475 research outputs found
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Farthest-Polygon Voronoi Diagrams
Given a family of k disjoint connected polygonal sites in general position
and of total complexity n, we consider the farthest-site Voronoi diagram of
these sites, where the distance to a site is the distance to a closest point on
it. We show that the complexity of this diagram is O(n), and give an O(n log^3
n) time algorithm to compute it. We also prove a number of structural
properties of this diagram. In particular, a Voronoi region may consist of k-1
connected components, but if one component is bounded, then it is equal to the
entire region
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Symmetry in Regular Polyhedra Seen as 2D Mรถbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams
This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Mรถbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first
would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures
์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
Layered Fields for Natural Tessellations on Surfaces
Mimicking natural tessellation patterns is a fascinating multi-disciplinary
problem. Geometric methods aiming at reproducing such partitions on surface
meshes are commonly based on the Voronoi model and its variants, and are often
faced with challenging issues such as metric estimation, geometric, topological
complications, and most critically parallelization. In this paper, we introduce
an alternate model which may be of value for resolving these issues. We drop
the assumption that regions need to be separated by lines. Instead, we regard
region boundaries as narrow bands and we model the partition as a set of smooth
functions layered over the surface. Given an initial set of seeds or regions,
the partition emerges as the solution of a time dependent set of partial
differential equations describing concurrently evolving fronts on the surface.
Our solution does not require geodesic estimation, elaborate numerical solvers,
or complicated bookkeeping data structures. The cost per time-iteration is
dominated by the multiplication and addition of two sparse matrices. Extension
of our approach in a Lloyd's algorithm fashion can be easily achieved and the
extraction of the dual mesh can be conveniently preformed in parallel through
matrix algebra. As our approach relies mainly on basic linear algebra kernels,
it lends itself to efficient implementation on modern graphics hardware.Comment: Natural tessellations, surface fields, Voronoi diagrams, Lloyd's
algorith
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