author={Eliot F. Young and Richard P. Binzel and Keenan Crane}, number = {4}, }. We apply this representation to compute 3D cross fields that are as smooth as possible everywhere but on a prescribed network of singular curves—since these fields are adapted to curve tangents, they can be directly used as input for field-guided mesh generation algorithms. @article{Sharp:2019:NIT, Its basic functionality is to translate abstract statements written in familiar math-like notation into one or more possible visual representations. pages = {5:1--5:14}, However, formalizing diagrams in most general-purpose tools requires painstaking low-level manipulation of shapes and positions.
simplicial surfacequad-surfacesmooth surface Figure 1.1. Graph. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. By adding an infinite potential well to this (or any) operator we can also robustly handle surface patches with irregular boundary. title = {Symmetric Moving Frames}, This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. (This chapter is an extended version of course notes developed for the 2018 AMS Short Course on Discrete Differential Geometry. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. Three papers at SIGGRAPH 2019, with three terrific young coauthors: I will serve on the program committee for the 2019, January 2019—SIGGRAPH Asia Technical Papers Committee, I have been invited to give a talk at the IPAM workshop on, We will present five papers at SIGGRAPH this year! PROJECTS WITH APPLICATIONS OF DIFFERENTIAL EQUATIONS AND MATLAB David Szurley Francis Marion University Department of Mathematics PO Box 100547 Florence, SC 29502 dszurley@fmarion.edu I. }, @article{White:2007:CAO, year = {2018}, In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. volume={121}, pages={1525-1533}, Rather than changing the geometry, as in traditional remeshing, we consider intrinsic triangulations which connect vertices by straight paths along the exact geometry of the input mesh. We present a novel geometric characterization of telescoping curves, and explore how free-form surfaces can be approximated by networks of such curves.
This thesis develops a discrete theory of connections that naturally leads to practical, efficient numerical algorithms for geometry processing, including texture synthesis and quadrilateral remeshing.
@article{Sharp:2020:LNT,
volume = {32}, @article{Konakovic:2018:RDC, Creating mathematical diagrams is essential for both developing oneâs intuition and conveying it to others. }, }, @article{Konakovic:2016:BDC, publisher = {American Mathematical Society} articleno = {37}, Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra.
Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. year = {2013} title = {Computational Design of Telescoping Structures}, DEC provides a simple, flexible, and efficient framework within which one can build a unified platform for geometry processing.
Graph. number = {1},
The method can be applied in any dimension, and on any domain that admits a gradient and inner product—including regular grids, triangle meshes, and point clouds. journal = {ACM Trans. author = {White, Ryan and Crane, Keenan and Forsyth, D. title={A Two-Color Map of Pluto's Sub-Charon Hemisphere},
}. publisher = "Addison-Wesley",
Real-world performance is an order of magnitude faster than state-of-the-art methods, while maintaining a comparable level of accuracy.
Unlike previous bilateral filters developed for meshes and other irregular domains, our construction reduces exactly to the image bilateral on rectangular domains and comes with a rigorous foundation in both the smooth and discrete settings. year = {2019}, author = {Stein, Oded and Grinspun, Eitan and Crane, Keenan}, journal = {ACM Trans. volume = {76}, author={Katherine Ye and Keenan Crane and Joshua Sunshine and Jonathan Aldrich}, address = {New York, NY, USA}, number = {4}, doi = {10.1145/3132705},
Numerical viscosity has long been a problem in fluid animation. title = {Monte Carlo Geometry Processing: A Grid-Free Approach to PDE-Based Methods on Volumetric Domains}, title = {Conformal Geometry Processing}, We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard sparse linear systems. }, publisher = {ACM}, location = {Boston, Massachusetts}, year = {2007}, We find that the dark albedo feature that forms a band below Pluto's equator is comprised of several distinct color units. Likewise, simulation suffers from long computation times and complications such as tangling.
A basic challenge in field-guided hexahedral meshing is to find a spatially-varying frame that is adapted to the domain geometry and is continuous up to symmetries of the cube. year = {2011}, }. In an effort to resolve some of these outstanding issues, we have developed two new techniques; the multiscale reference grid and position pegs that form a consistent foundation for presenting relative scale and position information to the user.Our design of a multiscale reference grid consistently subdivides and coalesces gridlines, based on the computation of a closeness metric, while ensuring that there are neither too many nor too few subdivisions. address = {New York, NY, USA},
title = {Navigating Intrinsic Triangulations}, We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. title = {Rapid Deployment of Curved Surfaces via Programmable Auxetics},
}. We use a novel data driven hole-filling technique to fill occluded regions. Centering is implemented via elementary operations on triangle meshes in R3, and minimizes area distortion. booktitle = {ACM SIGGRAPH 2007 sketches}, Graph. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations.
}.
August 2020—SIGGRAPH Technical Papers Committee, I will serve on the technical papers committee for, I have been invited to give a talk at the, I will serve on the program committee for the 2020, January 2020—SIGGRAPH Asia Technical Papers Committee, I will speak at a workshop on computational geometry and topology at the.
pages = {203--209}, title = {The Heat Method for Distance Computation},
booktitle = {Proceedings of Graphics Interface 2006}, }, We present a computational method for interactive 3D design and rationalization of surfaces via. Hence, by using derivatives as the principal representation (and only later recovering the field itself), one avoids the need to explicitly account for symmetry during optimization. In contrast to tools that specify diagrams via direct manipulation or low-level graphics programming, Penrose enables rapid creation and exploration of diagrams that faithfully preserve the underlying mathematical meaning. author={Alex Baden and Keenan Crane and Misha Kazhdan}, They hence do not discretize space, time, nor even function spaces, and provide the exact solution (in expectation) even on extremely challenging models. While our source of data is cloth animation captured using video cameras, the method is equally applicable to simulation data. }, In contrast, grid-free Monte Carlo methods avoid mesh generation entirely, and instead just evaluate closest point queries.
When a guidance field is present, finding the optimal field amounts to solving a single linear Poisson problem. }. Such connections can be used to design rotationally symmetric direction fields, which are essential in applications such as quadrilateral remeshing and texture synthesis. This paper develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult.
}, @article{Solomon:2014:GFB, Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. However, despite their ubiquity in 3D graphics authoring applications, little research has gone into important design considerations of the 3D reference grids themselves, which directly impact their usefulness. address = {New York, NY, USA}, year = {2020},
.
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