An Excursion Through Discrete Differential Geometry
Crane (ed.)
Symp. App. Math. (2020)
Abstract
Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data. In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume illustrates the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.
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Project
BibTeX
@book{Crane:2020:ETD,
editor = {Keenan Crane},
title = {An Excursion Through Discrete Differential Geometry},
journal = {Proceedings of Symposia in Applied Mathematics},
volume = {76},
year = {2020},
publisher = {American Mathematical Society}
}
You Can Find Geodesic Paths in Triangle Meshes by Just Flipping Edges
Sharp, Crane
ACM Trans. on Graph. (2020)
Abstract
This paper introduces a new approach to computing geodesics on polyhedral surfaces—the basic idea is to iteratively perform edge flips, in the same spirit as the classic Delaunay flip algorithm. This process also produces a triangulation conforming to the output geodesics, which is immediately useful for tasks in geometry processing and numerical simulation. More precisely, our FlipOut algorithm transforms a given sequence of edges into a locally shortest geodesic while avoiding selfcrossings (formally: it finds a geodesic in the same isotopy class). The algorithm is guaranteed to terminate in a finite number of operations; practical runtimes are on the order of a few milliseconds, even for meshes with millions of triangles. The same approach is easily applied to curves beyond simple paths, including closed loops, curve networks, and multiplycovered curves. We explore how the method facilitates tasks such as straightening cuts and segmentation boundaries, computing geodesic Bézier curves, extending the notion of constrained Delaunay triangulations (CDT) to curved surfaces, and providing accurate boundary conditions for partial differential equations (PDEs). Evaluation on challenging datasets such as Thingi10k indicates that the method is both robust and efficient, even for lowquality triangulations.
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Project
BibTeX
@article{Sharp:2019:NIT,
author = {Sharp, Nicholas and Crane, Keenan},
title = {You Can Find Geodesic Paths in Triangle Meshes by Just Flipping Edges},
journal = {ACM Trans. Graph.},
volume = {39},
number = {6},
year = {2020},
publisher = {ACM},
address = {New York, NY, USA},
}
Repulsive Curves
Yu, Schumacher, Crane
ACM Trans. on Graph. (2020) (conditionally accepted)
Abstract
Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or selfintersections. This paper develops efficient algorithms for (self)repulsion of plane and space curves that are wellsuited to problems in computational design. Our starting point is the socalled tangentpoint energy, which provides an infinite barrier to selfintersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent, based on a SobolevSlobodeckij inner product enables us to make rapid progress toward local minima—independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the perstep cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, noncrossing spline interpolation, flow visualization, and robotic path planning.
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arXiv
Video
A Laplacian for Nonmanifold Triangle Meshes
Sharp, Crane
Symp. on Geom. Proc. (2020)
Abstract
We describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust dropin replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poorquality meshes. The key idea is to build what we call a “tufted cover” over the input domain, which has nonmanifold vertices but manifold edges. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation. This construction also provides a highquality point cloud Laplacian, via a nonmanifold triangulation of the point set. We validate our Laplacian on a variety of challenging examples (including all models from Thingi10k), and a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.
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BibTeX
@article{Sharp:2020:LNT,
author={Nicholas Sharp and Keenan Crane},
title={{A Laplacian for Nonmanifold Triangle Meshes}},
journal={Computer Graphics Forum (SGP)},
volume={39},
number={5},
year={2020}
}
Monte Carlo Geometry Processing
Sawhney, Crane
ACM Trans. on Graph. (2020)
Abstract
This paper explores how core problems in PDEbased geometry processing can be efficiently and reliably solved via gridfree Monte Carlo methods. Modern geometric algorithms often need to solve Poissonlike equations on geometrically intricate domains. Conventional methods most often mesh the domain, which is both challenging and expensive for geometry with fine details or imperfections (holes, selfintersections, etc.). In contrast, gridfree Monte Carlo methods avoid mesh generation entirely, and instead just evaluate closest point queries. They hence do not discretize space, time, nor even function spaces, and provide the exact solution (in expectation) even on extremely challenging models. More broadly, they share many benefits with Monte Carlo methods from photorealistic rendering: excellent scaling, trivial parallel implementation, viewdependent evaluation, and the ability to work with any kind of geometry (including implicit or procedural descriptions). We develop a complete “black box” solver that encompasses integration, variance reduction, and visualization, and explore how it can be used for various geometry processing tasks. In particular, we consider several fundamental linear elliptic PDEs with constant coefficients on solid regions of R^{n}. Overall we find that Monte Carlo methods significantly broaden the horizons of geometry processing, since they easily handle problems of size and complexity that are essentially hopeless for conventional methods.
