Approximation of surface-to-surface intersection curves within a prescribed error bound satisfying G2 continuity

doi:10.1016/j.cad.2008.11.008

Computer-Aided Design

Volume 41, Issue 1, January 2009, Pages 37-46

https://doi.org/10.1016/j.cad.2008.11.008 Get rights and content

Abstract

We present a new method of approximating the intersection curves of two parametric surfaces. Our approximation satisfies $G^{2}$ continuity conditions, and is located within a prescribed distance from the exact intersection curve. First we identify the topology of the pre-image of the intersection in the domain of surface using extended [Grandine TA, Klein FW. A new approach to the surface intersection problem. Computer Aided Geometric Design 1997;14(2):111–34]’s topology determination method [Hur S, Oh M-J, Kim T-W. Classification and resolution of critical cases in Grandine–Klein’s topology determination using a perturbation method. Computer Aided Geometric Design (2008). in press [doi:10.1016/j.cagd.2008.03.003]]. Then, we choose a segment of the pre-image, identify the tangential directions and the signed curvatures at its two end-points, and construct a $G^{2}$ Hermite interpolation using a rational cubic Bézier curve. We go on to find the exact maximum error of the approximation, and the point at which that error occurs, using the Hausdorff distance function. If the error is larger than a prescribed error bound $ε$ , we subdivide the segment at the point of maximum error, and apply the $G^{2}$ interpolation process to the two new segments. We continue this recursive process until we have an approximation of the pre-image with an error smaller than $ε$ . We also find the maximum error bound of the approximation in the model space $R^{3}$ ; and the bound on the distance, in model space, between the approximations which come from the domain of each surface. We verify our method with several examples.

Introduction

The SSI (surface–surface intersection) problem is one of the classical problem in CAGD and Computational Geometry which is to find the intersection curve of two surfaces. There are two main types of surface representation: rational polynomial parametric (RPP) and implicit algebraic (IA). So in general there are three cases of the SSI problem: RPP–RPP, RPP–IA and IA–IA. Reviews of the general SSI problem can be found elsewhere [1], [2], [3].

We will consider the intersection of two RPP surfaces $F (u, v)$ and $G (s, t)$ . It is well known that exist exact implicit representations $f (x, y, z) = 0$ and $g (x, y, z) = 0$ of two surfaces $F (u, v)$ and $G (s, t)$ respectively, but the degrees of $f$ and $g$ are much higher than the degrees of $F$ and $G$ . For instance, two general bicubic tensor product surfaces intersect at a curve of degree 324. The pre-image of this curve can be expressed as an implicit equation $h (u, v) = 0$ of degree 108 [4]. It is reasonable to consider finding the parametric representations of the intersection curve $(x (τ), y (τ), z (τ))$ and its pre-images $(u (τ), v (τ))$ and $(s (τ), t (τ))$ in each domain; but finding the exact parametric expressions of the intersection curve or its pre-images is in general impossible [5], [6].

In practice, therefore, the SSI problem for two RPP surfaces usually means finding approximations to the intersection curves that lie within a prescribed tolerance. This subject remains an important issue in many mathematical and engineering fields, such as computational geometry, computer-aided design (CAD), computer aided geometric design (CAGD), geometric and solid modelling, and numerically controlled machining [1], [2], [7], [8], [9], [10].

We will consider the intersection of two RPP surfaces $F (u, v)$ and $G (s, t)$ , and present an algorithm for approximating their intersection curve which can be applied both in parameter spaces, and also in model space. The algorithm produces an approximated curve which is within a prescribed distance from the actual intersection. Our approach is to approximate the pre-image $c^{u v} = F^{- 1} (c)$ of the intersection $c = F (u, v) \cap G (s, t)$ and then to derive an approximation of the intersection as an image of the mapping $F$ of this approximation. We will describe the algorithm for approximating the pre-image in the $(u, v)$ domain, but the procedure can equally be applied in the $(s, t)$ domain. The approximated pre-image is a piecewise rational cubic Bézier curve with $G^{2}$ continuity, and this can be replaced by a piecewise cubic Bézier curve after appropriate subdivision. We note that the image resulting from the mapping $F$ of an approximation in the $(u, v)$ domain will not in general coincide in model space with the mapping $G$ of an approximation in the $(s, t)$ domain. The objective of our algorithm is to localise these two images of the approximation within a prescribed distance $γ$ from the exact intersection, so that the maximum distance between the two images is smaller than $2 γ$ .

