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Summer Seminar. Lubin Fan 2011-07-07. Discrete Differential Geometry. Circular arc structures Discrete Laplacians on General Polygonal Meshes HOT: Hodge-Optimized Triangulations Spin Transformations of Discrete Surfaces. Example-Based Simulation. Frame-based Elastic Models (TOG)

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Summer Seminar

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Summer seminar

Summer Seminar

Lubin Fan


Summer seminar

Discrete Differential Geometry

  • Circular arc structures

  • Discrete Laplacians on General Polygonal Meshes

  • HOT: Hodge-Optimized Triangulations

  • Spin Transformations of Discrete Surfaces

Example-Based Simulation

  • Frame-based Elastic Models (TOG)

  • Sparse Meshless Models of Complex Deformable Objects

  • Example-Based Elastic Materials

Circular arc structures

Circular Arc Structures

Pengbo Bo1,2 Helmut Pottmann2,3 Martin Kilian2

Wenping Wang1 Johannes Wallner2,4

1Univ. Hong Kong

2TU Wien


4TU Graz



Helmut Pottmann


Vienna University of Technology

Pengbo Bo

Postdoctoral Fellow

Univ. Hong Kong

Martin KilianRA

Vienna University of Technology

Wenping Wang


Univ. Hong Kong

Johannes Wallner


Graz University of Technology

Vienna University of Technology

Architectural geometry

Architectural Geometry

  • The most important guiding principle for freeform architecture

    • Balance

      • Cost efficiency

      • Adherence to the design intent

    • Key issue

      • Simplicity of supporting and connecting elements as well as repetition of costly parts

Node complexity

Previous work

Previous Work

  • Nodes optimization

    • [Liu et al. 2006; Pottmann et al. 2007] for quad meshes

    • [Schiftner et al. 2009] for hexagonal meshes

  • Rationalization with single-curved panel

    • [Pottmann et al. 2008]

  • Repetitive elements

    • [Eigensatz et al. 2010]

    • [Singh and Schaefer 2010] and [Fu et al. 2010]

    • The aesthetic quality is reduced if the number of repetitions increases.

This work

This Work

  • Propose the class of Circular Arc Structures (CAS)

  • Properties

    • Smooth appearance, congruent nodes, and the simplest possible elements for the curved edges

    • Do not interfere with an optimized skin panelization.

  • Contributions

    • freeform surfaces may be rationalized using CAS

    • repetitions not only in nodes, but also in radii of circular edges

    • extend to fully three-dimensional structures

    • have nice relations to discrete differential geometry and to the sphere geometries

Circular arc structures1

Circular Arc Structures

  • Definition

    A circular arc structure consists of 2D mesh combinatorics (V, E), where edges are realized as circular arcs, such that in each vertex the adjacent arcs touch a common tangent plane.

    We require congruence of interior vertices, and we consider the following three cases:

    • Hexagonal CAS have valence 3 vertices. Angles between edges equal 120 degrees;

    • Quadrilateral CAS have valence 4 vertices. Angles between edges have values α, π − α, α, π − α, if one walks around a vertex;

    • Triangular CAS have valence 6 vertices. Angles between edges equal 60 degrees.

Circular arc structures2

Circular Arc Structures

  • Data Structure

  • Target Functional

    • Deviation

    • Smoothness

    • Geometric consistency

    • Regularization

    • Angles

Circular arc structures3

Circular Arc Structures

  • Generalizations

    • Singularities

  • Supporting Elements

    • Add condition

Cas with repetitive elements

CAS with Repetitive Elements

  • Radius Repetitive

  • Definition

    A quadrilateral CAS is radius-repetitive along a flow line, if the radius of its edges is constant. It is transversely radius-repetitive for a pair of neighboring ‘parallel’ flow lines, if the edges which connect these flow lines have constant radius.

  • Condition

Cyclidic structure

Cyclidic Structure

  • Cyclidic CAS

  • Offsets

    • Offsetting operation of cyclidic CAS is well defined





  • Limitations

    • Loss of shape flexibility when additional geometric conditions are imposed.