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PDF
BibTeX
@article{Sawhney:2020:MCG,
author = {Sawhney, Rohan and Crane, Keenan},
title = {Monte Carlo Geometry Processing: A GridFree Approach to PDEBased Methods on Volumetric Domains},
journal = {ACM Trans. Graph.},
volume = {39},
number = {4},
year = {2020},
publisher = {ACM},
address = {New York, NY, USA},
}
Penrose: From Mathematical Notation to Beautiful Diagrams
Ye, Ni, Krieger, Ma'ayan, Wise, Aldrich, Sunshine, Crane
ACM Trans. on Graph. (2020)
Abstract
We introduce a system called Penrose for creating mathematical diagrams. Its basic functionality is to translate abstract statements written in familiar mathlike notation into one or more possible visual representations. Rather than rely on a fixed library of visualization tools, the visual representation is userdefined in a constraintbased specification language; diagrams are then generated automatically via constrained numerical optimization. The system is userextensible to many domains of mathematics, and is fast enough for iterative design exploration. In contrast to tools that specify diagrams via direct manipulation or lowlevel graphics programming, Penrose enables rapid creation and exploration of diagrams that faithfully preserve the underlying mathematical meaning. We demonstrate the effectiveness and generality of the system by showing how it can be used to illustrate a diverse set of concepts from mathematics and computer graphics.
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BibTeX
@article{Ye:2020:PFM,
author = {Ye, Katherine and Ni, Wode and Krieger, Max and Ma'ayan, Dor and Wise, Jenna and Aldrich, Jonathan and Sunshine, Joshua and Crane, Keenan},
title = {Penrose: From Mathematical Notation to Beautiful Diagrams},
journal = {ACM Trans. Graph.},
volume = {39},
number = {4},
year = {2020},
publisher = {ACM},
address = {New York, NY, USA},
}
Navigating Intrinsic Triangulations
Sharp, Soliman, Crane
ACM Trans. on Graph. (2019)
Abstract
We present a data structure that makes it easy to run a large class of algorithms from computational geometry and scientific computing on extremely poorquality surface meshes. 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. Our key insight is that such a triangulation can be encoded implicitly by storing the direction and distance to neighboring vertices. The resulting signpost data structure then allows geometric and topological queries to be made ondemand by tracing paths across the surface. Existing algorithms can be easily translated into the intrinsic setting, since this data structure supports the same basic operations as an ordinary triangle mesh (vertex insertions, edge splits, etc.). The output of intrinsic algorithms can then be stored on an ordinary mesh for subsequent use; unlike previous data structures, we use a constant amount of memory and do not need to explicitly construct an overlay mesh unless it is specifically requested. Working in the intrinsic setting incurs little computational overhead, yet we can run algorithms on extremely degenerate inputs, including all manifold meshes from the Thingi10k data set. To evaluate our data structure we implement several fundamental geometric algorithms including intrinsic versions of Delaunay refinement and optimal Delaunay triangulation, approximation of Steiner trees, adaptive mesh refinement for PDEs, and computation of Poisson equations, geodesic distance, and flipfree tangent vector fields.
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BibTeX
@article{Sharp:2019:NIT,
author = {Sharp, Nicholas and Soliman, Yousuf and Crane, Keenan},
title = {Navigating Intrinsic Triangulations},
journal = {ACM Trans. Graph.},
volume = {38},
number = {4},
year = {2019},
publisher = {ACM},
address = {New York, NY, USA},
}
Symmetric Moving Frames
Corman, Crane
ACM Trans. on Graph. (2019)
Abstract
A basic challenge in fieldguided hexahedral meshing is to find a spatiallyvarying frame that is adapted to the domain geometry and is continuous up to symmetries of the cube. We introduce a fundamentally new representation of such 3D cross fields based on Cartan's method of moving frames. Our key observation is that cross fields and ordinary frame fields are locally characterized by identical conditions on their Darboux derivative. 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. At the discrete level, derivatives are encoded by skewsymmetric matrices associated with the edges of a tetrahedral mesh; these matrices encode arbitrarily large rotations along each edge, and can robustly capture singular behavior even on coarse meshes. 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 fieldguided mesh generation algorithms. Optimization amounts to an easy nonlinear least squares problem that behaves like a convex program in the sense that it always appears to produce the same result, independent of initialization. We study the numerical behavior of this procedure, and perform some preliminary experiments with mesh generation.
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BibTeX
@article{Corman:2019:SMF,
author = {Corman, Etienne and Crane, Keenan},
title = {Symmetric Moving Frames},
journal = {ACM Trans. Graph.},
volume = {38},
number = {4},
year = {2019},
publisher = {ACM},
address = {New York, NY, USA},
}
The Vector Heat Method
Sharp, Soliman, Crane
ACM Trans. on Graph. (2019)
Abstract
This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vectorvalued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a shorttime heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, \etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.
