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11.6.1 Introduction

Pipe surfaces were first introduced by Monge [271] and
are defined as follows: Given a space curve
and a positive
number
, the pipe surface with spine curve
is defined to be the envelope of the set of spheres with radius
which are centered at
. Pipe surfaces can be considered as
the natural generalization of the offset of a space curve in 3-D
space. Pipe surfaces have many practical applications, such as in
shape reconstruction [383], construction of blending surfaces
[304,113], transition surfaces between pipes
[304], and in NC verification [438,25]. They also
have theoretical applications as well; for example, doCarmo uses them
in the proof of two very important theorems in Differential Geometry
concerning the total curvature of simple space curves,
[76], pp. 399-402.

If we assume that the spine curve
is regular, i.e.
is simple and
,
there exist
two kinds of singularities on pipe surfaces: those that arise from
local differential geometry properties of the surface and those
that come from global distance properties of the surface. The first
type of singularity occurs when the radius of the pipe surface
exceeds the minimum radius of curvature of the spine curve; we refer
to this singularity as local self-intersection. The second one
happens, for example, when twice the radius
of the pipe surface is
larger than the minimum distance between two interior points
(excluding the two end points) on the spine curve; we refer to this
singularity as global self-intersection. Kreyszig
[206], doCarmo [76] and Rossignac
[351] derive the condition for local self-intersection of
a pipe surface and Shani and Ballard [383] describe a method
to prevent local self-intersection of a generalized cylinder. So far
the discussion was based on ``Given a spine curve and a radius, does
the pipe surface self-intersect? If so, where does it
self-intersect?'' However in some applications, we may encounter the
case ``Given a spine curve, what is the maximum radius such that the
pipe surface does not self-intersect?'' More precisely, given a
regular space curve
we want to find the maximum
, so
that the pipe surface
is nonsingular, whenever
.
In [256] it is discussed how to find this maximum possible
radius
.

One immediate application lies in the area of finding a
topologically reliable approximation of a space curve
[57]. More precisely,
suppose we are given a regular space curve
, and would
like to approximate
with another curve
-within a prescribed tolerance- in a natural way; that is,
there is a space homeomorphism
that carries
onto
[256]. One
important consequence of such a homeomorphism is that
and
have the same knot type. To do this, we first
construct a nonsingular pipe surface
. Then, we construct
a curve
that lies inside
, and ``looks like''
. By taking
to be the tolerance we have a reliable
approximation of the given curve. Sakkalis and Charitos
[359] apply the concepts of pipe surfaces and alpha
shapes [84] to ambiently approximate a nonsingular
space curve with a piecewise linear curve.

Next: 11.6.2 Local self-intersection of
Up: 11.6 Pipe surfaces
Previous: 11.6 Pipe surfaces
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December 2009