11.6.3.1 End circle to end circle global self-intersection

Let us consider the plane which contains the end point and is perpendicular to . If we denote a point on the plane as , then the equation of that plane becomes

Similarly the equation of the plane that contains the other end of the pipe is given by

The self-intersection occurs along the intersection of these two planes as shown in Fig. 11.33. It also lies on the bisecting plane of the line segment . Thus if is a self-intersection point, then

Equations (11.116), (11.117 ), (11.118) form a system of three linear equations with the three components of as unknowns as follows:

where superscripts denote , , and components, and

The determinant of the matrix is readily computed as

We now consider the following cases:
Case 1. . In that case, if , then , where is the unique solution of the above system. If , and the system has no solution, we take . If the system has an infinte number of solutions, then we take . This minimum is always positive since .
Case 2. . In that case, if , the pipe is always singular for every , and thus . If , we take .

Figure 11.33: Two end circles globally self-intersecting at point (adapted from [256])

Example 11.6.2. Figure 11.34 illustrates the case when end circles are touching each other. The control points of the spine curve, which is a cubic integral Bézier curve, are given by (2.9, 3.0, 4.1), (0.0, 1.0, 2.0), (5.0, -2.0, 1.0) and (3.0, 3.1, 4.0). The linear system (11.116), (11.117), (11.118) gives us the intersection point as (2.918, 3.055, 4.023) with radius .

Figure 11.34: End circle to end circle global self-intersection ( =0.0963) (adapted from [256])