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BibTeX
@article{Sharp:2019:VHM,
author = {Sharp, Nicholas and Soliman, Yousuf and Crane, Keenan},
title = {The Vector Heat Method},
journal = {ACM Trans. Graph.},
volume = {38},
number = {3},
year = {2019},
publisher = {ACM},
address = {New York, NY, USA},
}
Conformal Geometry of Simplicial Surfaces
Crane
Proceedings of Symposia in Applied Mathematics (to appear)
Abstract
Conformal geometry studies geometric properties that are invariant with respect to anglepreserving transformations. In the discrete setting of polyhedral surfaces, the idea of naively preserving angles leads to a notion of conformal geometry that is far too rigid, i.e., it does not capture the rich behavior found in the smooth setting. These notes explore several alternative notions of discrete conformal structure, culminating with a recent uniformization theorem for general simplicial surfaces. Topics covered include circle packings, circle patterns, inversive distance, discrete Yamabe flow, and connections to variational principles for ideal hyperbolic polyhedra. (This chapter is an extended version of course notes developed for the 2018 AMS Short Course on Discrete Differential Geometry.)
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BibTeX
@incollection{Crane:2020:DCG,
author = "Keenan Crane",
title = {{Discrete Conformal Geometry}},
booktitle = "Proceedings of Symposia in Applied Mathematics",
publisher = "American Mathematical Society",
year = 2020,
}
Developability of Triangle Meshes
Stein, Grinspun, Crane
ACM Trans. on Graph. (2018)
Abstract
Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing. We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling—we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.
BibTeX
@article{Stein:2018:DSF,
author = {Stein, Oded and Grinspun, Eitan and Crane, Keenan},
title = {Developability of Triangle Meshes},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},
}
Variational Surface Cutting
Sharp, Crane
ACM Trans. on Graph. (2018)
Abstract
This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself (e.g., its length or curvature), we formulate a flow that directly optimizes the distortion induced by cutting and flattening. Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on the cut curve itself. We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate userdefined data such as importance or occlusion. 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. We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution.
BibTeX
@article{Sharp:2018:VSC,
author = {Sharp, Nick and Crane, Keenan},
title = {Variational Surface Cutting},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},
}
Optimal Cone Singularities for Conformal Flattening
Soliman, Slepčev, Crane
ACM Trans. on Graph. (2018)
Abstract
Anglepreserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply FenchelRockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports userdefined notions of importance, constraints on cone angles (e.g., positive, or within a given range), and sophisticated boundary conditions (e.g., convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.
BibTeX
@article{Soliman:2018:OCS,
author = {Soliman, Yousuf and Slep\v{c}ev, Dejan and Crane, Keenan},
title = {Optimal Cone Singularities for Conformal Flattening},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},
}
Rapid Deployment of Curved Surfaces via Programmable Auxetics
Konaković, Panetta, Crane, Pauly
ACM Trans. on Graph. (2018)
Abstract
Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doublycurved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on twodimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a userprogrammable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures—in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to largescale deployable architecture.
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BibTeX
@article{Konakovic:2018:RDC,
author = {Konakovi\'{c}, Mina and Panetta, Julian and Crane, Keenan and Pauly, Mark},
title = {Rapid Deployment of Curved Surfaces via Programmable Auxetics},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},
}
Möbius Registration
Baden, Crane, Kazhdan
Symp. on Geom. Proc. (2018)
Abstract
Conformal parameterizations over the sphere provide highquality maps between genus zero surfaces, and are essential for applications such as data transfer and comparative shape analysis. However, such maps are not unique: to define correspondence between two surfaces, one must find the Möbius transformation that best aligns two parameterizations—akin to picking a translation and rotation in rigid registration problems. We describe a simple procedure that canonically centers and rotationally aligns two spherical maps. Centering is implemented via elementary operations on triangle meshes in R^{3}, and minimizes area distortion. Alignment is achieved using the FFT over the group of rotations. We examine this procedure in the context of spherical conformal parameterization, orbifold maps, nonrigid symmetry detection, and dense pointtopoint surface correspondence.
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BibTeX
@article{Baden:2018:MR,
author={Alex Baden and Keenan Crane and Misha Kazhdan},
title={{M\"{o}bius Registration}},
journal={Computer Graphics Forum (SGP)},
volume={37},
number={5},
year={2018}
}
Boundary First Flattening
Sawhney, Crane
ACM Trans. on Graph. (2017)
Abstract
A conformal flattening maps a curved surface to the plane without distorting angles—such maps have become a fundamental building block for problems in geometry processing, numerical simulation, and computational design. Yet existing methods provide little direct control over the shape of the flattened domain, or else demand expensive nonlinear optimization. Boundary first flattening (BFF) is a linear method for conformal parameterization which is faster than traditional linear methods, yet provides control and quality comparable to sophisticated nonlinear schemes. The key insight is that the boundary data for many conformal mapping problems can be efficiently constructed via the Cherrier formula together with a pair of PoincaréSteklov operators; once the boundary is known, the map can be easily extended over the rest of the domain. Since computation demands only a single factorization of the real Laplace matrix, the amortized cost is about 50x less than any previously published technique for boundarycontrolled conformal flattening. As a result, BFF opens the door to realtime editing or fast optimization of highresolution maps, with direct control over boundary length or angle. We show how this method can be used to construct maps with sharp corners, cone singularities, minimal area distortion, and uniformization over the unit disk; we also demonstrate for the first time how a surface can be conformally flattened directly onto any given target shape.