    • The introduction of T-junctions

  • This Work

    • Shown the applicability of CAS

    • Demonstrated special CAS have more properties which are relevant for freeform building construction

  • Future Work

    • Explore more application

Discrete laplacians on general polygonal meshes

Discrete Laplacians on General Polygonal Meshes

Marc Alexa1 Max Wardetzky2

1TU Berlin

2Universitaat Gottingen



Marc Alexa


Electrical Engineering and Computer Science

TU Berlin

Max Wardetzky

Assistant Professor

Heading the Discrete Differential Geometry Lab

Universitaat Gottingen

This work1

This Work

  • Discrete Laplacianon surface with arbitrary polygonal faces

    • Non-planar & non-convex polygons

  • Mimic structural properties of the smooth Laplace-Beltrami operator

  • Motivation

    • Non-triangular polygons are widely used in geometry processing

Related work

Related Work

  • Geometric discrete Laplacians

    • Cotan formula [Pinkall and Polthier 1993]

    • The last decade has brought forward several parallel developments…

  • Application

    • Mesh parameterization

    • Fairing

    • Denoising

    • Manipulation

    • Compression

    • Shape analysis

Discrete laplacian framework

Discrete Laplacian Framework

  • Setup

  • An oriented 2-manifold mesh M, possibly with boundary, with vertex set V , edge set E, and face set F . We allow for faces that are simple, but possibly non-planar, polygons in R3.

    • Work with oriented halp-edge

    • EI, inner edges; EB boundary edge

  • Algebraic approach to discrete Laplacian

    • M0

    • M1



  • Locality

    • Maintain locality by only working with diagonal matrices M0 and by requiring that M1 is defined per face in the sense that

  • Symmetry : L = LT

  • Positive semi-definiteness

    • M0 & Mf are positive definiteness.

  • Linear precision

  • Scale invariance

  • Convergence

Vector area maximal projection

Vector Area & Maximal Projection

  • Vector Area

  • Maximal Projection

  • Mean Curvature

Maximal Projcetion

A family of discrete laplacians

A family of discrete Laplacians

  • [Perot and Suvramanian 2007]

—— pre-Laplacians

—— positive semi-definite



  • Construct 3 matrices

    • Diagonal matrx, M0

    • Coboundary matrix, d

      • dep = ±1 if e = ±eqp and dep = 0

    • M1

      • Assembled per face: Mf

Results application

Results & Application

  • Implicit mean curvature flow

  • Parameterization

Results application1

Results & Application

  • A planarizing flow

Results application2

Results & Application

  • Thin plate bending



  • This Work

    • presents here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons.

  • Feature Work

    • How to replace this combinatorial term by a more geometric one

Spin transformation of discrete surface

Spin Transformation of Discrete Surface

Keenan Crane1 Ulrich Pinkall2 Peter Schroder1

1California Institute of Technology

2TU Berlin



Ulrich Pinkall

Geometry Group

Institute of mathematics

TU Berlin

Keenan Crane

PhD Student

California Institute of Technology

Peter SchroderProfessor

Director of the Multi-Res Modeling Group

California Institute of Technology

This work2

This Work

  • Spin Transformation

    • A new method for computing conformal transformations of triangle meshes in R3

    • Consider maps into the quaternions H

Related work1

Related Work

  • Deformation

    • Local coordinate frame [Lipman et al. 2005, Paries et al. 2007]

    • Cage-based editing [Lipman et al. 2008]

  • Surface parametrization

    • Prescribe values at vertices that directly control the rescaling of the metric[Ben-Chen et al. 2008; Yang et al. 2008; Springborn et al. 2008].



  • Definition

    • The quaternions H can be viewed as a 4D real vector space with basis {1, i, j, k} along with the non-commutative Hamilton product, which satisfies the relationships i2 = j2 = k2 = ijk = −1.

    • The imaginary quaternions Im H are elements of the 3D subspace spanned by {i, j, k}.

    • q = a + bi + cj + dk, q = a - bi - cj – dk

    • Rotation of a vector , , (Similarity Transformation)

  • Calculus

    • Map f : M -> ImH

    • Differential df : TM -> ImH

Spin transformations

Spin Transformations

Spin transformations1

Spin Transformations

  • Integrable Condition [Kamberov et al. 1998]

    • D , Quaternionic Dirac Operator

  • Eigenvalue Problem

Spin transformations2

Spin Transformations

  • Procedure

    • Pick a scalar function ρ on M

    • Solve an eigenvalue problem

      for the similarity transformation λ

    • Sovle a linear system

      for the new surface



  • Discrete Dirac Operator



  • Scalar Multiplication

  • Discretized Spin Transformations



  • Painting Curvature



  • Arbitrary Deformation



  • This Work

    • Our discretization of the integrability condition (D − ρ)λ = 0 provides a principled, efficient way to construct conformal deformations of triangle meshes in R3.

