1,947 research outputs found
Polyhedral Geometry and the Two-plane Parameterization
Recently the light-ïŹeld and lumigraph systems have been proposed as general methods of representing the visual information present in a scene. These methods represent this information as a 4D function of light over the domain of directed lines. These systems use the intersection points of the lines on two planes to parameterize the lines in space.
This paper explores the structure of the two-plane parameterization in detail. In particular we analyze the association between the geometry of the scene and subsets of the 4D data. The answers to these questions are essential to understanding the relationship between a lumigraph, and the geometry that it attempts to represent. This knowledge is potentially important for a variety of applications such as extracting shape from lumigraph data, and lumigraph compression.Engineering and Applied Science
Minimal surfaces from circle patterns: Geometry from combinatorics
We suggest a new definition for discrete minimal surfaces in terms of sphere
packings with orthogonally intersecting circles. These discrete minimal
surfaces can be constructed from Schramm's circle patterns. We present a
variational principle which allows us to construct discrete analogues of some
classical minimal surfaces. The data used for the construction are purely
combinatorial--the combinatorics of the curvature line pattern. A
Weierstrass-type representation and an associated family are derived. We show
the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended
introduction. Minor changes in presentation. v3: revision according to the
referee's suggestions, improved & expanded exposition, references added,
minor mistakes correcte
The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion
We deal with a notion of weak binormal and weak principal normal for
non-smooth curves of the Euclidean space with finite total curvature and total
absolute torsion. By means of piecewise linear methods, we first introduce the
analogous notation for polygonal curves, where the polarity property is
exploited, and then make use of a density argument. Both our weak binormal and
normal are rectifiable curves which naturally live in the projective plane. In
particular, the length of the weak binormal agrees with the total absolute
torsion of the given curve. Moreover, the weak normal is the vector product of
suitable parameterizations of the tangent indicatrix and of the weak binormal.
In the case of smooth curves with positive curvature, the weak binormal and
normal yield (up to a lifting) the classical notions of binormal and normal.Comment: 18 pages, 2 figure
Implicitization of surfaces via geometric tropicalization
In this paper we further develop the theory of geometric tropicalization due
to Hacking, Keel and Tevelev and we describe tropical methods for
implicitization of surfaces. More precisely, we enrich this theory with a
combinatorial formula for tropical multiplicities of regular points in
arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev
regarding sufficient combinatorial conditions to compute tropical varieties via
geometric tropicalization. Using these two results, we extend previous work of
Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and
we provide methods for approaching the non-generic cases.Comment: 20 pages, 6 figures. Mayor reorganization and exposition improved.
The results on geometric tropicalization have been extended to any dimension.
In particular, Conjecture 2.8 is now Theorem 2.
Discrete curvature approximations and segmentation of polyhedral surfaces
The segmentation of digitized data to divide a free form surface into patches is one of the key steps required to perform a reverse engineering process of an object. To this end, discrete curvature approximations are introduced as the basis of a segmentation process that lead to a decomposition of digitized data into areas that will help the construction of parametric surface patches. The approach proposed relies on the use of a polyhedral representation of the object built from the digitized data input. Then, it is shown how noise reduction, edge swapping techniques and adapted remeshing schemes can participate to different preparation phases to provide a geometry that highlights useful characteristics for the segmentation process. The segmentation process is performed with various approximations of discrete curvatures evaluated on the polyhedron produced during the preparation phases. The segmentation process proposed involves two phases: the identification of characteristic polygonal lines and the identification of polyhedral areas useful for a patch construction process. Discrete curvature criteria are adapted to each phase and the concept of invariant evaluation of curvatures is introduced to generate criteria that are constant over equivalent meshes. A description of the segmentation procedure is provided together with examples of results for free form object surfaces
A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6
The invention of the "dual resonance model" N-point functions BN motivated
the development of current string theory. The simplest of these models, the
four-point function B4, is the classical Euler Beta function. Many standard
methods of complex analysis in a single variable have been applied to elucidate
the properties of the Euler Beta function, leading, for example, to analytic
continuation formulas such as the contour-integral representation obtained by
Pochhammer in 1890. Here we explore the geometry underlying the dual five-point
function B5, the simplest generalization of the Euler Beta function. Analyzing
the B5 integrand leads to a polyhedral structure for the five-crosscap surface,
embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120
in PGL(6). We find a Pochhammer-like representation for B5 that is a contour
integral along a surface of genus five. The symmetric embedding of the
five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the
surface of genus four in R6 that has a polyhedral structure with 24 pentagonal
faces and a symmetry group of order 240 in O(6). The methods appear
generalizable to all N, and the resulting structures seem to be related to
associahedra in arbitrary dimensions.Comment: 43 pages and 44 figure
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