965 research outputs found

    A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

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    We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-conjugate spherical CR uniformizations of the figure eight knot complement

    The Relation Between Offset and Conchoid Constructions

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    The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided

    Complex Hyperbolic Structures on Disc Bundles over Surfaces

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    We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\chi+e)=3\tau; satisfy the inequality \chi/2<e; and induce discrete and faithful representations \pi_1\Sigma->PU(2,1) with fractional Toledo invariant; where {\chi} is the Euler characteristic of \Sigma, e denotes the Euler number of M, and {\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\chi+e)=3\tau, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.Comment: 52 pages, 12 pictures, 10 tables, 20 references. Changes: final versio

    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

    Algebro-geometric analysis of bisectors of two algebraic plane curves

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    In this paper, a general theoretical study, from the perspective of the algebraic geometry, of the untrimmed bisector of two real algebraic plane curves is presented. The curves are considered in C2, and the real bisector is obtained by restriction to R2. If the implicit equations of the curves are given, the equation of the bisector is obtained by projection from a variety contained in C7, called the incidence variety, into C2. It is proved that all the components of the bisector have dimension 1. A similar method is used when the curves are given by parametrizations, but in this case, the incidence variety is in C5. In addition, a parametric representation of the bisector is introduced, as well as a method for its computation. Our parametric representation extends the representation in Farouki and Johnstone (1994b) to the case of rational curves

    On odd-periodic orbits in complex planar billiards

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    The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory

    ์˜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
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