9,371 research outputs found
Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations
We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1989). Our attention is focused on fields with nodes of order two,
which can be regarded as continuum ensembles of non-intersecting contours in
the plane, sharing a number of features with the two-dimensional Ising model.
We introduce non-homogeneous version of polygonal fields in anisotropic
enviroment. For these fields we provide a class of new graphical constructions
and random dynamics. These include a generalised dynamic representation,
generalised and defective disagreement loop dynamics as well as a generalised
contour birth and death dynamics. Next, we use these constructions as tools to
obtain new exact results on the geometry of higher order correlations of
polygonal Markov fields in their consistent regime.Comment: 54 page
Puncture of gravitating domain walls
We investigate the semi-classical instability of vacuum domain walls to
processes where the domain walls decay by the formation of closed string loop
boundaries on their worldvolumes. Intuitively, a wall which is initially
spherical may `pop', so that a hole corresponding to a string boundary
component on the wall, may form. We find instantons, and calculate the rates,
for such processes. We show that after puncture, the hole grows exponentially
at the same rate that the wall expands. It follows that the wall is never
completely thermalized by a single expanding hole; at arbitrarily late times
there is still a large, thin shell of matter which may drive an exponential
expansion of the universe. We also study the situation where the wall is
subjected to multiple punctures. We find that in order to completely annihilate
the wall by this process, at least four string loops must be nucleated. We
argue that this process may be relevant in certain brane-world scenarios, where
the universe itself is a domain wall.Comment: 13 pages REVTeX, 3 .ps figures, added some references - version to
appear in Physics Letters
Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter
We study the entanglement entropy of gapped phases of matter in three spatial
dimensions. We focus in particular on size-independent contributions to the
entropy across entanglement surfaces of arbitrary topologies. We show that for
low energy fixed-point theories, the constant part of the entanglement entropy
across any surface can be reduced to a linear combination of the entropies
across a sphere and a torus. We first derive our results using strong
sub-additivity inequalities along with assumptions about the entanglement
entropy of fixed-point models, and identify the topological contribution by
considering the renormalization group flow; in this way we give an explicit
definition of topological entanglement entropy in (3+1)D,
which sharpens previous results. We illustrate our results using several
concrete examples and independent calculations, and show adding "twist" terms
to the Lagrangian can change in (3+1)D. For the generalized
Walker-Wang models, we find that the ground state degeneracy on a 3-torus is
given by in terms of the topological
entanglement entropy across a 2-torus. We conjecture that a similar
relationship holds for Abelian theories in dimensional spacetime, with
the ground state degeneracy on the -torus given by
.Comment: 34 pages, 16 figure
ConTesse: Accurate Occluding Contours for Subdivision Surfaces
This paper proposes a method for computing the visible occluding contours of
subdivision surfaces. The paper first introduces new theory for contour
visibility of smooth surfaces. Necessary and sufficient conditions are
introduced for when a sampled occluding contour is valid, that is, when it may
be assigned consistent visibility. Previous methods do not guarantee these
conditions, which helps explain why smooth contour visibility has been such a
challenging problem in the past. The paper then proposes an algorithm that,
given a subdivision surface, finds sampled contours satisfying these
conditions, and then generates a new triangle mesh matching the given occluding
contours. The contours of the output triangle mesh may then be rendered with
standard non-photorealistic rendering algorithms, using the mesh for visibility
computation. The method can be applied to any triangle mesh, by treating it as
the base mesh of a subdivision surface.Comment: Accepted to ACM Transactions on Graphics (TOG
Numerical equilibrium analysis for structured consumer resource models
In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs
Minimal stretch maps between hyperbolic surfaces
This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces
analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps
(minimal stretch maps) and geodesics for the `Lipschitz metric' are
constructed. The extremal Lipschitz constant equals the maximum ratio of
lengths of measured laminations, which is attained with probability one on a
simple closed curve. Cataclysms are introduced, generalizing earthquakes by
permitting more violent shearing in both directions along a fault. Cataclysms
provide useful coordinates for Teichmuller space that are convenient for
computing derivatives of geometric function in Teichmuller space and measured
lamination space.Comment: 53 pages, 11 figures, version of 1986 preprin
BVH와 토러스 패치를 이용한 곡면 교차곡선 연산
학위논문(박사) -- 서울대학교대학원 : 공과대학 컴퓨터공학부, 2021.8. 김명수.두 변수를 가지는 B-스플라인 자유곡면의 곡면간 교차곡선과 자가 교차곡선, 그리고 오프셋 곡면의 자가 교차곡선을 구하는 효율적이고 안정적인 알고리즘을 개발하는 새로운 접근 방법을 제시한다. 새로운 방법은 최하단 노드에 최대 접촉 토러스를 가지는 복합 바운딩 볼륨 구조에 기반을 두고 있다.