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BibTeX
@article{Sawhney:2017:BFF,
author = {Sawhney, Rohan and Crane, Keenan},
title = {Boundary First Flattening},
journal = {ACM Trans. Graph.},
volume = {37},
number = {1},
month = dec,
year = {2017},
issn = {07300301},
pages = {5:15:14},
articleno = {5},
numpages = {14},
url = {http://doi.acm.org/10.1145/3132705},
doi = {10.1145/3132705},
acmid = {3132705},
publisher = {ACM},
address = {New York, NY, USA}
}
The Heat Method for Distance Computation
Crane, Weischedel, Wardetzky
Communications of the ACM (2017)
Abstract
We introduce the heat method for solving the single or multiplesource shortest path problem on both flat and curved domains. A key insight is that distance computation can be split into two stages: first find the direction along which distance is increasing, then compute the distance itself. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard sparse linear systems. These systems can be factored once and subsequently solved in nearlinear time, substantially reducing amortized cost. Realworld performance is an order of magnitude faster than stateoftheart methods, while maintaining a comparable level of accuracy. 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. Numerical evidence indicates that the method converges to the exact distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where greater regularity is desired.
Project
BibTeX
@article{Crane:2017:HMD,
author = {Crane, Keenan and Weischedel, Clarisse and Wardetzky, Max},
title = {The Heat Method for Distance Computation},
journal = {Commun. ACM},
issue_date = {November 2017},
volume = {60},
number = {11},
month = oct,
year = {2017},
issn = {00010782},
pages = {9099},
numpages = {10},
url = {http://doi.acm.org/10.1145/3131280},
doi = {10.1145/3131280},
acmid = {3131280},
publisher = {ACM},
address = {New York, NY, USA},
}
A Glimpse Into Discrete Differential Geometry
Crane, Wardetzky
Notices of the American Mathematical Society (2017)
Abstract
The emerging field of discrete differential geometry (DDG) studies discrete analogues of smooth geometric objects, providing an essential link between analytical descriptions and computation. In recent years it has unearthed a rich variety of new perspectives on applied problems in computational anatomy/biology, computational mechanics, industrial design, computational architecture, and digital geometry processing at large. The basic philosophy of discrete differential geometry is that a discrete object like a polyhedron is not merely an approximation of a smooth one, but rather a differential geometric object in its own right. In contrast to traditional numerical analysis which focuses on eliminating approximation error in the limit of refinement (e.g., by taking smaller and smaller finite differences), DDG places an emphasis on the socalled “mimetic” viewpoint, where key properties of a system are preserved exactly, independent of how large or small the elements of a mesh might be. Just as algorithms for simulating mechanical systems might seek to exactly preserve physical invariants such as total energy or momentum, structurepreserving models of discrete geometry seek to exactly preserve global geometric invariants such as total curvature. More broadly, DDG focuses on the discretization of objects that do not naturally fall under the umbrella of traditional numerical analysis. This article provides an overview of some of the themes in DDG.
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BibTeX
@article{Crane:2017:GID,
author={Keenan Crane and Max Wardetzky},
title={A Glimpse Into Discrete Differential Geometry},
journal={Notices of the American Mathematical Society},
month={November},
volume={64},
number={10},
pages={11531159},
year={2017}
}
SUBSTANCE and STYLE: DomainSpecific Languages for Mathematical Diagrams
Ni, Ye, Sunshine, Aldrich, Crane
DSLDI (2017)
Abstract
Creating mathematical diagrams is essential for both developing one’s intuition and conveying it to others. However, formalizing diagrams in most generalpurpose tools requires painstaking lowlevel manipulation of shapes and positions. We report on early work on PENROSE, a system we are building to automatically visualize mathematics from notation. PENROSE comprises two languages: SUBSTANCE, a domainspecific language that mimics the declarativeness of mathematical notation, and STYLE, a styling language that concisely specifies the visual semantics of the notation. Our system can automatically visualize set theory expressions with userdefined styles, and it can visualize abstract definitions of functions by producing concrete examples. We plan to extend the system to more domains of math.
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BibTeX
@inproceedings{Ni:2017:SSD,
author={Wode Ni and Katherine Ye and Joshua Sunshine and Jonathan Aldrich and Keenan Crane},
title={SUBSTANCE and STYLE: domainspecific languages for mathematical diagrams},
booktitle={DSLDI (DomainSpecific Language Design and Implementation)},
year={2017}
}
A Dirac Operator for Extrinsic Shape Analysis
Liu, Jacobson, Crane
Symp. on Geom. Proc. (2017)
Abstract
The eigenfunctions and eigenvalues of the LaplaceBeltrami operator have proven to be a powerful tool for digital geometry processing, providing a description of geometry that is essentially independent of coordinates or the choice of discretization. However, since LaplaceBeltrami is purely intrinsic it struggles to capture important phenomena such as extrinsic bending, sharp edges, and fine surface texture. We introduce a new extrinsic differential operator called the relative Dirac operator, leading to a family of operators with a continuous tradeoff between intrinsic and extrinsic features. Previous operators are either fully or partially intrinsic. In contrast, the proposed family spans the entire spectrum: from completely intrinsic (depending only on the metric) to completely extrinsic (depending only on the Gauss map). By adding an infinite potential well to this (or any) operator we can also robustly handle surface patches with irregular boundary. We explore use of these operators for a variety of shape analysis tasks, and study their performance relative to operators previously found in the geometry processing literature.
