6 research outputs found

    International Journal of Mathematical Combinatorics, Vol.7A

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    The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

    Euclidean Offset and Bisector Approximations of Curves over Freeform Surfaces

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    The computation of offset and bisector curves/surfaces has always been considered a challenging problem in geometric modeling and processing. In this work, we investigate a related problem of approximating offsets of curves on surfaces (OCS) and bisectors of curves on surfaces (BCS). While at times the precise geodesic distance over the surface between the curve and its offset might be desired, herein we approximate the Euclidean distance between the two. The Euclidean distance OCS problem is reduced to a set of under-determined non-linear constraints, and solved to yield a univariate approximated offset curve on the surface. For the sake of thoroughness, we also establish a bound on the difference between the Euclidean offset and the geodesic offset on the surface and show that for a C2 surface with bounded curvature, this difference vanishes as the offset distance is diminished. In a similar way, the Euclidean distance BCS problem is also solved to generate an approximated bisector curve on the surface. We complete this work with a set of examples that demonstrates the effectiveness of our approach to the Euclidean offset and bisector operations

    ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ฒ€์ถœ ๋ฐ ์ œ๊ฑฐ

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    ํ•™์œ„๋…ผ๋ฌธ(๋ฐ•์‚ฌ)--์„œ์šธ๋Œ€ํ•™๊ต ๋Œ€ํ•™์› :๊ณต๊ณผ๋Œ€ํ•™ ์ปดํ“จํ„ฐ๊ณตํ•™๋ถ€,2020. 2. ๊น€๋ช…์ˆ˜.Offset curves and surfaces have many applications in computer-aided design and manufacturing, but the self-intersections and redundancies must be trimmed away for their practical use. We present a new method for offset curve and surface trimming that detects the self-intersections and eliminates the redundant parts of an offset curve and surface that are closer than the offset distance to the original curve and surface. We first propose an offset trimming method based on constructing geometric constraint equations. We formulate the constraint equations of the self-intersections of an offset curve and surface in the parameter domain of the original curve and surface. Numerical computations based on the regularity and intrinsic properties of the given input curve and surface is carried out to compute the solution of the constraint equations. The method deals with numerical instability around near-singular regions of an offset surface by using osculating tori that can be constructed in a highly stable way, i.e., by offsetting the osculating torii of the given input regular surface. We reveal the branching structure and the terminal points from the complete self-intersection curves of the offset surface. From the observation that the trimming method based on the multivariate equation solving is computationally expensive, we also propose an acceleration technique to trim an offset curve and surface. The alternative method constructs a bounding volume hierarchy specially designed to enclose the offset curve and surface and detects the self-collision of the bounding volumes instead. In the case of an offset surface, the thickness of the bounding volumes is indirectly determined based on the maximum deviations of the positions and the normals between the given input surface patches and their osculating tori. For further acceleration, the bounding volumes are pruned