Rotational symmetry field design on surfaces

Palacios, Jonathan; Zhang, Eugene

doi:10.1145/1275808.1276446

Cited by 80 publications

(115 citation statements)

References 31 publications

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“…A new family of methods to generate quadrilateral meshes on surfaces have been developed by the computer graphics community [101][102][103][104][105][106]. These methods initially sought a globally smooth parameterization of the surface that does not require any previous partition of the geometry.…”

Section: Cross-field Based Methodsmentioning

confidence: 99%

Unstructured and Semi-Structured Hexahedral Mesh Generation Methods

Ramos¹,

Ruiz‐Gironés²,

Navarro³

2014

Comp. Tech. Rev.

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Discretization techniques such the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in applied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to generate. However, in many applications such as boundary layers in computational fluid dynamics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.

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Section: Cross-field Based Methodsmentioning

confidence: 99%

Unstructured and Semi-Structured Hexahedral Mesh Generation Methods

Ramos¹,

Ruiz‐Gironés²,

Navarro³

2014

Comp. Tech. Rev.

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“…This result was reformulated in [de Goes and Crane 2010;Crane et al 2013] as a series of sparse linear solves. If one wishes to automatically determine the locations and the indices of singularities that lead to a smooth face-based field, various non-convex optimizations can be used, involving a trigonometry-based energy [Hertzmann and Zorin 2000], unit constraints [Ray et al 2006;Palacios and Zhang 2007;Ray et al 2009], or integer variables [Ray et al 2008]. Diamanti et al [2014] also recently introduced an energy that mixes angle and component differences in order to generate smooth polyvectors fields.…”

Section: Connection Parallel Transport and Singularitiesmentioning

confidence: 99%

Vector field processing on triangle meshes

Goes¹,

Desbrun

Tong

2016

ACM SIGGRAPH 2016 Courses

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While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application.This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finitedimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians. PrefaceT hese course notes provide a review of the various representations of tangent vector fields on triangulated surfaces. Over the past decades, a number of approaches to express, design, and analyze vector fields on arbitrary meshes have been proposed in geometry processing, targeting a series of applications varying from rendering to animation. Some describe vector fields through values on vertices, some on edges, and some on faces; some treat vector fields as piecewise constant, some as piecewise linear, and some even involve non-linear finite elements. This lack of a unified approach to vector field representation and manipulation can be confusing for practitioners, and rare are the references that discuss the pros and cons of these various representations. The chapters in this document were written with this fact in mind, as a way to offer an introduction to the motivations, benefits, and shortcomings of the most common discretizations of vector fields in graphics-along with their associated discrete differential operators (curl, divergence, Laplacian, etc). Emphasis is placed on the tradeoffs between simplicity, efficiency, and accuracy of these geometry processing tools for applications ranging from texture synthesis to meshing.Course rationale. In many ways, this course is a continuation of ACM SIGGRAPH courses on Discrete Differential Geometry (SIGGRAPH '05 and '06, and SIGGRAPH Asia '09) and Discrete Exte...

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“…Various ways of reducing distortion of conformal parametrization are considered in [Jin et al 2004;Ben-Chen et al 2008;Springborn et al 2008]; these works do not consider alignment constraints. The method of [Ray et al 2009], based on [Palacios and Zhang 2007], generates smooth fields with fewer cones on geometrically intricate or noisy data by smoothing surface curvature. In contrast, our method trades off between distortion minimization and field smoothness, but could also, in principle, use curvature smoothing to reduce cones further.…”

Section: Related Workmentioning

confidence: 99%

Controlled-distortion constrained global parametrization

Myles

Zorin

2013

ACM Trans. Graph.

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The quality of a global parametrization is determined by a number of factors, including amount of distortion, number of singularities (cones), and alignment with features and boundaries. Placement of cones plays a decisive role in determining the overall distortion of the parametrization; at the same time, feature and boundary alignment also affect the cone placement. A number of methods were proposed for automatic choice of cone positions, either based on singularities of cross-fields and emphasizing alignment, or based on distortion optimization. In this paper we describe a method for placing cones for seamless global parametrizations with alignment constraints. We use a close relation between variation-minimizing cross-fields and related 1-forms and conformal maps, and demonstrate how it leads to a constrained optimization problem formulation. We show for boundaryaligned parametrizations metric distortion may be reduced by cone chains, sometimes to an arbitrarily small value, and the trade-off between the distortion and the number of cones can be controlled by a regularization term. Constrained parametrizations computed using our method have significantly lower distortion compared to the state-of-the art field-based method, yet maintain feature and boundary alignment. In the most extreme cases, parametrization collapse due to alignment constraints is eliminated.

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Rotational symmetry field design on surfaces

Cited by 80 publications

References 31 publications

Unstructured and Semi-Structured Hexahedral Mesh Generation Methods

Unstructured and Semi-Structured Hexahedral Mesh Generation Methods

Vector field processing on triangle meshes

Controlled-distortion constrained global parametrization

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