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BibTeX
@article{Liu:2017:DOE,
author={Derek Liu and Alec Jacobson and Keenan Crane},
title={A Dirac Operator for Extrinsic Shape Analysis},
journal={Computer Graphics Forum (SGP)},
volume={36},
number={5},
year={2017}
}
Designing Extensible, DomainSpecific Languages for Mathematical Diagrams
Ye, Crane, Aldrich, Sunshine
OBT (2017)
Abstract
In science, a wellchosen illustration can turn bafflement into enlightenment. Yet technical exposition remains largely textual, due to the tremendous expertise required to produce highquality figures. We propose PENROSE, a system to automatically generate professionalquality mathematical illustrations from highlevel, purely semantic descriptions of mathematical objects. Unlike lowlevel tools where diagrams are specified via graphical primitives, a mathematicallyinclined user should not require any graphic design skill to create beautiful diagrams. PENROSE comprises two extensible domainspecific languages (DSLs): SUBSTANCE, which diagrammers use to specify mathematical objects and relationships, and STYLE, which implementers use to encode various ways of realizing these relationships visually, akin to the separation of content and style in modern HTML/CSS. To compile diagrams, we are developing a sophisticated constraint solver incorporating techniques from optimization and computer graphics.
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BibTeX
@inproceedings{Ye:2017:DED,
author={Katherine Ye and Keenan Crane and Joshua Sunshine and Jonathan Aldrich},
title={Designing Extensible, DomainSpecific Languages for Mathematical Diagrams},
booktitle={OBT (Off the Beaten Track)},
year={2017}
}
Computational Design of Telescoping Structures
Yu, Crane, Coros
ACM Trans. on Graph. (2017)
Abstract
Telescoping structures are valuable for a variety of applications where mechanisms must be compact in size and yet easily deployed. So far, however, there has been no systematic study of the types of shapes that can be modeled by telescoping structures, nor practical tools for telescopic design. We present a novel geometric characterization of telescoping curves, and explore how freeform surfaces can be approximated by networks of such curves. In particular we consider piecewise helical space curves with torsional impulses, which significantly generalize the linear telescopes found in typical engineering designs. Based on this principle we develop a system for computational design and fabrication which allows users to explore the space of telescoping structures; inputs to our system include user sketches or arbitrary meshes, which are then converted to a curve skeleton. We prototype applications in animation, fabrication, and robotics, using our system to design a variety of both simulated and fabricated examples.
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BibTeX
@article{Yu:2017:CDT,
author = {Yu, Chris and Crane, Keenan and Coros, Stelian},
title = {Computational Design of Telescoping Structures},
journal = {ACM Trans. Graph.},
volume = {36},
number = {4},
year = {2017},
publisher = {ACM},
address = {New York, NY, USA},
}
Beyond Developable: Computational Design and Fabrication with Auxetic Materials
Konaković, Crane, Deng, Bouaziz, Piker, Pauly
ACM Trans. on Graph. (2016)
Abstract
We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doublycurved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatiallyvarying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable nonlinear optimization. We demonstrate that this global approach can handle nontrivial topology and nonlocal dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications.
BibTeX
@article{Konakovic:2016:BDC,
author = {Konakovi\'{c}, Mina and Crane, Keenan and Deng, Bailin and Bouaziz, Sofien and Piker, Daniel and Pauly, Mark},
title = {Beyond Developable: Computational Design and Fabrication with Auxetic Materials},
journal = {ACM Trans. Graph.},
volume = {35},
number = {4},
year = {2016},
publisher = {ACM},
address = {New York, NY, USA},
}
Stripe Patterns on Surfaces
Knöppel, Crane, Pinkall, Schröder
ACM Trans. on Graph. (2015)
Abstract
Stripe patterns are ubiquitous in nature, describing macroscopic phenomena such as stripes on plants and animals, down to material impurities on the atomic scale. We propose an algorithm for synthesizing stripe patterns on triangulated surfaces, where branch points are automatically inserted in order to achieve userspecified orientation and line spacing. Patterns are characterized as global minimizers of a simple convexquadratic energy which is welldefined in the smooth setting. Computation amounts to finding the smallest eigenvector of a symmetric positivedefinite matrix with the same sparsity pattern as the standard graph Laplacian. The resulting patterns are globally continuous, and can be applied to a variety of tasks in design and texture synthesis.
