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Three Moves on Signed Surface Triangulations

Shalom Eliahou, Sylvain Gravier, Charles Payan
2002 Journal of combinatorial theory. Series B (Print)  
We introduce three moves on signed surface triangulations which preserve the Heawood property.  ...  We consider finite triangulations of surfaces with signs attached to the faces.  ...  INTRODUCTION Let G=(V, E) be a triangulation of a closed surface S.  ... 
doi:10.1006/jctb.2001.2066 fatcat:mug4rnrvbrbunlregmxucd6nam

Extending triangulations of the 2-sphere to the 3-disk preserving a 4-coloring

Rui Carpentier
2012 Pacific Journal of Mathematics  
We prove that any triangulation of a 2-dimensional sphere with a proper strict 4-coloring on its vertices can be seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices in  ...  /or divisions of a triangle into three triangles (by adding a vertex v inside the triangle and edges joining v to the vertices of the triangle) with the opposite sign (move II). move I: + + ←→ - - move  ...  on their surface, and (5) gluing two triangulated cones of 2-disks along two adjacent shared triangles on their surface.  ... 
doi:10.2140/pjm.2012.257.257 fatcat:pmwdjqkoazfapppfvwy6dvs2ee

Extending triangulations of the 2-sphere to the 3-disk preserving a 4-coloring [article]

Rui Pedro Carpentier
2011 arXiv   pre-print
In this paper we prove that any triangulation of a 2-dimensional sphere with a strict 4-coloring on its vertices can seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices  ...  of signed diagonal flips (move I) and/or divisions of a triangle into three triangles (by adding a vertex v inside the triangle and edges joining v to the vertices of the triangle) with opposite signs  ...  a shared triangle on their surface, and 5) glueing two triangulated cones of 2-disks along two adjacent shared triangles on their surface.  ... 
arXiv:1101.5537v2 fatcat:4qpsvo6hbbdjlhijwsm6oot2u4

Veering triangulations admit strict angle structures

Craig D Hodgson, J Hyam Rubinstein, Henry Segerman, Stephan Tillmann
2011 Geometry and Topology  
Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure.  ...  We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures.  ...  Normal surfaces and angle structures A normal surface in a triangulation T of M is a surface that intersects each tetrahedron in a collection of normal triangles and normal quadrilaterals (quads) as shown  ... 
doi:10.2140/gt.2011.15.2073 fatcat:gxlpwy5ofzg7jejms4q6n2bbyy

State sum construction of two-dimensional topological quantum field theories on spin surfaces [article]

Sebastian Novak, Ingo Runkel
2015 arXiv   pre-print
The behaviour of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces.  ...  Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions  ...  A 2d-Pachner move on a complex C changes at most three adjacent triangles.  ... 
arXiv:1402.2839v2 fatcat:73t3tkvrzvh67jtdiee67mlrha

Combined signed distance calculation algorithm for numerical simulation of physical processes and visualization of solid bodies movement
Комбинированный алгоритм вычисления расстояния со знаком для задач численного моделирования физических процессов и визуализации движения твердых тел

S.A. Soukov
2020 Scientific Visualization  
The article deals with the problem of initializing the field of the signed distance function to the surface of a moving solid body of arbitrary shape.  ...  The idea is to interpolate the function over the cells of an adaptive grid with local switching to find the distance to surface triangulation.  ...  Signed distance to triangulation Spatial triangulation is one of the main approaches for body boundaries discretization.  ... 
doi:10.26583/sv.12.5.08 fatcat:5ppeaeax5vflpfirilbztx6yra

Freezing a fluid random surface

C. F. Baillie, D. A. Johnston
1993 Physical Review D, Particles and fields  
Such a term thus provides a way of interpolating between dynamically triangulated (ie fluid) and crystalline random surfaces.  ...  We investigate a dynamically triangulated random surface action that consists of a gaussian term plus the modulus of the intrinsic scalar curvature.  ...  Even on these larger surfaces S f reeze is most effective at freezing out the flip moves on the surface, which reassures us that we are not seeing a finite size effect.  ... 
doi:10.1103/physrevd.48.5025 pmid:10016160 fatcat:xtdb3yg7fbcjphhbdex7bshcae

Taut ideal triangulations of 3–manifolds

Marc Lackenby
2000 Geometry and Topology  
A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions  ...  For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai's result that the singular genus of a knot in the 3-sphere is equal to its genus.  ...  The evaluation of the cohomology class e on the oriented surface F is equal to the number of saddle singularities, counted with sign, the sign depending on whether the transverse orientations of F and  ... 
doi:10.2140/gt.2000.4.369 fatcat:nbqzbe2zgbdihduvuamcccp2k4