이 바운딩 볼륨 구조는 곡면간 교차나 자가 교차가 발생할 가능성이 있는 작은 곡면 조각 쌍들의 기하학적 검색을 가속화한다. 최대 접촉 토러스는 자기가 근사한 C2-연속 자유곡면과 2차 접촉을 가지므로 주어진 곡면에서 다양한 기하 연산의 정밀도를 향상시키는데 필수적인 역할을 한다.
효율적인 곡면간 교차곡선 계산을 지원하기 위해, 미리 만들어진, 최하단 노드에 최대 접촉 토러스가 있으며 구형구면 트리를 가지는 복합 이항 바운딩 볼륨 구조를 설계하였다.
최대 접촉 토러스는 거의 모든 곳에서 접선교차가 발생하는, 자명하지 않은 곡면간 교차곡선 계산 문제에서도 효율적이고 안정적인 결과를 제공한다.
곡면의 자가 교차 곡선을 구하는 문제는 주로 마이터 점 때문에 곡면간 교차곡선을 계산하는 것 보다 훨씬 더 어렵다. 자가 교차 곡면은 마이터 점 부근에서 법선 방향이 급격히 변하며, 마이터 점은 자가 교차 곡선의 끝점에 위치한다.
따라서 마이터 점은 자가 교차 곡면의 기하 연산 안정성에 큰 문제를 일으킨다. 마이터 점을 안정적으로 감지하여 자가 교차 곡선의 계산을 용이하게 하기 위해, 자유곡면을 위한 복합 바운딩 볼륨 구조에 적용할 수 있는 삼항 트리 구조를 제시한다. 특히, 두 변수를 가지는 곡면의 매개변수영역에서 마이터 점을 충분히 작은 사각형으로 감싸는 특별한 표현 방법을 제시한다.
접선교차와 마이터 점을 가지는, 아주 자명하지 않은 자유곡면 예제를 사용하여 새 방법이 효과적임을 입증한다. 모든 실험 예제에서, 기하요소들의 정확도는 하우스도르프 거리의 상한보다 낮음을 측정하였다.We present a new approach to the development of efficient and stable algorithms for intersecting freeform surfaces, including the surface-surface-intersection and the surface self-intersection of bivariate rational B-spline surfaces. Our new approach is based on a hybrid Bounding Volume Hierarchy(BVH) that stores osculating toroidal patches in the leaf nodes.
The BVH structure accelerates the geometric search for the potential pairs of local surface patches that may intersect or self-intersect. Osculating toroidal patches have second-order contact with C2-continuous freeform surfaces that they approximate, which plays an essential role in improving the precision of various geometric operations on the given surfaces.
To support efficient computation of the surface-surface-intersection curve, we design a hybrid binary BVH that is basically a pre-built Rectangle-Swept Sphere(RSS) tree enhanced with osculating toroidal patches in their leaf nodes.
Osculating toroidal patches provide efficient and robust solutions to the problem even in the non-trivial cases of handling two freeform surfaces intersecting almost tangentially everywhere.
The surface self-intersection problem is considerably more difficult than computing the intersection of two different surfaces, mainly due to the existence of miter points.
A self-intersecting surface changes its normal direction dramatically around miter points, located at the open endpoints of the self-intersection curve.
This undesirable behavior causes serious problems in the stability of geometric algorithms on self-intersecting surfaces. To facilitate surface self-intersection computation with a stable detection of miter points, we propose a ternary tree structure for the hybrid BVH of freeform surfaces. In particular, we propose a special representation of miter points using sufficiently small quadrangles in the parameter domain of bivariate surfaces and expand ideas to offset surfaces.
