coordinates using (11.17) form cuspidal edges of the offset surface. These curves can be viewed as contour lines of constant principal curvatures of . The detailed formulation and a robust method for tracing constant curvature lines was discussed in Chap. 8. Isolated points and cusps can be treated as a special case of cuspidal edges. When the surface is part of a sphere with radius , then everywhere. Therefore if the offset distance is , offset surface degenerates to a point which is the center of the sphere. At an umbilic , where is the normal curvature at the umbilic, and if the offset is , the offset surface becomes singular at the point corresponding to the umbilic and forms a cusp. Therefore to detect a cusp on the offset, we need to locate all the umbilics on the progenitor surface. A robust method to locate umbilics is described in Chap. 9.

Self-intersections of an offset surface are defined by finding pairs of distinct parameter values such that

Chen and Ravani [52] present a marching algorithm to compute the self-intersection curve of an offset surface. The algorithm generates a straight-line approximation to a small portion of the intersection curve by looking at the intersection of two triangular elements representing the two tangent planes to the self-intersecting surfaces. The starting line segment is obtained by searching only the bounding curves of the patch. Aomura and Uehara [11] also developed a marching method to compute the self-intersection curves on the offset surface of a uniform bicubic B-spline surface patch. The starting points for marching are obtained by the Powell-Zangwill method based on a dense grid of points. Then the self-intersection curves are traced by integrating a system of ordinary differential equations using the Runge-Kutta-Gill method. Visualization of self-intersecting offsets of Bézier patches by means of ray tracing was studied in [430]. Self-intersection of offsets of regular Bézier surface patches due to local differential geometry and global distance function properties is investigated by Maekawa et al. [253]. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the Interval Projected Polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance (see Sect. 11.3.5). Since it is an essential task for many practical applications to detect all components of the self-intersection curves and to trace them correctly for generating the trimmed offset, we will discuss this topic in greater detail in the following three sections. In Sect. 11.3.3 we discuss the self-intersection of offsets of implicit quadratic surfaces, while in Sect. 11.3.4 we discuss the self-intersection of offsets of explicit quadratic surfaces. In Sect. 11.3.5 we introduce a method to find the self-intersections of offsets of more general polynomial parametric surface patches.