BibTeX
@article{Knoppel:2015:SPS,
author = {Kn\"{o}ppel, Felix and Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Stripe Patterns on Surfaces},
journal = {ACM Trans. Graph.},
volume = {34},
number = {4},
year = {2015},
publisher = {ACM},
address = {New York, NY, USA},
}
A General Framework for Bilateral and Mean Shift Filtering
Solomon, Crane, Butscher, Wojtan
arXiv:1405.4734 (2014)
Abstract
We present a generalization of the bilateral filter that can be applied to featurepreserving smoothing of signals on images, meshes, and other domains within a single unified framework. Our discretization is competitive with stateoftheart smoothing techniques in terms of both accuracy and speed, is easy to implement, and has parameters that are straightforward to understand. 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. These guarantees allow us to construct unconditionally convergent meanshift schemes that handle a variety of extremely noisy signals. We also apply our framework to geometric edgepreserving effects like feature enhancement and show how it is related to local histogram techniques.
BibTeX
@article{Solomon:2014:GFB,
author = {Solomon, Justin and Crane, Keenan and Butscher, Adrian and Wojtan, Chris},
title = {A General Framework for Bilateral and Mean Shift Filtering},
journal = {ArXiv eprints},
volume = {32},
archivePrefix = "arXiv",
eprint = {1405.4734},
primaryClass = "cs.GR",
keywords = {Computer Science  Graphics},
year = {2014},
month = {April}
}
Robust Fairing via Conformal Curvature Flow
Crane, Pinkall, Schröder
ACM Trans. on Graph. (2013)
Abstract
This paper presents a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in curvature space—we develop the precise conditions under which curvature is allowed to evolve. The practical outcome is a highly efficient algorithm that naturally preserves texture and does not require remeshing during the flow. We apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces. We also present a new algorithm for lengthpreserving flow on planar curves, which provides a valuable analogy for the surface case.
BibTeX
@article{Crane:2013:RFC,
author = {Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Robust Fairing via Conformal Curvature Flow},
journal = {ACM Trans. Graph.},
volume = {32},
number = {4},
year = {2013},
publisher = {ACM},
address = {New York, NY, USA},
}
Globally Optimal Direction Fields
Knöppel, Crane, Pinkall, Schröder
ACM Trans. on Graph. (2013)
Abstract
This paper presents a method for constructing smooth unit ndirection fields (line fields, cross fields, etc.) on surfaces that is an order of magnitude faster than stateoftheart methods, while still producing fields of equal or better quality. The method is based on a simple quadratic energy whose minimizers are globally optimal in the sense that they produce the smoothest fields over all possible configurations of singularities (number, location, and index). The method is fully automatic and can optionally produce fields aligned with a given guidance field, for example, principal curvature directions. Computationally the smoothest field is found via a sparse eigenvalue problem involving a matrix similar to the cotanLaplacian. When a guidance field is present, finding the optimal field amounts to solving a single linear Poisson problem.
BibTeX
@article{Knoppel:2013:GOD,
author = {Kn\"{o}ppel, Felix and Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Globally optimal direction fields},
journal = {ACM Trans. Graph.},
volume = {32},
number = {4},
year = {2013},
publisher = {ACM},
address = {New York, NY, USA},
}
Digital Geometry Processing with Discrete Exterior Calculus
Crane, de Goes, Desbrun, Schröder
SIGGRAPH 2013 Course Notes
Abstract
This course provides an introduction to geometry processing using discrete exterior calculus (DEC). DEC provides a simple, flexible, and efficient framework within which one can build a unified platform for geometry processing. The course provides essential mathematical background as well as a large array of realworld examples. It also provides a short survey of the most relevant recent developments in digital geometry processing and discrete differential geometry.
BibTeX
@inproceedings{Crane:2013:DGP,
author = {Keenan Crane and Fernando de Goes and Mathieu Desbrun and Peter Schr\"{o}der},
title = {Digital Geometry Processing with Discrete Exterior Calculus},
booktitle = {ACM SIGGRAPH 2013 courses},
series = {SIGGRAPH '13},
year = {2013},
location = {Anaheim, California},
numpages = {126},
publisher = {ACM},
address = {New York, NY, USA},
}
Spin Transformations of Discrete Surfaces
Crane, Pinkall, Schröder
ACM Trans. on Graph. (2011)
Abstract
This paper introduces a new method for computing conformal transformations of triangle meshes in R^{3}. 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. 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. We instead consider maps into the quaternions, which allows us to work directly with surfaces sitting in R^{3}. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.
BibTeX
@article{Crane:2011:STD,
author = {Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Spin Transformations of Discrete Surfaces},
journal = {ACM Trans. Graph.},
volume = {30},
number = {4},
year = {2011},
publisher = {ACM},
address = {New York, NY, USA},
}
Discrete Connections for Geometry Processing
Keenan Crane
Caltech Master's Thesis (2010)
Abstract
Connections provide a way to compare local quantities defined at different points of a geometric space. 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. Our formulation is motivated by realworld applications where meshes may be noisy or coarsely discretized. Further, because our discrete framework closely parallels the smooth theory, we can draw upon a huge wealth of existing knowledge to develop and interpret mesh processing algorithms. The solutions we produce are globally optimal in the sense that they describe the trivial connection closest to LeviCivita among all solutions with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra.
