Surface slicing algorithm based on topology transition
Introduction
Surface/surface intersection (SSI) is one of the most fundamental yet difficult problems in various applications involving geometry such as CAD/CAM, geographic information systems, computer graphics, and so on. Hence, there have been extensive efforts to solve this problem exactly, efficiently, and robustly in a general setting [1], [6], [13], [22], [23], and these efforts can be classified into four major categories, namely the algebraic, the lattice evaluation, the marching, and the recursive subdivision [3], [4], [25].
Since there are several important applications and special properties that help to ease the general problem, several researchers have addressed the problem of surface/plane intersection (SPI), a special case of SSI. Lee and Fredericks [18] applied a recursive subdivision technique to this problem. Fu investigated the necessary and sufficient conditions for non-empty surface intersection [12]. Lee and Chang obtained intersection points by a lattice search method and connected the points by a contour growth method [19]. Farouki [9] introduced a semi-algebraic method for sectioning a parametric surface with a conic surface. Blinn described a heuristic method for determining the local extremum perpendicular distances from the surface to the section plane, and employed these to identify all closed loops of the section curve [5]. Hoitsma and Roche later refined this technique by locating the extrema via algebraic method [14].
The intersection of a surface with a set of parallel planes, a special case of SPI, has also several applications as follows: surface slicing for laminated object manufacturing (LOM) and stereolithography [11], [8], NC tool path generation [21], intersections of hunting planes to evaluate machining information [19], [20], contour lines for geographical models, and so on.
While this problem may be solved via applying an ordinary SPI algorithm for a single plane as many times as needed, the efficiency of the algorithm can be improved by considering the spatial coherency among the planes as was suggested by Dutta [17], [24]. If we investigate the intersection curves obtained by sectioning a surface with two parallel planes, we may find not only geometric changes but also topological changes. The topology change involves the creation of new curves, the removal of existing curves, the merge of two curves, the merge of two ends of an open curve to form a closed curve, and the split of a curve. For example, suppose that a section plane passes down through a local maximum point interior to a surface. Then, a new closed curve will be created. In this example, the topological change is caused by the local extreme point, which is called a topology transition point TTP, on the surface. Using the properties of TTP's on a surface, we have developed a new and efficient algorithm for finding all the intersection curves between a parametric regular surface and a series of planes parallel to XY-plane. The proposed algorithm consists of three major steps.
- 1.
Detection of all TTP's on a surface and sorting them in a descending order with respect to the height of the points.
- 2.
Computing the starting points of intersection curves using the properties of the TTP's and the section curves of the previous section plane.
- 3.
Tracing the intersection curves from the starting points.
This paper mainly focuses on the first and the second steps: the investigation of the properties of TTP's and the use of TTP's to locate the starting points efficiently and robustly. For the third step, we adopted a method similar to the one described in [3]. For the details of the tracing of intersection curve, readers are recommended to refer to the reference. After the idea of the proposed algorithm is presented, a few examples will be provided with discussions.
Section snippets
Properties of topology transition points of surface
Suppose that a surface r=r(u,v), x,n∈E3. Then, a section plane Pi is defined as:where n is the normal vector of a section plane, and Np is the number of the section planes. The section planes are assumed to be ordered such as h1>h2>…>hNp. The intersection between r(u,v) and Pi is defined in u,v-domain as:Ci may be a collection of several disjoint curves. The half space Hi constructed by a section plane Pi is defined by:
Detection of topology transition points
The seven types of topology transition points may exist either interior to, on the boundary curve, or at the vertex of a surface. Detection of a TTP and the decision of the type of a TTP should be done separately for each case as follows.
Finding starting points of intersection curves
The entire domain is partitioned into several intersection and non-intersection regions. Each separate region is bounded by one outer loop and possibly a number of inner loops. Fig. 5 shows intersection loops and loop structure, which is adopted from the trimmed surface loop structure [7], [10]. Note that outer loops of intersection regions are counter-clockwise oriented and hole loops (or inner loops) are clockwise oriented.