  • Future Work

    • D is expressed in terms of extrinsic geometry it can be used to compute normal information, mean curvature, and the shape operator.

California institute of technology

California Institute of Technology

HOT: Hodge-Optimized Triangulations

Patrick Mullen Pooran Memari Fernando de Goes Mathieu Desbrun

This work3

This Work

  • “Good” dual

  • Motivation

    • Fluid simulation

  • This work

    • Hodge-optimized triangulation

Previous work1

Previous Work

  • Delaunay / Voronoi pairs

    • [Meyer et al. 2003]

    • [Perot and Subramanian 2007]

    • [Elcott et al. 2007]

    • Drawbacks

      • Circumcenter lies outside its associated tetrahedron

      • Inability to choose the position of dual mesh

      • Too restrictive in many practical situations





1 university of british columbia vancouver canada 2 university of grenoble 3 inria 4 ljk cnrs

1University of British Columbia, Vancouver, CANADA

2University of Grenoble



Frame-based Elastic Models

Benjamin Gilles1 Guillaume Bousquet2,3,4 Francois Faure2,3,4 Dinesh K. Pai1



Guillaume Bousquet

Second year PhD student

University of Grenoble

Laboratoire Jean KuntzmannINRIA

Benjamin Gilles

Post-doctoral Fellow

Sensorimotor Systems Lab

Department of Computer ScienceUniversity of British Columbia

François Faure

Assistant Professor

University of Grenoble

Laboratoire Jean KuntzmannINRIA

Dinesh K. Pai


Sensorimotor Systems Lab

Department of Computer ScienceUniversity of British Columbia

Deformable models terzopoulos et al 1988

Deformable Models [Terzopoulos et al. 1988]

  • Application

    • Computer animation

      • Animating characters, Soft objects, …

  • Approaches

    • Physically based deformation

    • Skinning

Physically based deformation nealen et al 2005

Physically based deformation [Nealen et al. 2005]

  • Finite Element Method

  • Lagrangian models of deformable objects

  • Two main method

    • Mesh-based methods

    • Meshless methods

  • Pros

    • Physical realism

  • Cons

    • Expensive

    • Difficult to use

Physically based deformation

Physically based deformation

  • Lagrangian mechanics

  • Simulation loop



  • Vertex blending / skeletal subspace deformation

  • Interpolating rigid transformation

  • Point is computed as

  • Pros

    • Sparse sampling

    • Efficient

  • Cons

    • Physically realistic dynamic deformation



  • Dual quaternion blending [Kavan et al. 2007]

    • Linear interpolation of screws

    • Reasonable cost

    • Well suited for parameterizing a physically based deformable model

This work4

This Work

  • New type of deformable model

  • Combination

    • Physically based continuum mechanics models

    • Frame-based skinning methods

This work5

This Work

  • Contribution

    • Creation models with sparse and intuitive sampling

    • on-the-fly adaptation to create local deformations

    • Effective

    • Integrated in SOFA

Modeling objects

Modeling Objects

  • Weight (Shape function)

  • Sampling

    • voxelization

Modeling objects1

Modeling Objects

  • Volume integrals

    • Compute the integral by regularly discrediting the volume inside the bounding box of the undeformed object

  • Fast pre-computed models

  • Adaptive

Validation results

Validation & Results

  • Implementation

    • Integrated in the SOFA (Simulation Open Framework Architecture)