BibTeX
@mastersthesis{caltechthesis5880,
title = {Discrete connections for geometry processing},
school = {California Institute of Technology},
author = {Keenan Crane},
year = {2010},
url = {http://resolver.caltech.edu/CaltechTHESIS:05282010102307125}
}
Trivial Connections on Discrete Surfaces
Crane, Desbrun, Schröder
Symp. on Geometry Proc. (2010)
Abstract
This paper presents a straightforward algorithm for constructing connections on surfaces that are as smooth as possible everywhere but on a prescribed set of isolated singularities with given index. Such connections can be used to design rotationally symmetric direction fields, which are essential in applications such as quadrilateral remeshing and texture synthesis. We compute connections by solving a single sparse linear system built from standard operators. Our solutions are globally optimal in the sense that they describe the trivial connection closest to LeviCivita among all connections with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra.
BibTeX
@article{Crane:2010:TCD,
author={Keenan Crane and Mathieu Desbrun and Peter Schr\"{o}der},
title={Trivial Connections on Discrete Surfaces},
journal={Computer Graphics Forum (SGP)},
volume={29},
number={5},
pages={15251533},
year={2010}
}
Lie Group Integrators for Animation and Control of Vehicles
Kobilarov, Crane, Desbrun
ACM Trans. on Graph. (2009)
Abstract
This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their statespace geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary userspecified constraints.
BibTeX
@article{Kobilarov:2009:LGI,
author = {Kobilarov, Marin and Crane, Keenan and Desbrun, Mathieu},
title = {Lie group integrators for animation and control of vehicles},
journal = {ACM Trans. Graph.},
volume = {28},
number = {2},
month = {May},
year = {2009},
}
EnergyPreserving Integrators for Fluid Animation
Mullen, Crane, Pavlov, Tong, Desbrun
ACM Trans. on Graph. (2009)
Abstract
Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive timestepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a timereversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energypreserving HarlowWelch / CrankNicolson scheme to simplicial grids.
BibTeX
@article{Mullen:2009:EIF,
author = {Mullen, Patrick and Crane, Keenan and Pavlov, Dmitry and Tong, Yiying and Desbrun, Mathieu},
title = {Energypreserving integrators for fluid animation},
journal = {ACM Trans. Graph.},
volume = {28},
number = {3},
month = {July},
year = {2009},
}
Multiscale 3D Reference Visualization
Glueck, Crane, Anderson, Rutnik, Khan
i3D (2009)
Abstract
Reference grids are commonly used in design software to help users to judge distances and to understand the orientation of the virtual workspace. 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. 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. Position pegs extend the grid so that objects that are lying above or below the ground plane can be brought into a common environmental frame of reference without interfering with the grid or object data. We introduce an analytical system, similar to MIP mapping, to provide a stable viewpointdetermined result. Our design solves several depth cue problems in a way that is independent of viewing projection.
BibTeX
@inproceedings{Glueck:2009:MRV,
author = {Glueck, Michael and Crane, Keenan and Anderson, Sean and Rutnik, Andres and Khan, Azam},
title = {Multiscale 3D reference visualization},
booktitle = {Proceedings of the 2009 symposium on Interactive 3D graphics and games},
series = {I3D '09},
year = {2009},
isbn = {9781605584294},
location = {Boston, Massachusetts},
pages = {225232},
numpages = {8},
url = {http://doi.acm.org/10.1145/1507149.1507186},
doi = {http://doi.acm.org/10.1145/1507149.1507186},
acmid = {1507186},
publisher = {ACM},
address = {New York, NY, USA},
}
Capturing and Animating Occluded Cloth
White, Crane, Forsyth
ACM Trans. on Graph. (2007)
Abstract
We capture the shape of moving cloth using a custom set of color markers printed on the surface of the cloth. The output is a sequence of triangle meshes with static connectivity and with detail at the scale of individual markers in both smooth and folded regions. We compute markers' coordinates in space using marker correspondence across multiple synchronized video cameras. Correspondence is determined from color information in small neighborhoods and refined using a novel strain pruning process. Final correspondence does not require neighborhood information. We use a novel data driven holefilling technique to fill occluded regions. Our results include several challenging examples: a wrinkled shirt sleeve, a dancing pair of pants, and a rag tossed onto a cup. Finally, we demonstrate that cloth capture is reusable by animating a pair of pants using human motion capture data.
BibTeX
@article{White:2007:CAO,
author = {White, Ryan and Crane, Keenan and Forsyth, D. A.},
title = {Capturing and animating occluded cloth},
journal = {ACM Trans. Graph.},
volume = {26},
number = {3},
month = {July},
year = {2007},
}
Data Driven Cloth Animation
White, Crane, Forsyth
SIGGRAPH 2007 Technical Sketches
Abstract
We present a new method for cloth animation based on data driven synthesis. In contrast to approaches that focus on physical simulation, we animate cloth by manipulating short sequences of existing cloth animation. While our source of data is cloth animation captured using video cameras, the method is equally applicable to simulation data. The approach has benefits in both cases: current cloth capture is limited because small tweaks to the data require filming an entirely new sequence. Likewise, simulation suffers from long computation times and complications such as tangling. In this sketch we create new animations by fitting cloth animation to human motion capture data, i.e., we drive the cloth with a skeleton.