On the other hand, the non-intersection region, which is the whole
Treatment of singular cases
In the geometric view point, we have discussed point type TTP's. However, the shape of a TTP can vary in several aspects. Fig. 7(a), for example, illustrates some examples of such degeneracies: a line segment or curve, a bounded region, a closed curve, a bounded region with a hole(s), and the combination of the previous cases. In general, a TTP can be in any shape that can result from a general surface/plane intersection problem, and a point TTP is just a special case in this viewpoint. In
Examples
The proposed algorithm has been implemented in C language on a Windows NT machine and applied to some examples. Fig. 9(a) shows a portion of a sphere. It has an IMAX, four BMAXSAD's, and four BMIN's. The highest plane passes through the IMAX exactly, and an isolated point is made. The hyperbolic paraboloid shown in Fig. 9(b) has two BMAX's, two BMIN's, and an ISAD. Two intersection curves meet at the ISAD because a section plane passes through it. Note that none of the corner points are TTP's
Conclusions and future researches
We have presented an algorithm for slicing a general parametric surface with a series of parallel planes. When slicing the surface with parallel planes, the change of the topology of the intersection curves is caused by characteristic points of the surface, which are called topology transition points. In this paper, the properties of TTP's are studied, an algorithm to detect TTP's is presented, and an efficient method to locate the starting points to trace intersection curves are provided.
The
Acknowledgements
The work of C-S Jun and Dong-Soo Kim was supported financially by the Korea Science and Engineering Foundation through the Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Korea.
Cha-Soo Jun is a professor in the Division of Industrial and Systems Engineering, Gyeongsang National University, Chinju, Korea. He received a BS in mechanical engineering from Pusan National University in 1983, and an MS and a PhD, both in industrial engineering from KAIST (Korea Advanced Institute of Science and Technology) in 1985 and 1989, respectively. He spent a year at Purdue University as a visiting scholar in 1993. His research interests include surface modeling, multi-axis machining,
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Cha-Soo Jun is a professor in the Division of Industrial and Systems Engineering, Gyeongsang National University, Chinju, Korea. He received a BS in mechanical engineering from Pusan National University in 1983, and an MS and a PhD, both in industrial engineering from KAIST (Korea Advanced Institute of Science and Technology) in 1985 and 1989, respectively. He spent a year at Purdue University as a visiting scholar in 1993. His research interests include surface modeling, multi-axis machining, CAD/CAM, and CAPP.
Dong-Soo Kim is a PhD candidate in the Department of Industrial and Systems Engineering, Gyeongsang National University, Korea. He received a BS and MS in the Industrial Engineering from the same University in 1996 and 1998 respectively. His research interests include computer graphics, geometric modeling and CAD/CAM.
Deok-Soo Kim is an associate professor in the Department of Industrial Engineering, Hanyang University, Korea. Before he joined the university in 1995, he worked at Applicon, USA, and Samsung Advanced Institute of Technology, Korea. He received a BS from Hanyang University, Korea, an MS from the New Jersey Institute of Technology, USA, and a PhD from The University of Michigan, USA, in 1982, 1985 and 1990, respectively. His current research interests are in the streaming of 3D shapes on Internet, computational geometry, and geometric modeling and its applications.
Hyun-Chan Lee received the BS degree from Seoul National University in 1978, the MS degree from Korea Advanced Institute of Science and Technology in 1980, and the PhD degree in Industrial and Operations Engineering from the University of Michigan, Ann arbor in 1988. Before he joined the University of Michigan, he worked for the Pusan Steel Pipe Inc. for three years in the strategic planning department. In 1988, he joined the Korea Electronics and Telecommunications Research Institute as a head of design automation section. He is now an associate professor of Hongik University, Seoul, Korea, in the department of Information and Industrial Engineering. His research interests include CAD/CAM, surface modeling, computer graphics, computational geometry, engineering database, and product information management.
Ji-Seon Hwang is the manager of the Die Engineering R&D Team of Hyundai Motor Company, Korea. He received an MS from KAIST in 1987 and a PhD from Purdue University in 1997, both in Industrial Engineering. His research interests include CAD/CAM, geometric modeling, and sculptured surface machining. He is also interested in the advanced technology developments for automotive body manufacturing comprising virtual manufacturing and production planning and control.
Dr Chang is a professor of Industrial Engineering at Purdue University. He earned a BS degree in Industrial Engineering with minor in Mechanical Engineering from Chung Yuan University, Taiwan, an MS and a PhD degree in Industrial Engineering from Virginia Tech. In 1982 he joined the School of Industrial Engineering at Purdue University where he is now a full professor. Dr Chang is an author and co-author of five books and more than 100 technical papers in journals and conferences. His interests include automated process planning, numerical control, computer-aided manufacturing, and geometric modeling.