  • Accuracy

Validation results1

Validation & Results

  • Deformation modeling

    • Using a reduced number of control primitives







  • This work

    • A new type of deformable model

    • Robust to large displacement and deformations

  • Future work

    • Hardware implementation

    • The relation between stiffness and weight functions could be exploited

Sparse meshless models of complex deformable solids

Sparse Meshless Models of Complex Deformable Solids

Francois Faure2,3,4 Benjamin Gilles1 Guillaume Bousquet2,3,4Dinesh K. Pai1

1University of British Columbia, Vancouver, CANADA

2University of Grenoble





Guillaume Bousquet

Second year PhD student

University of Grenoble

Laboratoire Jean KuntzmannINRIA

Benjamin Gilles

Post-doctoral Fellow

Sensorimotor Systems Lab

Department of Computer ScienceUniversity of British Columbia

François Faure

Assistant Professor

University of Grenoble

Laboratoire Jean KuntzmannINRIA

Dinesh K. Pai


Sensorimotor Systems Lab

Department of Computer ScienceUniversity of British Columbia

This work6

This Work

  • Goal

    • Deform objects with heterogeneous material properties and complex geometries.

Previous work2

Previous Work

  • Frame-based Method

  • Nodes

    • A discrete number of independent DOFs

    • Kernel functions (RBF)

    • Shape functions

      • Geometrically designed

      • Independent of the material

  • Displacement function

  • Problem

    • Impossible in interactive application

This work7

This Work

  • Novel: Material-aware shape function

  • Input

    • Volumetric map of the material properties

    • An arbitrary number of control nodes

  • Output

    • A distribution of the nodes

    • A associated shape function

  • Contributions

    • Material-aware shape function

    • Automatically model a complex object

    • High frame rates using small number of control nodes

Work flow

Work Flow

Material aware shape functions

Material-aware shape functions

  • Compliance Distance

Local compression:

Displacement function:

Shape function:

Compliance distance:

Slope of shape function:

Affine function!

Voronoi kernel functions

Voronoi kernel functions

  • Goal

    • Interpolating, smooth, linear and decreasing function

  • Voronoi subdivision

  • Dijkstra’ shortest path algorithm

RBF kernels

Our kernels

Node distribution: farthest point sampling [Martin et al. 2010]

Deformable model computation

Deformable model computation



  • Validation

    • Integrated in the SOFA

  • Performance





  • This Work

    • Novel, anisotropic kernel functions using a new definition of distance based on compliance, which allow the encoding of detailed stiffness maps in coarse meshless models. They can be combined with the popular skinning deformation method.

  • Future Work

    • Dynamic adaptivity of the models

    • Local deformations

Example based elastic materials

Example-based Elastic Materials

Sebastian Martin1 Bernhard Thomaszewski1,2Eitan Grinspunt3 Markus Gross1,2

1ETH Zurich

2Disney Research Zurich

3Columbia University



Bernhard Thomaszewski

Post-doctoral Researcher

Disney Research Zurich

Sebastian Martin

RA, PhD. Student


Eitan GrinspunAssociate Professor

Computer Science Dept.Columbia University

Markus Gross



Disney Research Zurich

This work8

This Work

  • An example-based approach simulating complex elastic material behavior

  • Due to its example-based, this method promotes an art-directed approach to solid simulation.

Related work2

Related Work

  • Material Models

    • The groundbreaking works [Terzopoulos et al. 1988]

    • Elastic models [Irving et al. 2004]

    • Plasticity and viscoelasticity [Bargteil et al. 2007]

    • Learning material properties from experiments [Bickel et al. Sig 2009]

  • Directing animations

    • Explicit control forces [Thurey et al. 2006]

    • Space-time constraints [Barbic et al. 2009]

  • Example-based graphical methods

    • State of the Art in Example-based Texture Synthesis [Wei et al. EG2009]

    • Example-Based Facial Rigging [Li et al. Sig 2010]

Work flow1

Work Flow

  • Interpolation

    • Construct a space of characteristic shapes by means of interpolation

  • Projection

    • Project configurations onto it by solving a minimization problem

  • Simulation

    • Define an elastic potential that attracts an object to its space of preferable deformations

Example manifold

Example Manifold

  • Example manifold by example interpolation

  • Interpolation Energy

Example projection

Example Projection

  • Projection Problem

  • Summary

Example design implementation

Example Design & Implementation

  • Example design

    • Same topology

    • What kind of examples should be used (3)

  • Embedding Triangle Meshes

    • High-quality surface details

  • Local and Global Examples





  • This Work

    • Intuitive and direct method for artistic design and simulation of complex material behavior.

  • Future Work

    • Optimization scheme should be increased

    • Develop methods to assist users to provide appropriate examples

    • Automatically select example poses from input animation

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