Meshes

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Meshes
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Triangular Mesh (or mesh)

• A mesh is a set of adjacent triangles

edges
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Topological properties of a mesh

• Two Manifold or not
• yes if each edge is shared by max 2 faces
• two manifold good
• not two manifold bad
• most algorithms on meshes assume it is
  two-manifold

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Topological properties of a mesh

• Closed or open
• When closed, each edge is shared by exactly two
  faces

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Topological properties of a mesh

• Coherent orientation
• Is it possible to assign to each face an
  orientation so that the overall assignment is
  coherent?
• Coherent orientation coherent normals!

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How can we define a mesh?

• A mesh is a set of adjacent triangles
• How can we define it?
• Direct way
• An array of triangles
• and for each triangle three vertices
• and for each vertex three coordinate

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How can we define a mesh?

• Indexed mode
• Ordered list of vertices
• for each vertex its position
• Ordered list of faces
• for each face, 3 indices to vertices

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What about the attributes?

• Can be defined
• per vertex
• Define an attribute for each vertex
• per face
• Define an attribute for each face
• per wedge (vertex of face)
• Define three attributes for each face
• Most common attributes
• color
• Texture coordinates
• Normals

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Example PLY format

• Digital format for meshes
• Can be either binary, or ASCII (text)
• binary more compact and easy to machine-read
• ascii human-readable with a text-editor
• In both cases, it starts with an ASCII header.

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Example PLY format
cube.ply
ply format ascii 1.0 comment a simple
cube element vertex 8 property float x property
float y property float z element face 12 property
list uchar int vertex_indices end_header
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Example OFF format
LetterL.off
1 5 10 5 14 3 2 1 04 5 4 3 04 6 7 8 94 6 9
10 114 0 1 7 64 1 2 8 74 2 3 9 84 3 4 10 94
4 5 11 104 5 0 6 11
OFF12 10 400 0 03 0 03 1 01 1 01 5 00 5
00 0 13 0 13 1 11 1 1
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Mesh common tasks

• Given a mesh compute the bounding box
• Very useful in many applications, e.g. collisions
• How to compute it?
• We cycle on the vertices compute the max and
  the min of all x, y and z

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Mesh common tasks

• Given a mesh compute the normals per vertex
• How?

1 Set all the vertex normals to 0 2 cycle on each
face compute the normal normalize it add it
to the vertex normal 3 cycle on each
vertex normalize the normal
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More difficult tasks

• Compute a Bounding sphere
• Stripification
• Parametrize
• Automated simplification
• Detail recovery
• ...

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More difficult tasks

• Stripification
• Organize the triangles in triangle strips
• As long as possible

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More difficult tasks

• Parameterization
• assign a pair of texture coordinates to each
  wedge
• When there are seams (lines appearing in two
  separate images)
• replicate the vertices
• Save the texture coordinates for each wedge

v
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More difficult tasks

• Automated simplification
• Parameters A maximal error or a given number of
  faces

automatically
mesh originale 500K triangles
mesh semplificata 2K triangles
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Automated simplification
p e r f o r m a n c e
q u a l i t y
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Automated simplification
A hierarchy of Levels of Detail
LOD 2
LOD 3
LOD 4
LOD 1
To be used when viewer is far away
To be used when viewer is nearby
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Detail preservation (or texture for geometry)

• Idea
• Synthesize a texture
• To compensate for the detail lost during
  simplification

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500k triangles
Automated simplification
2k triangles
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simplified 2K triangles
originale 500K triangles