We demonstrate the effectiveness of the proposed new approach using some highly non-trivial examples of freeform surfaces with tangential intersections and miter points. In all the test examples, the closeness of geometric entities is measured under the Hausdorff distance upper bound.Chapter 1 Introduction 1
1.1 Background 1
1.2 Surface-Surface-Intersection 5
1.3 Surface Self-Intersection 8
1.4 Main Contribution 12
1.5 Thesis Organization 14
Chapter 2 Preliminaries 15
2.1 Differential geometry of surfaces 15
2.2 Bezier curves and surfaces 17
2.3 Surface approximation 19
2.4 Torus 21
2.5 Summary 24
Chapter 3 Previous Work 25
3.1 Surface-Surface-Intersection 25
3.2 Surface Self-Intersection 29
3.3 Summary 32
Chapter 4 Bounding Volume Hierarchy for Surface Intersections 33
4.1 Binary Structure 33
4.1.1 Hierarchy of Bilinear Surfaces 34
4.1.2 Hierarchy of Planar Quadrangles 37
4.1.3 Construction of Leaf Nodes with Osculating Toroidal Patches 41
4.2 Ternary Structure 44
4.2.1 Miter Points 47
4.2.2 Leaf Nodes 50
4.2.3 Internal Nodes 51
4.3 Summary 56
Chapter 5 Surface-Surface-Intersection 57
5.1 BVH Traversal 58
5.2 Construction of SSI Curve Segments 59
5.2.1 Merging SSI Curve Segments with G1-Biarcs 60
5.2.2 Measuring the SSI Approximation Error Using G1-Biarcs 63
5.3 Tangential Intersection 64
5.4 Summary 65
Chapter 6 Surface Self-Intersection 67
6.1 Preprocessing 68
6.2 BVH Traversal 69
6.3 Construction of Intersection Curve Segments 70
6.4 Summary 72
Chapter 7 Trimming Offset Surfaces with Self-Intersection Curves 74
7.1 Offset Surface and Ternary Hybrid BVH 75
7.2 Preprocessing 77
7.3 Merging Intersection Curve Segments 81
7.4 Summary 84
Chapter 8 Experimental Results 85
8.1 Surface-Surface-Intersection 85
8.2 Surface Self-Intersection 97
8.2.1 Regular Surfaces 97
8.2.2 Offset Surfaces 100
Chapter 9 Conclusion 106
Bibliography 108
초록 120박
Self-Intersection Computation for Freeform Surfaces Based on a Regional Representation Scheme for Miter Points
We present an efficient and robust algorithm for computing the self-intersection of a freeform surface, based
on a special representation of miter points, using sufficiently small quadrangles in the parameter domain.
A self-intersecting surface changes its normal direction quite dramatically around miter points, located at
the open endpoints of the self-intersection curve. This undesirable behavior causes serious problems in the
stability of geometric algorithms on the surface. To facilitate a stable detection of miter points, we employ
osculating toroidal patches and their intersections, and consider a gradual change to degenerate intersections
as a signal for the detection of miter points. The exact location of each miter point is bounded by a tiny ball
in the Euclidean space and is also represented as a small quadrangle in the parameter space. The surface
self-intersection curve is then constructed, using a hybrid Bounding Volume Hierarchy (BVH), where the
leaf nodes contain osculating toroidal patches and miter quadrangles. We demonstrate the effectiveness of
our approach by using test examples of computing the self-intersection of freeform surfaces
Intermittent heating in the solar corona employing a 3D MHD model
We investigate the spatial and temporal evolution of the heating of the
corona of a cool star such as our Sun in a three-dimensional
magneto-hydrodynamic (3D MHD) model. We solve the 3D MHD problem numerically in
a box representing part of the (solar) corona. The energy balance includes
Spitzer heat conduction along the magnetic field and optically thin radiative
losses. The self-consistent heating mechanism is based on the braiding of
magnetic field lines rooted in the convective photosphere. Magnetic stress
induced by photospheric motions leads to currents in the atmosphere which heat
the corona through Ohmic dissipation. While the horizontally averaged
quantities, such as heating rate, temperature or density, are relatively
constant in time, the simulated corona is highly variable and dynamic, on
average reaching temperatures and densities as found in observations. The
strongest heating per particle is found in the transition region from the
chromosphere to the corona. The heating is concentrated in current sheets
roughly aligned with the magnetic field and is transient in time and space.
This supports the idea that numerous small heating events heat the corona,
often referred to a nanoflares
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