339 research outputs found
Exact Geosedics and Shortest Paths on Polyhedral Surface
We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.National Institute for Biomedical Imaging and Bioengineering (R01 EB001550
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
We extend the notion of star unfolding to be based on a quasigeodesic loop Q
rather than on a point. This gives a new general method to unfold the surface
of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut
along one shortest path from each vertex of P to Q, and cut all but one segment
of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and
adds references. v3 improves two figures and their captions. New version v4
offers a completely different proof of non-overlap in the quasigeodesic loop
case, and contains several other substantive improvements. This version is 23
pages long, with 15 figure
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
A framework for working with digitized cultural heritage artefacts
In this paper, we present our work in designing, implementing, and evaluating a set of 3D interactive spatial measurement tools in the context of Cultural Heritage Toolbox (CH Toolbox), a framework for computer-aided cultural heritage research. Our application utilizes a bi-manual, spaceball and mouse driven user interface to help the user manage visualized 3D models digitized from real artifacts. We have developed a virtual radius estimator, useful for analyzing incomplete pieces of radial artifacts, and a virtual tape measure, useful in measurement of geodesic distances between two points on the surface of an artifact. We tested the tools on the special case of pottery analysis
Geodesic computation on implicit surfaces
Geodesics have a wide range of applications in CAD, shape design and machine learning. Current research on geodesic computation focuses primarily on parametric surfaces and mesh representations. There is little work on implicit surfaces. In this paper, we present a novel algorithm able
to compute the exact geodesics on implicit surfaces. Although the existing Fast Marching Method can generate a geodesic path on a Cartesian grid that envelopes the implicit surface in question, this method, as well as
other existing methods, is unable to compute a geodesic on the original surface. The computed geodesic path is actually a polyline offsetting from the surface. Our approach provides a solution to two existing fundamental
problems, which are (1) to produce a Cartesian grid that can tightly embed the implicit surface concerned, which remains challenging; and (2) to formulate exact geodesics on the original implicit surface itself. Our algorithm consists of two steps, Cartesian grid based geodesic
computation and geodesic correction. The later corrects an approximate geodesic path so that it can be on the implicit surface. In addition, in comparison with other existing work, our methods can handle both low dimensional and high dimensional surfaces (hyper-surfaces)
A Note on the Unsolvability of the Weighted Region Shortest Path Problem
Let S be a subdivision of the plane into polygonal regions, where each region
has an associated positive weight. The weighted region shortest path problem is
to determine a shortest path in S between two points s, t in R^2, where the
distances are measured according to the weighted Euclidean metric-the length of
a path is defined to be the weighted sum of (Euclidean) lengths of the
sub-paths within each region. We show that this problem cannot be solved in the
Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one
can compute exactly any number that can be obtained from the rationals Q by
applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for
any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's
technique.Comment: 6 pages, 1 figur
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