We use the extended Grandine and Klein’s [11] algorithm to obtain the topological structure of SSI curves in the domain of the parameters. The [11]’s algorithm has recently been extended for the critical situations by the authors [12], and consequently, this topology determination method can be applied for all situations. (We note that an alternative method of doing this has recently been published [13].) Our algorithm therefore starts with the assumption that we know the topology of the pre-image of the intersection, and we choose a segment of that pre-image in parameter space. The two end-points of a segment are characteristic points which may be boundary points or turning points, or both. We calculate the tangential directions and the signed curvatures of the two end-points of the chosen segment. Then we find an approximation to the segment using a two-point $G^{2}$ Hermite interpolation with a rational cubic Bézier curve which has the same tangential directions and signed curvatures as the segment at its end-points. We will show the existence of such an interpolation, and we will also show how to find the point of maximum error and the exact value of the maximum error between the approximation and the true segment. For the measurement of the error of approximations, we will use a Hausdorff distance between two curves. If the error is outside the prescribed bound, we subdivide the segment at the point of maximum error and apply the same approximation algorithm to the two new segments. This procedure can be expected to reduce the error significantly, and we can repeat it until the error is within the prescribed bound. We will also verify the error bounds in model space.

The rest of this paper is organised in the following manner: in Section 2 we present a brief review of previous work on SSI which is related to our study, and we also introduce some necessary background material. In Section 3, we describe how to calculate the tangential directions and the signed curvatures of the two end-points of a segment. In Section 4, we show the existence of a rational cubic Bézier curve which satisfies the $G^{2}$ continuity conditions at the two end-points of a segment. In Section 5, we present an error check, in which we obtain the Hausdorff distance between the exact and the approximation curves, and we describe how to find the point of maximum error and the exact global maximum error, which are prerequisites for the subdivision process. The last section contains several examples which demonstrate the effectiveness of our method, and then we conclude this paper. There is also an Appendix which contains some details omitted from Section 4, including a direct proof of the existence of a rational cubic Bézier curve satisfying the $G^{2}$ continuity conditions. There is also a discussion of the selection of appropriate weights for this rational curve.

•
$F (u, v), G (s, t)$ : Two parametric representations of two surfaces $S_{1}, S_{2}$ , respectively, in $R^{3}$ .
•
$c = F (u, v) \cap G (s, t)$ : The intersection of two surface in $R^{3}$ .
•
$c^{u v} = F^{- 1} (c), c^{s t} = G^{- 1} (c)$ : The inverse images of $c$ in $(u, v)$ and $(s, t)$ domains, respectively.
•
$c_{k}^{u v}, 1 \leq k \leq n$ : The segments of $c^{u v} = ⋃_{1 \leq k \leq n} c_{k}^{u v}$ .
•
$s_{k}^{u v}, e_{k}^{u v}$ : Two end-points of $c_{k}^{u v}$ . $s_{k}^{u v}$ is starting and $e_{k}^{u v}$ is ending point according to the extend G-K method.
•
$c_{a, k}^{u v}$ : Approximation of $c_{k}^{u v}$ .
•
$t_{k}^{s}, t_{k}^{e}$ : The unit tangent vectors at $s_{k}^{u v}, e_{k}^{u v}$ , respectively.
•
$κ_{k}^{s}, κ_{k}^{e}$ : The signed curvatures at $s_{k}^{u v}, e_{k}^{u v}$ , respectively.
•
$(a, b) \land (c, d) = a d - b c$ .
•
$ε$ : Error bound of approximations in $(u, v)$ domain.

