On 48 and Quasi 48 Meshes Luiz Velho Jonas Gomes Visgraf - - PowerPoint PPT Presentation

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on 4 8 and quasi 4 8 meshes
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On 48 and Quasi 48 Meshes Luiz Velho Jonas Gomes Visgraf - - PowerPoint PPT Presentation

Sep 17, 2023 535 likes •773 views

On 48 and Quasi 48 Meshes Luiz Velho Jonas Gomes Visgraf Laboratory IMPA Instituto de Matem atica Pura e Aplicada Rio de Janeiro, Brasil 1 Outline Background on 4-8 Meshes Hierarchical Quasi 4-8 Meshes Representation

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SLIDE 1

On 4–8 and Quasi 4–8 Meshes

Luiz Velho Jonas Gomes Visgraf Laboratory IMPA – Instituto de Matem´ atica Pura e Aplicada Rio de Janeiro, Brasil

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SLIDE 2

Outline

Background on 4-8 Meshes Hierarchical Quasi 4-8 Meshes Representation and Operations Construction Methods Recent Developments

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Background

Origins of Classical 4–8 Meshes

– Leaves Tilings – 4–direction Meshes – Quincunx Lattices

Generalizations of 4–8 Meshes

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SLIDE 4

Uniform Partitions of

R 2 Leaves Tilings [4 4 ] [6 3 ] [4:8 2 ]

* Symmetry: Crystallographic Groups

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Four Direction Meshes

Generated by the vectors (e 1 ; e 2 ; e 1 + e 2 ; e 1
  • e
2 )

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SLIDE 6

Quincunx Lattices

  • Q
= fM x; x 2 Z
  • Z
g, where M is the quincunx matrix M = 1 1 1 1 ! Two Interleaved Quincunx Lattices

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SLIDE 7

Four Direction Box Splines

  • C
1 basis function generated by the convolution set 1 1 1 1 1 1 !

* Defined on Quadrilaterals

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SLIDE 8

Generalization of 4–8 Meshes

Regular 4–8 Meshes

– 4–8 Topology

(

deg (v ) = 4; 8) Semi-Regular 4–8 Meshes

– Isolated Extraordinary Vertices

(

deg (v ) 6= 4; 8)

* Most of the properties of 4–8 Meshes

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SLIDE 9

Hierarchical 4–8 Meshes

Construction Methods

– Direct Refinement – Red-Black Refinement

Representation

– Multiresolution Representation – Variable Resolution Representation

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Direct Refinement

Quaternary Subdivision Template

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Red-Black Refinement

Recursive Binary Subdivision Template

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Hierarchy of Right Triangles

Binary Tree Representation

* Regular Case (Terrain)

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Variable Resolution Multi-Triangulation

Dependency Graph (DAG)

* Semi-Regular Case

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Query Operations

Cut in the DAG Definition:

– Threshold Function – Focus Set

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Mesh Extraction

Adapted Meshes

– Geometry Approximation – View-Dependent – Etc.

* Consistency by Construction

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Construction Methods

Local Modifications

– Refinement

(Edge Bisection)

– Simplification

(Half Edge Collapse)

refinement

# " simplification

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SLIDE 17

New Developments

Refinement Methods

– Topology-Based

(Generalized 4 direction Box Splines)

– Geometry-Sensitive

(Quasi 4–8 Subdivision Surfaces)

– Adaptive

(Tesselation of Parametric and Implicit Surfaces)

Simplification Methods

– Topology Preserving

(2-Manifold Triangle Meshes)

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Four Direction Box Splines

Interleaved Edge Subdivision

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Quasi 4–8 Subdivision Surfaces

Longest Edge Bisection

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Adaptive Surface Tesselation

Multiresolution Edge Sampling

parametric surface implicit surface

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SLIDE 21

Surface Simplification

Half-Edge Collapse and Edge Swap

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SLIDE 22

Future Work

Hierarchical Parametrizations Multiresolution Decomposition Space-Filling Hamiltonian Paths Integrated Framework

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