The 3D Ising model represented as random surfaces

J. Ambjørn, A. Sedrakyan, G. Thorleifsson
1993 Physics Letters B  
We consider a random surface representation of the three-dimensional Ising model.The model exhibit scaling behaviour and a new critical index which relates _string for the bosonic string to the exponent  ...  From this point of view it seems reasonable to use triangulated piecewise linear surfaces immersed in three dimensions as our statistical ensemble of surfaces.  ...  standard moves.  ... 
doi:10.1016/0370-2693(93)91440-x fatcat:qjgxommvhnemhf4pr7l3iq7r64

Survey of Surface Reconstruction Algorithms

Amin Alqudah
2014 Journal of Signal and Information Processing  
In this paper, we discuss some computational geometry preliminaries, and then move on to a summary of some different techniques used to address the surface reconstruction problem.  ...  To date, the primary application of surface reconstruction algorithms has been in computer graphics, where physical models are digitized in three dimensions with laser range scanners or mechanical digitizing  ...  three points on P, or those with exactly two faces, five edges and four points on P."  ... 
doi:10.4236/jsip.2014.53009 fatcat:effryesmyzcvfony4oeoid5pvi

Convex contouring of volumetric data

Tao Ju, Scott Schaefer, Joe Warren
2003 The Visual Computer  
Unlike Marching Cubes or other cell-based algorithms, the proposed polygonization generates convex negative space inside individual cells, enabling fast collision detection on the triangulated isosurface  ...  The algorithm is driven by an automatically constructed look-up table that stores compact decision trees by sign configurations.  ...  Note that the three vertices of the triangle can move only along three fixed cell edges, this half-space is always contained in the union of the halfspaces formed when the three vertices are at the ends  ... 
doi:10.1007/s00371-003-0216-0 fatcat:ocwagabblnfmlm5m3jghnn67ua

Geometric torsions and invariants of manifolds with a triangulated boundary

I. G. Korepanov
2009 Theoretical and mathematical physics  
We use geometric torsions to construct invariants for a manifold with a triangulated boundary.  ...  These invariants can be naturally united in a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector.  ...  Consider a chain of Pachner moves transforming the old boundary triangulation into the new one.  ... 
doi:10.1007/s11232-009-0006-6 fatcat:k2erhhwzrjcjnkr5b2zkauvfzi

Smooth random surfaces from tight immersions?

C. F. Baillie, D. A. Johnston
1994 Physical Review D, Particles and fields  
We investigate actions for dynamically triangulated random surfaces that consist of a gaussian or area term plus the modulus of the gaussian curvature and compare their behavior with both gaussian plus  ...  triangulations T means that we have, in effect, a fluid surface.  ...  To get the equivalent of 1 2π M |K|dS ≥ 4 − χ(M ) (7) with the equality holding when the surface is imbedded as a convex surface in three dimensional space.  ... 
doi:10.1103/physrevd.49.4139 pmid:10017419 fatcat:aq76pwxi2vhi7ax6z3geiuvrqa

Automatic reconstruction of surfaces and scalar fields from 3D scans

Chandrajit L. Bajaj, Fausto Bernardini, Guoliang Xu
1995 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques - SIGGRAPH '95  
The simplicity and efficiency of our approach is based on several novel uses of appropriate sub-structures (alpha shapes) of a three-dimensional Delaunay Triangulation, its dual the three-dimensional Voronoi  ...  A possible application is the rapid computer model reconstruction of an existing part or prototype from a three dimensional (3D) points scan of its surface.  ...  Such a tetrahedralization is called a (3D) Delaunay triangulation and, under non degeneracy assumptions (no three points on a line, etc.) it is unique.  ... 
doi:10.1145/218380.218424 dblp:conf/siggraph/BajajBX95 fatcat:qwcewxvljnefto7k276mofhnym

A new combinatorial class of 3-manifold triangulations [article]

Feng Luo, Stephan Tillmann
2015 arXiv   pre-print
As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.  ...  We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares.  ...  An example is given by the 4 tetrahedron triangulation of S 3 defined by the following face pairings: can be simplified to a 1-tetrahedron triangulation of S 3 via a sequence of three Pachner moves.  ... 
arXiv:1312.5087v2 fatcat:vg36w7plkfg63fgb3ehjc7k4ly
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