Section snippets

Previous work

Although there has been much research on the SSI problem, there is as yet no established general algorithm for this problem. Each of the different approaches that were mentioned in the previous section has its own advantages and limits, and improving these methods is a topic of active research. At present the marching method [14], [15], [16], [17] is probably the most widely used approach to the SSI problem, and we will now review this method briefly. Reviews of the algebraic, lattice, and

Unit tangent vectors and signed curvatures of the pre-image of the intersection curve

In this section we present our method of identifying the unit tangent vectors and the signed curvatures at two end-points of a segment which is obtained by the extended G-K topology determination method. These information and two end-points will be used for the $G^{2}$ Hermite interpolation with which we obtain a piecewise $G^{2}$ approximation to $c^{u v}$ . Let $c_{k}^{u v}$ be a segment of intersection curve $c^{u v}$ which is composed of two end-points, one $s_{k}^{u v}$ is called starting point and the other $e_{k}^{u v}$ is called ending

Approximation of an intersection curve by piecewise rational cubic Bézier curves with $G^{2}$ continuity

We now construct a piecewise $G^{2}$ approximation $c_{a}^{u v}$ of the pre-image $c^{u v}$ of the intersection $c$ . The pre-image $c^{u v}$ is composed of a finite number of segments $c_{k}^{u v}, 1 \leq k \leq n$ , together with isolated points. We will only consider the segments, and we want to find an approximation $c_{a, k}^{u v}$ for each segment $c_{k}^{u v}$ which satisfies the $G^{2}$ continuity conditions at the two end points $s_{k}^{u v}, e_{k}^{u v}$ of $c_{k}^{u v}$ . This means that the two curves $c_{a, k}^{u v}$ and $c_{k}^{u v}$ have

(i)
the same end points $s_{k}^{u v}$ and $e_{k}^{u v}$ ,
(ii)
the same tangential

The exact global maximum error in parameter space

We will now find the maximum error of the approximation in the parameter spaces $(u, v)$ and $(s, t)$ , and later in the model space $R^{3}$ . We start by identifying the exact maximum error between a segment $c_{k}^{u v}$ of $c^{u v}$ and its approximation $c_{a, k}^{u v} (σ) = (u (σ), v (σ)), 0 \leq σ \leq 1$ , which was considered in the previous section. Suppose that $(u (τ), v (τ)), 0 \leq τ \leq 1$ , is a parametrization of the segment $c_{k}^{u v}$ , which we do not know exactly.

For the measurement the error of approximations, we use Hausdorff distance function. Let $a (τ$

Examples

We present several examples which show the effectiveness of our method of approximation. We choose $θ = π / 4$ for the examples. The weights of the control points $b_{1}$ and $b_{2}$ are determined by choosing ${\tilde{R}}_{0} = {\tilde{R}}_{3} = 1 / 2$ as described in the Appendix, and $ε$ is the error bound on each approximation. The first example is presented in more detail than the others.

Example 4

The intersection of an algebraic surface $S_{1}$ and a plane $S_{2}$ in $R^{3}$ (see Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10): $S_{1} = {(u, v, f (u, v)) : 0 \leq u, v \leq 1}$ $S_{2} = {(s, t$

Acknowledgements

This work was supported by grant No. R01-2005-000-11257-0 from the Basic Research Program of the Korea Science and Engineering Foundation, by grant No. KRF-2006-070-C00014 from the Korea Research Foundation Grant funded by the Korean Government, and in part by the Seoul R&BD Program.

The authors would like to express their gratitude to T.A. Grandine, F.W. Klein IV and G. Elber for their suggestions during this research.

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¹: Also holds a post in the Department of Mathematical Science, Seoul National University, Seoul 151-747, South Korea.

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Approximation of surface-to-surface intersection curves within a prescribed error bound satisfying G2 continuity

Abstract

Introduction

Section snippets

Previous work

Unit tangent vectors and signed curvatures of the pre-image of the intersection curve

Approximation of an intersection curve by piecewise rational cubic Bézier curves with G2 continuity

The exact global maximum error in parameter space

Examples

Acknowledgements

Computer Vision, Graphics, and Image Processing

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer-Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Fundamentals of Computer Aided Geometric Design

Surface-to-surface intersections

IEEE Computer Graphics and Applications

An SSI bibliography

Nurbs approximation of surface/surface intersection curves

Advances in Computational Mathematics

Approximation of surface-to-surface intersection curves within a prescribed error bound satisfying $G^{2}$ continuity

Approximation of an intersection curve by piecewise rational cubic Bézier curves with $G^{2}$ continuity