1,030 research outputs found
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise
to a circle pattern, consisting of the circles which are circumscribed to the
facets. We treat the problem whether there exists a Delaunay cell decomposition
for a given (topological) cell decomposition and given intersection angles of
the circles, whether it is unique and how it may be constructed. Somewhat more
generally, we allow cone-like singularities in the centers and intersection
points of the circles. We prove existence and uniqueness theorems for the
solution of the circle pattern problem using a variational principle. The
functionals (one for the euclidean, one for the hyperbolic case) are convex
functions of the radii of the circles. The analogous functional for the
spherical case is not convex, hence this case is treated by stereographic
projection to the plane. From the existence and uniqueness of circle patterns
in the sphere, we derive a strengthened version of Steinitz' theorem on the
geometric realizability of abstract polyhedra.
We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and
Rivin for circle packings and circle patterns from our variational principles.
In the case of Br\"agger's and Rivin's functionals. Leibon's functional for
hyperbolic circle patterns cannot be derived directly from our functionals. But
we construct yet another functional from which both Leibon's and our
functionals can be derived.
We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics
Minkowski Actions of Quaternion Sets and their Applications
Applications of Laguerre geometry to computer aided geometric design are presented, realized through Minkowski actions. Basic Laguerre geometry is first discussed. Then quaternions, set multiplication, and the use of quaternion sets to define Minkowski actions are described and used to achieve results used in geometric design
Tangent-ball techniques for shape processing
Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes. Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing. Many applications of shape processing can be found in the entertainment and medical industries.
In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes.
We propose a set of ball-based operators and discuss their properties, implementations, and applications. We divide the group of ball-based operations into unary and binary as follows:
Unary operators include:
* Identifying details (sharp, salient features, constrictions)
* Smoothing shapes by removing such details, replacing them by fillets and roundings
* Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures
Binary operators include:
* Measuring the local discrepancy between two shapes
* Computing the average of two shapes
* Computing point-to-point correspondence between two shapes
* Computing circular trajectories between corresponding points that meet both shapes at right angles
* Using these trajectories to support smooth morphing (inbetweening)
* Using a curve morph to construct surfaces that interpolate between contours on consecutive slices
The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis. These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon
Extracting curve-skeletons from digital shapes using occluding contours
Curve-skeletons are compact and semantically relevant shape descriptors, able to summarize both topology and pose of a wide range of digital objects. Most of the state-of-the-art algorithms for their computation rely on the type of geometric primitives used and sampling frequency. In this paper we introduce a formally sound and intuitive definition of curve-skeleton, then we propose a novel method for skeleton extraction that rely on the visual appearance of the shapes. To achieve this result we inspect the properties of occluding contours, showing how information about the symmetry axes of a 3D shape can be inferred by a small set of its planar projections. The proposed method is fast, insensitive to noise, capable of working with different shape representations, resolution insensitive and easy to implement
Medial Axis Isoperimetric Profiles
Recently proposed as a stable means of evaluating geometric compactness, the
isoperimetric profile of a planar domain measures the minimum perimeter needed
to inscribe a shape with prescribed area varying from 0 to the area of the
domain. While this profile has proven valuable for evaluating properties of
geographic partitions, existing algorithms for its computation rely on
aggressive approximations and are still computationally expensive. In this
paper, we propose a practical means of approximating the isoperimetric profile
and show that for domains satisfying a "thick neck" condition, our
approximation is exact. For more general domains, we show that our bound is
still exact within a conservative regime and is otherwise an upper bound. Our
method is based on a traversal of the medial axis which produces efficient and
robust results. We compare our technique with the state-of-the-art
approximation to the isoperimetric profile on a variety of domains and show
significantly tighter bounds than were previously achievable.Comment: Code and supplemental available here:
https://github.com/pzpzpzp1/isoperimetric_profil
Doctor of Philosophy
dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest
Environment Characterization for Non-Recontaminating Frontier-Based Robotic Exploration
This paper addresses the problem of obtaining a concise description of a physical environment for robotic exploration. We aim to determine the number of robots required to clear an environment using non-recontaminating exploration. We introduce the medial axis as a configuration space and derive a mathematical representation of a continuous environment that captures its underlying topology and geometry. We show that this representation provides a concise description of arbitrary environments, and that reasoning about points in this representation is equivalent to reasoning about robots in physical space. We leverage this to derive a lower bound on the number of required pursuers. We provide a transformation from this continuous representation into a symbolic representation. Finally, we present a generalized pursuit-evasion algorithm. Given an environment we can compute how many pursuers we need, and generate an optimal pursuit strategy that will guarantee the evaders are detected with the minimum number of pursuers.Singapore-MIT Alliance for Research and Technology Center (Future Urban Mobility Project)United States. Air Force Office of Scientific Research (Award FA9550-08-1-0159)National Science Foundation (U.S.) (Award CNS-0715397)National Science Foundation (U.S.) (Award CCF-0726514)National Science Foundation (U.S.) (Grant 0735953
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