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11.6.2 Local self-intersection of pipe surfaces

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

(11.111)

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

(11.112)

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

(11.113)

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

(11.114)

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

(11.115)

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])

Next: 11.6.3 Global self-intersection of
Up: 11.6 Pipe surfaces
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December 2009