11.6.3.3 End circle to body global self-intersection

Finally we consider the case of end circle to body global self-intersection. This case can be considered as a special case of body to body global self-intersection. We can substitute into (11.125) which gives [256]

If the spine curve is a rational Bézier curve, (11.132) will become a univariate polynomial equation. This equation contains the trivial solution and therefore should be factored out. Similarly, we can substitute into (11.125) and factor out . Notice that the line connecting and the end point is orthogonal to the spine curve at but not necessarily orthogonal at . Therefore, with the radius equal to half the distance between and , the pipe surface may not self-intersect. In the limiting case of tangential self-intersection, at the intersection point, using the parametrization of (11.111), the following equations hold:

Equation (11.134) comes from the fact that the end circle tangentially self-intersects to the body (see Fig. 11.37). This system consists of four scalar equations with four unknowns, namely , , and . We can also form the four scalar equations in terms of polynomials using the rational parametrization of the pipe surface [256]. However we cannot factor out the trivial solution from the system. Maekawa et al. [253] developed a method to handle such a case (see Sect. 11.3.5). But in this specific case we do not need to use this, as we can easily solve the system using Newton's method, since there is only one solution and we can provide a very accurate initial approximation as follows: We consider a circle at , i.e. using the solution of (11.132) as . By considering this circle as one of the end circles, we can use the end circle to end circle global self-intersection technique, that we just introduced, to find the intersection point between the two end circles. From this intersection point we can evaluate the radius and the two angles and for the initial values, using coordinate transformations. In case when the spine curve is planar, we cannot solve the linear system, since it becomes singular. In such case we will use the solution of (11.132) as and half the distance between and (or ) as , and and as 0 or as initial approximation. Let us now denote the resulting radius from Newton's method by .

Example 11.6.4. The 3-D quartic spine curve with control points (-0.3, 0.8, 0.1), (0.24, 0.15, -0.45), (0,0,0.2), (-0.24, 0.12, 0.96) and (-2, 0.6, 0) and weights 1, 2, 0.5, 2.5, 1 respectively, has minimum distance 0.0595918 between two points and . However with , the pipe surface does not self-intersect, since the vector is not orthogonal to the spine curve at . Using Newton's method we obtain the touching radius as . The spine curve also has a global maximum curvature at with =31.272916. Therefore the pipe surface starts to self-intersect locally when =0.031977 and globally when = 0.041829. Figure 11.37 shows the pipe surface with = 0.041829.

Figure 11.37: End circle tangentially intersecting the body and the local self-intersection is occurring at = 0.761 ( ) (adapted from [256])