next up previous contents index
Next: 11.6.2 Local self-intersection of Up: 11.6 Pipe surfaces Previous: 11.6 Pipe surfaces   Contents   Index


11.6.1 Introduction

Pipe surfaces were first introduced by Monge [271] and are defined as follows: Given a space curve $ {\bf c}(t)$ and a positive number $ r$ , the pipe surface with spine curve $ {\bf c}(t)$ is defined to be the envelope of the set of spheres with radius $ r$ which are centered at $ {\bf c}(t)$ . 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 $ {\bf c}(t)$ is regular, i.e. $ {\bf c}(t)$ is simple and $ \vert\dot{\bf c}(t)\vert \neq 0$ , 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 $ r$ 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 $ r$ 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 $ {\bf c}(t)$ we want to find the maximum $ R, R>0$ , so that the pipe surface $ {\bf p}(r)$ is nonsingular, whenever $ r<R$ . In [256] it is discussed how to find this maximum possible radius $ R$ .

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 $ {\bf c}(t)$ , and would like to approximate $ {\bf c}(t)$ with another curve $ {\bf g}(t)$ -within a prescribed tolerance- in a natural way; that is, there is a space homeomorphism $ h: {\bf R}^3 \to {\bf R}^3 $ that carries $ {\bf c}(t)$ onto $ {\bf g}(t)$ [256]. One important consequence of such a homeomorphism is that $ {\bf c}(t)$ and $ {\bf g}(t)$ have the same knot type. To do this, we first construct a nonsingular pipe surface $ {\bf p}(r)$ . Then, we construct a curve $ {\bf g}(t)$ that lies inside $ {\bf p}(r)$ , and ``looks like'' $ {\bf c}(t)$ . By taking $ r$ 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 up previous contents index
Next: 11.6.2 Local self-intersection of Up: 11.6 Pipe surfaces Previous: 11.6 Pipe surfaces   Contents   Index
December 2009