Modifying free-formed NURBS curves and surfaces for offsetting without local self-intersection

doi:10.1016/j.cad.2003.11.002

Computer-Aided Design

Volume 36, Issue 12, October 2004, Pages 1161-1169

https://doi.org/10.1016/j.cad.2003.11.002 Get rights and content

Abstract

This paper presents an algorithm of modifying free-formed NURBS curve/surface for offsetting without local self-intersecting. The method consists of (1) sampling a number of points from a progenitor curve/surface based on second derivatives; (2) checking the curvature or maximum curvature of the progenitor curve/surface at the sampled points; (3) inserting corresponding knots of sampled points; (4) repositioning control points till the curvature/maximum curvature of the curve/surface everywhere are less than the reciprocal of offset distance. The method is efficient and is able to obtain better offsetting results.

Introduction

The technique of computing offset curves/surfaces is widely used in many applications like numerical control (NC) tool path planning and robot path planning. Offsetting operations are also gaining importance in tolerance analysis, mould- and die-making applications, and rapid prototyping applications and in finite element mesh generation. Offset curves/surfaces are normally more complex than their progenitors except for special cases such as lines, circles, planes, spherical and cylindrical surfaces. Offset curves/surfaces may have tangent discontinuities, cusps and self-intersections, local or global, even if the progenitor is regular. It is necessary to fill in the discontinuities and trim off the loops arising from self-intersections in practical applications.

Many approaches have been presented for offsetting of curves and surfaces [1], [2], [3], [4], [5], [6], [7], [8], [9]. Filip et al. [10] developed theorems of approximating curves and surfaces with a given tolerance using the bounds of second derivatives of curves and surfaces. Based on their theorems, Piegl and Tiller [11] proposed algorithms for offsetting free-formed NURBS curve/surface. In their approach, points were sampled based on bounds on second derivatives and interpolated to obtain the offset curve/surface. However, most existing algorithms do not ensure non-self-intersections which are common in curve/surface offsetting. Ravi Kumar et al. [12] developed algorithms for computing non-self-intersecting offset of surfaces, including free-formed NURBS surface, based on Piegl and Tiller's approach. Before being interpolated, the sampled points on a row or a column are connected into a poly-line for checking of any self-intersection. After cleaning of each row and column poly-lines, the output surface is non-self-intersecting. The advantage of this method is its ability to eliminate both local and global self-intersections. However, this method is complex and takes a longer time to compute the offset. Computational times are consumed by scanning every row and column at least once and inserting points into self-intersecting rows and columns to maintain the number of sample points.

Surface offset in 3-D modeling software often fails or leads to unacceptable results when there are any self-intersections. The tangent discontinuities resulted from trimming may preferably need to be eliminated. However, it can be difficult, at least cumbersome, to trim the self-intersections of offset surface and to repair the tangent discontinuities. In 3-D modeling, many offsets of free-formed surface are used for profile design without stringent tolerance requirements. Such surface offset in the applications of interactive interface requires rapid response which in turn requires efficient offset algorithm. Offset curve/surface may self-intersect locally when the absolute value of the offset distance exceeds the minimum radius of curvature in the concave regions¹ [13]. Namely, for the offset distance d, local self-intersections are caused by large signed curve curvature (κ) or maximum principal curvatures of surface (κ_max) that κ/κ_max>1/d, where the sign of curvature is defined as positive when the offset direction is same as curve/surface normal vector. Hereinafter, curvature/maximum curvature refers to the signed one. Isolated points and cusps occur when κ or κ_max or the signed minimum principal curvature (κ_min) equals to 1/d [13]. The offset curve/surface is locally self-intersections free when the κ/κ_max at any curve/surface point is less than 1/d. Therefore, modifying the curvature of a progenitor curve/surface not only can eliminate local self-intersection of offset curve/surface but can also apply smooth tangent continuity in the self-intersection regions.

Theorems developed by Filip et al. [10] can be used to ensure this curvature requirement. The equivalent of the curvature requirement is that the maximum κ/κ_max of the curve/surface is less than 1/d. Farouki [2] suggested using both univariate and bivariate Newton–Raphson iteration to find the extremum κ/κ_max. For a bivariate function κ_max(u,v), the maximum value can be found among three sets: (1) the set of corner values; (2) the set of border extrema; and (3) the set of surface extrema. Patrikalakis and Maekawa [13] proposed Interval Projected Polyhedron (IPP) algorithm which is able to solve non-linear polynomial system robustly. They also applied IPP in finding extremum κ/κ_max.

In this paper, the local self-intersection of offsetting is concerned. The proposed algorithm is to decrease κ/κ_max by repositioning control points (CPs) iteratively till the curvature requirement or its equivalent is satisfied. Only the regions where κ/κ_max>1/d and their neighboring regions, which will also lead to local self-intersections, will be changed. Most of the regions where κ/κ_max<1/d remain unchanged.

