11.3.6 Tracing of self-intersection curves

Differential equations for tracing self-intersection curves of an offset surface were first derived by Aomura and Uehara [11]. They are formulated such that the self-intersection curve is arc length parametrized in the parameter domain of the progenitor surface. Here we derive a set of ordinary differential equations following the method we introduced in Sect. 5.8.2 for tracing the surface to surface intersection curves. The tracing direction coincides with the tangential direction of the self-intersection curve of the offset surface which is perpendicular to the two normal vectors at the corresponding foot points on the progenitor surfaces and where . Therefore, the tracing direction can be obtained as follows:

where and are the normal vectors

evaluated at and where and . The normalization of the tangent vector forces to be arc length parametrized in .

The self-intersection curve of an offset surface can be also viewed as a curve on the offset surface. If we denote the pair of the self-intersection curves in the parameter domain of the progenitor surface as , and , , where denotes the arc length on the offset surface, then the self-intersection curve on the offset can be expressed as

We can derive the unit tangent vector of the self-intersection curve as a curve on the offset surface using the chain rule as:

Since we know the unit tangent vector of the intersection curve from (11.87), we can find and as well as and by taking the dot product of both sides of (11.90) with and and of (11.91) with and , which leads to linear systems in , and , . The solutions to the two linear systems have the same form except that they are evaluated at different parameter values and . Using the relation between and (11.19), the ordinary differential equations for tracing the self-intersection curve of an offset surface are given by

Figure 11.28 illustrates the global self-intersection of an offset without loops. As depicted in Fig. 11.28 the surface has a global constriction between two corner points and the offset surface self-intersects globally without any internal loops. The self-intersection curves can be traced by starting at the surface boundary.

The next example, in Fig. 11.29, shows global self-intersection with loops. The surface also has 4 pairs of collinear normal point with distances 0.3757, 0.3945, 0.1367, 0.3757. Therefore if the magnitude of the offset distance exceeds , two self-intersection loops start to grow in the parameter domain enclosing the pair of collinear normal points whose distance in is 0.1367 [253].

Figure 11.28: Self-intersection curves of the offset of a bicubic surface patch when =0.09 (adapted from [253]): (a) pre-images of the self-intersection curves in parameter domain where the same symbols are mapped to the same points in the offset surface, (b) the mapping of the self-intersection curves in the parameter domain onto the progenitor surface, (c) the offset surface and the self-intersection curve

Figure 11.29: Self-intersection curves of the offset of a bisextic surface patch when =-0.08 (adapted from [253]): (a) pre-images of the self-intersection curves in parameter domain where the same symbols are mapped to the same points in the offset surface, (b) the mapping of the self-intersection curves in the parameter domain onto the progenitor surface, (c) the offset surface and the self-intersection curve