11.6.2 Local self-intersection of pipe surfaces

Next: 11.6.3 Global self-intersection of Up: 11.6 Pipe surfaces Previous: 11.6.1 Introduction Contents Index

11.6.2 Local self-intersection of pipe surfaces

The pipe surface ${\bf p}(r)$ can be parametrized using the Frenet-Serret trihedron $({\bf t}(t), {\bf n}(t), {\bf b}(t))$ [76,351] as follows:

$\displaystyle {\bf p}(t,\theta)={\bf c}(t)+r[\cos \theta {\bf n}(t)+\sin \theta {\bf b}(t)]\;,$

(11.111)

where $t \in [0,1]$ and $\theta \in [0, 2 \pi]$ . Its partial derivative with respect to is given by

$\displaystyle {\bf p}_{t}(t, \theta)= \dot{\bf c}(t)+r[\cos \theta \dot{\bf n}(t)+\sin \theta \dot{\bf b}(t)]\;.$

(11.112)

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

$\displaystyle {\bf p}_{t}(t, \theta)=\vert\dot{\bf c}(t)\vert(1-\kappa(t)r\cos ... ...in \theta {\bf n}(t)+r \vert\dot{\bf c}(t)\vert\tau(t)\cos \theta {\bf b}(t)\;,$

(11.113)

where $\kappa (t)$ and $\tau (t)$ are the curvature and torsion of the spine curve given by (2.26) and (2.48), respectively. Similarly we can derive ${\bf p}_{\theta}$ as

$\displaystyle {\bf p}_{\theta}(t, \theta)=r[-\sin \theta {\bf n}(t)+\cos \theta {\bf b}(t)]\;.$

(11.114)

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

$\displaystyle {\bf p}_{t} \times {\bf p}_{\theta}=-\vert\dot{\bf c}(t)\vert r [1-\kappa(t)r\cos\theta][ \sin\theta {\bf b}(t)+\cos \theta {\bf n}(t)]\;.$

(11.115)

It is easy to observe [206,76,351] that the pipe surface becomes singular when $1-\kappa(t)r\cos \theta =0$ . Since $\cos\theta$ varies between -1 and 1, there will be no local self-intersection if $\kappa(t)r<1$ . Therefore, to avoid local self-intersection we need to find the largest curvature $\kappa_{a}$ of the spine curve and set the radius of the pipe surface such that $r<1/\kappa_{a}$ .

The curvature $\kappa (t)$ of a space curve ${\bf c}(t)$ is given in (2.26). Thus, to find the largest curvature $\kappa_{a}$ we need to locate the critical points of $\kappa (t)$ , i.e. solve the equation $\dot\kappa (t)=0$ (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. $\kappa(0)$ and $\kappa (1)$ , 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 $\dot\kappa (t)=0$ 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 $\kappa=2$ at . Therefore in order to have no local self-intersection the radius should be $r<\frac{1}{2}$ . 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])
$\begin{figure}\vspace{5mm} \centerline{ \psfig{figure=fig/local.PS,height=4in} } \vspace{-5mm}\end{figure}$

Next: 11.6.3 Global self-intersection of Up: 11.6 Pipe surfaces Previous: 11.6.1 Introduction Contents Index

December 2009