Section snippets

Geometric properties

In this paper, lines, circles, planes and spheres are not considered. It is assumed that all the curves/surfaces are described in the NURBS form. The detailed mathematical description of the NURBS can be found in literature [14]. Given a NURBS curve $C (u)$ of degree p, $C (u)= ∑ i=0 n R_{i,p} (u) P_{i} R_{i,p} (u)= ω_{i} N_{i,p} (u) ∑ j=0 n ω_{j} N_{j,p} (u)$ or a NURBS surface $S (u,v)$ of degree p, in the u-direction and degree q, in the v-direction, $S (u,v)= ∑ i=0 n ∑ j=0 m R_{i,p;j,q} (u,v) P_{i,j} R_{i,p;j,q} (u,v)= ω_{i,j} N_{i,p} (u)N_{j,q} (v) ∑ k=0 n ∑ l=0 m ω_{k,l} N_{k,p} (u)N_{l,q}$

Methodology

For offsetting without local self-intersection, the curvature requirement is given by $∀u∈[a,b]:κ(u)<1/d or ∀u∈[a,b]∧v∈[g,h]:κ_{max} (u,v)<1/d$ It is equivalent to $(κ)_{max} <1/d or (κ_{max})_{max} <1/d$ The modifying approach consists of two main parts: algorithm to decrease the κ/κ_max and methodology to examine whether the curvature requirement is satisfied.

Results and discussions

Using Rhinoceros^® [15], a NURBS modeling software for Windows^®, for graphic display, Fig. 1 shows the control net of a non-rational bi-cubic free-formed NURBS surface with identical $u$ and $v$ knot vector {0, 0, 0, 0, 1, 2, 2, 2, 2}. The control areas of CPs $P_{2,2}$ is 0≤u<2 and 0≤v<2. Fig. 2 shows the same surface after knot insertion with knot vector {0, 0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2, 2, 2}. The control areas of CPs $P_{2,2}$ are reduced to 0≤u<0.6 and 0≤v<0.6. Obviously, the

Conclusions

In this paper, a modifying approach for eliminating local self-intersections of offset curve/surface was proposed based on the fact that larger curvature/maximum curvature is the cause of unwanted tangent discontinuities, cusps, and local self-intersections occurred in offsetting. This method is efficient because much fewer points in total are involved in the computation. It is more efficient when large $d ̂$ can be applied. It can get better result in case the offset surface is local

Yifeng Sun received his BE from the Xi'an Jiaotong University, P.R. China in 1993 and ME from Xiamen University, P.R. China in 1996. Later he worked as a CAD/CAM engineer in a mold-making and injection molding company in Hong Kong. He is currently a graduate student in the Department of Mechanical Engineering, National University of Singapore (NUS). His research interests include CAD/CAM and the application of FEM in mold cooling analysis and design.

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A.Y.C. Nee is a professor of manufacturing engineering, Department of Mechanical Engineering, National University of Singapore since 1989. He received his PhD and DEng from UMIST in 1973 and 2002, respectively. He is presently the Co-Director of the Singapore-MIT Alliance (SMA) Program and CEO of the Design Technology Institute (DTI). His research interest is in computer applications to tool, die, fixture design and planning, intelligent and distributed manufacturing systems, and application of VR and AR techniques in manufacturing. He has published four books and over 450 papers in refereed journals and conference presentations. He currently held regional and associate editorship, and member of editorial board of 14 international journals in the field of manufacturing engineering. He is an active member of CIRP and a Fellow of the Society of Manufacturing Engineers, both elected in 1990.

K. S. Lee obtained his BSc (1 Hons) from the University of Manchester in 1977, MSc and PhD from UMIST in 1978 and 1982, respectively. He joined the Department of Mechanical and Production Engineering, National University of Singapore in March 1982 as a Lecturer. He was promoted to Senior Lecturer in 1988 and Associate Professor in 1998, respectively. In 1989, he received the Outstanding Young Manufacturing Engineer Award from the Society of Manufacturing Engineers (USA). He was also awarded the NUS Faculty of Engineering, Innovative Teaching Award (Gold) in 2002 in recognition of his innovative contribution in e-education. His main research interests include intelligent injection mould design techniques, collaborative workflow management in plastic injection mould design environment, Design and manufacturing automation, E-Learning and in process tool wear monitoring in machining. He is a senior member of SME and a member of ASME and IES.

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Modifying free-formed NURBS curves and surfaces for offsetting without local self-intersection

Abstract

Introduction

Section snippets

Geometric properties

Methodology

Results and discussions

Conclusions

Comput Aided Geom Des

Comput Aided Geom Des

Comput Aided Des

Comput Aided Des

Comput Aided Geom Des

Comput Aided Des

Comput Aided Geom Des