Mesh Processing§
author: | David Coeurjolly |
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author: | David Coeurjolly |
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Rough definition
Def.
A mesh is a piecewise linear geometrical structure embedded in
Usually, we assume
For instance, we can consider
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Triangular meshes | Quadtrilateral meshes | Tetrahedral meshes |
Why meshes?
Cellular structure
Vertices (0-cell), edges (1-cell), triangles (2-cell), tetrahedon (3-cell) simplices
Def. Simplex
A k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 affinely independent vertices
Def. Simplicial Complex
A simplicial complex is a set of simplices that
satisfies the following conditions:
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Strong topological structure (order, boundary, co-boundary,
each simplex is topological ball)
Many topological operations on meshes can be described as k-simplices operations
Specific combinatoric makes implementation easy
We remove the constraint that k-cells are defined by k+1-vertices
..but..
Neighboring faces for triangular meshes can be stored in a explicit structure
AdjF:
…but… not efficient for cellular meshes
Double-linked list of edges
cgal.org
Core topological operations maintaining the structure
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cgal.org
Valid combinatorial model in any dimension
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cgal.org
Principle
Start from a control mesh and apply local subdivision rules to generate finer object.
Subdivision rule
Main idea: from the subdivison mechanism we can prove that the
limit surface is (,
,…)
Related to B-spline schemes
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(resp. Catmull-Clark, Loop, Doo-Sabin)
Principle
Optimize the size of input mesh removing unnecessary triangles
More formally
Compute with
and
What kind of shape metric d ?
Local metric evaluation
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Fast and easy implement but
Idea
Parametrized a complex mesh (2-manifold) on a 2D parameter space
Issues
Many applications
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Conformal Parametrization
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Discrete formulation
Defs.
Two meshes
with same
topology are (conformally) equivalent iff
such that
or equivalently (logspace)
linear formulation
Topological issues
Practical solution: Cut and Open
Idea
Half-edge or Vertex-Edge data structure contains redundant information that can be compressed
We want to
Example: Triangular strip
<see blackboard>