BibTeX
@inproceedings{White:2007:DDC,
author = {White, Ryan and Crane, Keenan and Forsyth, D. A.},
title = {Data driven cloth animation},
booktitle = {ACM SIGGRAPH 2007 sketches},
series = {SIGGRAPH '07},
year = {2007},
location = {San Diego, California},
articleno = {37},
acmid = {1278825},
publisher = {ACM},
address = {New York, NY, USA},
}
Real Time Simulation and Rendering of 3D Fluids
Crane, Llamas, Tariq
GPU Gems 3 (2007)
Abstract
Physically based animation of fluids such as smoke, water, and fire provides some of the most stunning visuals in computer graphics, but has historically been the domain of highquality offline rendering due to great computational cost. In this chapter we not only show how these effects can be simulated and rendered in real time, but also how they can be seamlessly integrated into real time applications.
BibTeX
@InBook{Crane07,
author = "Crane, Keenan and Llamas, Ignacio, and Tariq, Sarah",
title = "Real Time Simulation and Rendering of 3D Fluids",
booktitle = "GPUGems 3",
editor = "Hubert Nguyen",
chapter = "30",
publisher = "AddisonWesley",
year = "2007"
}
Rectangular MultiChart Geometry Images
Carr, Hoberock, Crane, Hart
Symp. on Geom. Proc. (2006)
Abstract
Many mesh parameterization algorithms have focused on minimizing distortion and utilizing texture area, but few have addressed issues related to processing a signal on the mesh surface. We present an algorithm which partitions a mesh into rectangular charts while preserving a onetoone texel correspondence across chart boundaries. This mapping permits any computation on the mesh surface which is typically carried out on a regular grid, and prevents seams by ensuring resolution continuity along the boundary. These features are also useful for traditional texture applications such as surface painting where continuity is important. Distortion is comparable to other parameterization schemes, and the rectangular charts yield efficient packing into a texture atlas. We apply this parameterization to texture synthesis, fluid simulation, mesh processing and storage, and locating geodesics.
BibTeX
@inproceedings{Carr:2006:RMG,
author = {Carr, Nathan A. and Hoberock, Jared and Crane, Keenan and Hart, John C.},
title = {Rectangular MultiChart Geometry Images},
booktitle = {Proceedings of the fourth Eurographics symposium on Geometry processing},
year = {2006},
}
Fast GPU Ray Tracing of Dynamic Meshes using Geometry Images
Carr, Hoberock, Crane, Hart
Graphics Interface 2006
Abstract
Using the GPU to accelerate ray tracing may seem like a natural choice due to the highly parallel nature of the problem. However, determining the most versatile GPU data structure for scene storage and traversal is a challenge. In this paper, we introduce a new method for quick intersection of triangular meshes on the GPU. The method uses a threaded bounding volume hierarchy built from a geometry image, which can be efficiently traversed and constructed entirely on the GPU. This acceleration scheme is highly competitive with other GPU ray tracing methods, while allowing for both dynamic geometry and an efficient level of detail scheme at no extra cost.
BibTeX
@inproceedings{Carr:2006:FGR,
author = {Carr, Nathan A. and Hoberock, Jared and Crane, Keenan and Hart, John C.},
title = {Fast GPU ray tracing of dynamic meshes using geometry images},
booktitle = {Proceedings of Graphics Interface 2006},
series = {GI '06},
year = {2006},
isbn = {1568813082},
location = {Quebec, Canada},
pages = {203209},
numpages = {7},
url = {http://portal.acm.org/citation.cfm?id=1143079.1143113},
acmid = {1143113},
publisher = {Canadian Information Processing Society},
address = {Toronto, Ont., Canada, Canada},
}
A TwoColor Map of Pluto's SubCharon Hemisphere
Young, Binzel, Crane
The Astronomical Journal (2001)
Abstract
Pluto and its satellite Charon regularly occulted or transited each other's disks from 1985 through 1990. The light curves resulting from these events (collectively called "mutual events") have been used to determine albedo maps of Pluto's subCharon hemisphere. We now use a data set of four light curves that were obtained in both B and V Johnson filters to construct a twocolor map of Pluto's surface. We are able to resolve the central part of Pluto's subCharon hemisphere. We find that the dark albedo feature that forms a band below Pluto's equator is comprised of several distinct color units. We detect ratios of Vfilter/Bfilter normal reflectances ranging from 1.15 to 1.39 on Pluto's subCharon hemisphere.
BibTeX
@article{153838811211552,
author={Eliot F. Young and Richard P. Binzel and Keenan Crane},
title={A TwoColor Map of Pluto's SubCharon Hemisphere},
journal={The Astronomical Journal},
volume={121},
number={1},
pages={552},
url={http://stacks.iop.org/15383881/121/i=1/a=552},
year={2001}
}
