965 research outputs found
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
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
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
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
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
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
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
์คํ์ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ฒ์ถ ๋ฐ ์ ๊ฑฐ
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ)--์์ธ๋ํ๊ต ๋ํ์ :๊ณต๊ณผ๋ํ ์ปดํจํฐ๊ณตํ๋ถ,2020. 2. ๊น๋ช
์.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.์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ computer-aided design (CAD)์ computer-aided manufacturing (CAM)์์ ๋๋ฆฌ ์ด์ฉ๋๋ ์ฐ์ฐ๋ค ์ค ํ๋์ด๋ค.
ํ์ง๋ง ์ค์ฉ์ ์ธ ํ์ฉ์ ์ํด์๋ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์์ ์๊ธฐ๋ ์๊ฐ ๊ต์ฐจ๋ฅผ ์ฐพ๊ณ ์ด๋ฅผ ๊ธฐ์ค์ผ๋ก ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์์ ์๋์ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ๊ฐ๊น์ด ๋ถํ์ํ ์์ญ์ ์ ๊ฑฐํ์ฌ์ผํ๋ค.
๋ณธ ๋
ผ๋ฌธ์์๋ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์์ ์๊ธฐ๋ ์๊ฐ ๊ต์ฐจ๋ฅผ ๊ณ์ฐํ๊ณ , ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์์ ์๊ธฐ๋ ๋ถํ์ํ ์์ญ์ ์ ๊ฑฐํ๋ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ํ๋ค.
๋ณธ ๋
ผ๋ฌธ์ ์ฐ์ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ์ ๋ค๊ณผ ๊ทธ ๊ต์ฐจ์ ๋ค์ด ๊ธฐ์ธํ ์๋ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์ ๋ค์ด ์ด๋ฃจ๋ ํ๋ฉด ์ด๋ฑ๋ณ ์ผ๊ฐํ ๊ด๊ณ๋ก๋ถํฐ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ์ ์ ์ ์ฝ ์กฐ๊ฑด์ ๋ง์กฑ์ํค๋ ๋ฐฉ์ ์๋ค์ ์ธ์ด๋ค.
์ด ์ ์ฝ์๋ค์ ์๋ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ๋ณ์ ๊ณต๊ฐ์์ ํํ๋๋ฉฐ, ์ด ๋ฐฉ์ ์๋ค์ ํด๋ ๋ค๋ณ์ ๋ฐฉ์ ์์ ํด๋ฅผ ๊ตฌํ๋ solver๋ฅผ ์ด์ฉํ์ฌ ๊ตฌํ๋ค.
์คํ์
๊ณก๋ฉด์ ๊ฒฝ์ฐ, ์๋ ๊ณก๋ฉด์ ์ฃผ๊ณก๋ฅ ์ค ํ๋๊ฐ ์คํ์
๋ฐ์ง๋ฆ์ ์ญ์์ ๊ฐ์ ๋ ์คํ์
๊ณก๋ฉด์ ๋ฒ์ ์ด ์ ์๊ฐ ๋์ง ์๋ ํน์ด์ ์ด ์๊ธฐ๋๋ฐ,
์คํ์
๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ด ์ด ๋ถ๊ทผ์ ์ง๋ ๋๋ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ ๊ณ์ฐ์ด ๋ถ์์ ํด์ง๋ค.
๋ฐ๋ผ์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ด ์คํ์
๊ณก๋ฉด์ ํน์ด์ ๋ถ๊ทผ์ ์ง๋ ๋๋ ์คํ์
๊ณก๋ฉด์ ์ ์ด ํ ๋ฌ์ค๋ก ์นํํ์ฌ ๋ ์์ ๋ ๋ฐฉ๋ฒ์ผ๋ก ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ ๊ตฌํ๋ค.
๊ณ์ฐ๋ ์คํ์
๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ผ๋ก๋ถํฐ ๊ต์ฐจ ๊ณก์ ์ -๊ณต๊ฐ์์์ ๋ง๋จ ์ , ๊ฐ์ง ๊ตฌ์กฐ ๋ฑ์ ๋ฐํ๋ค.
๋ณธ ๋
ผ๋ฌธ์ ๋ํ ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ ๊ธฐ๋ฐ์ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ๊ฒ์ถ์ ๊ฐ์ํํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค.
๋ฐ์ด๋ฉ ๋ณผ๋ฅจ์ ๊ธฐ์ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ๋จ์ํ ๊ธฐํ๋ก ๊ฐ์ธ๊ณ ๊ธฐํ ์ฐ์ฐ์ ์ํํจ์ผ๋ก์จ ๊ฐ์ํ์ ๊ธฐ์ฌํ๋ค.
์คํ์
๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์ ๊ตฌํ๊ธฐ ์ํ์ฌ, ๋ณธ ๋
ผ๋ฌธ์ ์คํ์
๊ณก๋ฉด์ ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ๋ฅผ ๊ธฐ์ ๊ณก๋ฉด์ ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ๊ณผ ๊ธฐ์ ๊ณก๋ฉด์ ๋ฒ์ ๊ณก๋ฉด์ ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ์ ๊ตฌ์กฐ๋ก๋ถํฐ ๊ณ์ฐํ๋ฉฐ ์ด๋ ๊ฐ ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ์ ๋๊ป๋ฅผ ๊ณ์ฐํ๋ค.
๋ํ, ๋ฐ์ด๋ฉ ๋ณผ๋ฅจ ์ค์์ ์ค์ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ์ ๊ธฐ์ฌํ์ง ์๋ ๋ถ๋ถ์ ๊น์ ์ฌ๊ท ์ ์ ์ฐพ์์ ์ ๊ฑฐํ๋ ์ฌ๋ฌ ์กฐ๊ฑด๋ค์ ๋์ดํ๋ค.
ํํธ, ์๊ฐ ๊ต์ฐจ๊ฐ ์ ๊ฑฐ๋ ์คํ์
๊ณก์ ๋ฐ ๊ณก๋ฉด์ ๊ธฐ์ ๊ณก์ ๋ฐ ๊ณก๋ฉด์ ๋ณด๋ก๋
ธ์ด ๊ตฌ์กฐ์ ๊น์ ๊ด๋ จ์ด ์๋ ๊ฒ์ด ์๋ ค์ ธ ์๋ค.
๋ณธ ๋
ผ๋ฌธ์์๋ ์์ ๊ณก๋ฉด์ ์ฐ์๋ ์คํ์
๊ณก๋ฉด๋ค๋ก๋ถํฐ ์์ ๊ณก๋ฉด์ ๋ณด๋ก๋
ธ์ด ๊ตฌ์กฐ๋ฅผ ์ ์ถํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค.
ํนํ, ์คํ์
๊ณก๋ฉด์ ์๊ฐ ๊ต์ฐจ ๊ณก์ ์์์ ๋ํ๋๋ ๊ฐ์ง ์ ์ด๋ ๋ง๋จ ์ ๊ณผ ๊ฐ์ ํน์ด์ ๋ค์ด ์์ ๊ณก๋ฉด์ ๋ณด๋ก๋
ธ์ด ๊ตฌ์กฐ์์ ์ด๋ป๊ฒ ํด์๋๋์ง ์ ์ํ๋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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