as much as possible during self-collision detection using various geometric constraints imposed on the offset surface. We demonstrate the effectiveness of the new trimming method using several non-trivial test examples of offset trimming. Lastly, we investigate the problem of computing the Voronoi diagram of a freeform surface using the offset trimming technique for surfaces. By trimming the offset surface with a gradually changing offset radius, we compute the boundary of the Voronoi cells that appear in the concave side of the given input surface. In particular, we interpret the singular and branching points of the self-intersection curves of the trimmed offset surfaces in terms of the boundary elements of the Voronoi diagram.์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์€ computer-aided design (CAD)์™€ computer-aided manufacturing (CAM)์—์„œ ๋„๋ฆฌ ์ด์šฉ๋˜๋Š” ์—ฐ์‚ฐ๋“ค ์ค‘ ํ•˜๋‚˜์ด๋‹ค. ํ•˜์ง€๋งŒ ์‹ค์šฉ์ ์ธ ํ™œ์šฉ์„ ์œ„ํ•ด์„œ๋Š” ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์—์„œ ์ƒ๊ธฐ๋Š” ์ž๊ฐ€ ๊ต์ฐจ๋ฅผ ์ฐพ๊ณ  ์ด๋ฅผ ๊ธฐ์ค€์œผ๋กœ ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์—์„œ ์›๋ž˜์˜ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์— ๊ฐ€๊นŒ์šด ๋ถˆํ•„์š”ํ•œ ์˜์—ญ์„ ์ œ๊ฑฐํ•˜์—ฌ์•ผํ•œ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์—์„œ ์ƒ๊ธฐ๋Š” ์ž๊ฐ€ ๊ต์ฐจ๋ฅผ ๊ณ„์‚ฐํ•˜๊ณ , ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์—์„œ ์ƒ๊ธฐ๋Š” ๋ถˆํ•„์š”ํ•œ ์˜์—ญ์„ ์ œ๊ฑฐํ•˜๋Š” ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•œ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์€ ์šฐ์„  ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ์ ๋“ค๊ณผ ๊ทธ ๊ต์ฐจ์ ๋“ค์ด ๊ธฐ์ธํ•œ ์›๋ž˜ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ ๋“ค์ด ์ด๋ฃจ๋Š” ํ‰๋ฉด ์ด๋“ฑ๋ณ€ ์‚ผ๊ฐํ˜• ๊ด€๊ณ„๋กœ๋ถ€ํ„ฐ ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ์ ์˜ ์ œ์•ฝ ์กฐ๊ฑด์„ ๋งŒ์กฑ์‹œํ‚ค๋Š” ๋ฐฉ์ •์‹๋“ค์„ ์„ธ์šด๋‹ค. ์ด ์ œ์•ฝ์‹๋“ค์€ ์›๋ž˜ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ๋ณ€์ˆ˜ ๊ณต๊ฐ„์—์„œ ํ‘œํ˜„๋˜๋ฉฐ, ์ด ๋ฐฉ์ •์‹๋“ค์˜ ํ•ด๋Š” ๋‹ค๋ณ€์ˆ˜ ๋ฐฉ์ •์‹์˜ ํ•ด๋ฅผ ๊ตฌํ•˜๋Š” solver๋ฅผ ์ด์šฉํ•˜์—ฌ ๊ตฌํ•œ๋‹ค. ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ๊ฒฝ์šฐ, ์›๋ž˜ ๊ณก๋ฉด์˜ ์ฃผ๊ณก๋ฅ  ์ค‘ ํ•˜๋‚˜๊ฐ€ ์˜คํ”„์…‹ ๋ฐ˜์ง€๋ฆ„์˜ ์—ญ์ˆ˜์™€ ๊ฐ™์„ ๋•Œ ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ๋ฒ•์„ ์ด ์ •์˜๊ฐ€ ๋˜์ง€ ์•Š๋Š” ํŠน์ด์ ์ด ์ƒ๊ธฐ๋Š”๋ฐ, ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์ด ์ด ๋ถ€๊ทผ์„ ์ง€๋‚  ๋•Œ๋Š” ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์˜ ๊ณ„์‚ฐ์ด ๋ถˆ์•ˆ์ •ํ•ด์ง„๋‹ค. ๋”ฐ๋ผ์„œ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์ด ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ํŠน์ด์  ๋ถ€๊ทผ์„ ์ง€๋‚  ๋•Œ๋Š” ์˜คํ”„์…‹ ๊ณก๋ฉด์„ ์ ‘์ด‰ ํ† ๋Ÿฌ์Šค๋กœ ์น˜ํ™˜ํ•˜์—ฌ ๋” ์•ˆ์ •๋œ ๋ฐฉ๋ฒ•์œผ๋กœ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์„ ๊ตฌํ•œ๋‹ค. ๊ณ„์‚ฐ๋œ ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์œผ๋กœ๋ถ€ํ„ฐ ๊ต์ฐจ ๊ณก์„ ์˜ xyzxyz-๊ณต๊ฐ„์—์„œ์˜ ๋ง๋‹จ ์ , ๊ฐ€์ง€ ๊ตฌ์กฐ ๋“ฑ์„ ๋ฐํžŒ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์€ ๋˜ํ•œ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ธฐ๋ฐ˜์˜ ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„  ๊ฒ€์ถœ์„ ๊ฐ€์†ํ™”ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ์€ ๊ธฐ์ € ๊ณก์„  ๋ฐ ๊ณก๋ฉด์„ ๋‹จ์ˆœํ•œ ๊ธฐํ•˜๋กœ ๊ฐ์‹ธ๊ณ  ๊ธฐํ•˜ ์—ฐ์‚ฐ์„ ์ˆ˜ํ–‰ํ•จ์œผ๋กœ์จ ๊ฐ€์†ํ™”์— ๊ธฐ์—ฌํ•œ๋‹ค. ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์„ ๊ตฌํ•˜๊ธฐ ์œ„ํ•˜์—ฌ, ๋ณธ ๋…ผ๋ฌธ์€ ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ๋ฅผ ๊ธฐ์ € ๊ณก๋ฉด์˜ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ๊ณผ ๊ธฐ์ € ๊ณก๋ฉด์˜ ๋ฒ•์„  ๊ณก๋ฉด์˜ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ์˜ ๊ตฌ์กฐ๋กœ๋ถ€ํ„ฐ ๊ณ„์‚ฐํ•˜๋ฉฐ ์ด๋•Œ ๊ฐ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ์˜ ๋‘๊ป˜๋ฅผ ๊ณ„์‚ฐํ•œ๋‹ค. ๋˜ํ•œ, ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ์ค‘์—์„œ ์‹ค์ œ ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ์— ๊ธฐ์—ฌํ•˜์ง€ ์•Š๋Š” ๋ถ€๋ถ„์„ ๊นŠ์€ ์žฌ๊ท€ ์ „์— ์ฐพ์•„์„œ ์ œ๊ฑฐํ•˜๋Š” ์—ฌ๋Ÿฌ ์กฐ๊ฑด๋“ค์„ ๋‚˜์—ดํ•œ๋‹ค. ํ•œํŽธ, ์ž๊ฐ€ ๊ต์ฐจ๊ฐ€ ์ œ๊ฑฐ๋œ ์˜คํ”„์…‹ ๊ณก์„  ๋ฐ ๊ณก๋ฉด์€ ๊ธฐ์ € ๊ณก์„  ๋ฐ ๊ณก๋ฉด์˜ ๋ณด๋กœ๋…ธ์ด ๊ตฌ์กฐ์™€ ๊นŠ์€ ๊ด€๋ จ์ด ์žˆ๋Š” ๊ฒƒ์ด ์•Œ๋ ค์ ธ ์žˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์ž์œ  ๊ณก๋ฉด์˜ ์—ฐ์†๋œ ์˜คํ”„์…‹ ๊ณก๋ฉด๋“ค๋กœ๋ถ€ํ„ฐ ์ž์œ  ๊ณก๋ฉด์˜ ๋ณด๋กœ๋…ธ์ด ๊ตฌ์กฐ๋ฅผ ์œ ์ถ”ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ํŠนํžˆ, ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„  ์ƒ์—์„œ ๋‚˜ํƒ€๋‚˜๋Š” ๊ฐ€์ง€ ์ ์ด๋‚˜ ๋ง๋‹จ ์ ๊ณผ ๊ฐ™์€ ํŠน์ด์ ๋“ค์ด ์ž์œ  ๊ณก๋ฉด์˜ ๋ณด๋กœ๋…ธ์ด ๊ตฌ์กฐ์—์„œ ์–ด๋–ป๊ฒŒ ํ•ด์„๋˜๋Š”์ง€ ์ œ์‹œํ•œ๋‹ค.1. Introduction 1 1.1 Background and Motivation 1 1.2 Research Objectives and Approach 7 1.3 Contributions and Thesis Organization 11 2. Preliminaries 14 2.1 Curve and Surface Representation 14 2.1.1 Bezier Representation 14 2.1.2 B-spline Representation 17 2.2 Differential Geometry of Curves and Surfaces 19 2.2.1 Differential Geometry of Curves 19 2.2.2 Differential Geometry of Surfaces 21 3. Previous Work 23 3.1 Offset Curves 24 3.2 Offset Surfaces 27 3.3 Offset Curves on Surfaces 29 4. Trimming Offset Curve Self-intersections 32 4.1 Experimental Results 35 5. Trimming Offset Surface Self-intersections 38 5.1 Constraint Equations for Offset Self-Intersections 38 5.1.1 Coplanarity Constraint 39 5.1.2 Equi-angle Constraint 40 5.2 Removing Trivial Solutions 40 5.3 Removing Normal Flips 41 5.4 Multivariate Solver for Constraints 43 5.A Derivation of f(u,v) 46 5.B Relationship between f(u,v) and Curvatures 47 5.3 Trimming Offset Surfaces 50 5.4 Experimental Results 53 5.5 Summary 57 6. Acceleration of trimming offset curves and surfaces 62 6.1 Motivation 62 6.2 Basic Approach 67 6.3 Trimming an Offset Curve using the BVH 70 6.4 Trimming an Offset Surface using the BVH 75 6.4.1 Offset Surface BVH 75 6.4.2 Finding Self-intersections in Offset Surface Using BVH 87 6.4.3 Tracing Self-intersection Curves 98 6.5 Experimental Results 100 6.6 Summary 106 7. Application of Trimming Offset Surfaces: 3D Voronoi Diagram 107 7.1 Background 107 7.2 Approach 110 7.3 Experimental Results 112 7.4 Summary 114 8. Conclusion 119 Bibliography iDocto

    The Bisector Surface of Rational Space Curves

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    Given a point and a rational curve in the plane, their bisector curve is rational [2]. However, in general, the bisector of two rational curves in the plane is not rational [3]. Given a point and a rational space curve, this paper shows that the bisector surface is a rational ruled surface. Moreover, given two rational space curves, we show that the bisector surface is rational (except for the degenerate case in which the two curves are coplanar). Key Words: Bisector, Voronoi surface, medial surface, skeleton. To appear in CVGIP: Graphical Models and Image Processing, Vol. 55, 1993. To appear in ACM Transactions on Graphics. Also available as Technical Report CIS-9619, Center for Intelligent Systems, Technion, Israel Institute of Technology, October 1996. This research was supported in part by the Korean Ministry of Science and Technology under Grants 96-NS-01-05-A-02-A and 96-NS-01-05-A-02-B of STEP 2000, by Korea Science and Engineering Foundation (KOSEF) under Grant 96-0..

    The bisector surface of rational space curves

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