11.6.2 Local self-intersection of pipe surfaces

The pipe surface can be parametrized using the Frenet-Serret trihedron [76,351] as follows:

where and . Its partial derivative with respect to is given by

Equation (11.112) can be rewritten using the Frenet-Serret formulae (2.57) as

where and are the curvature and torsion of the spine curve given by (2.26) and (2.48), respectively. Similarly we can derive as

The surface normal of the pipe surface can be obtained by taking the cross product of (11.113) and (11.114) yielding

It is easy to observe [206,76,351] that the pipe surface becomes singular when . Since varies between -1 and 1, there will be no local self-intersection if . Therefore, to avoid local self-intersection we need to find the largest curvature of the spine curve and set the radius of the pipe surface such that .

The curvature of a space curve is given in (2.26). Thus, to find the largest curvature we need to locate the critical points of , i.e. solve the equation (8.20), and decide whether they are local maxima (see Sect. 7.3.1). Then we compare these local maxima with the curvature at the end points, i.e. and , and obtain the largest curvature. This problem can be solved by elementary calculus. If the spine curve is given by a rational Bézier curve, equation reduces to a single univariate nonlinear polynomial equation (8.21) for a planar spine curve and (8.22) for a 3-D spine curve. In the case where the spine curve is a rational B-spline, we can extract the rational Bézier segments by knot insertion [175,314].

Example 11.6.1. The parabola has its largest curvature at . Therefore in order to have no local self-intersection the radius should be . Figure 11.32 shows the local self-intersection of the pipe surface with the above parabolic spine curve and with radius 0.8. Obviously, there is a local self-intersection on the pipe surface corresponding to the point at the spine curve.

Figure 11.32: Local self-intersection of a pipe surface ( ) (adapted from [256])