1. Meshes

2. Triangular Mesh (or mesh)
A mesh is a set of adjacent triangles
faces
vertices
edges

3. 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 NO
YES

4. Topological properties of a mesh
Closed or open
When closed, each edge is shared by exactly two faces

5. 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! A 1 2 3
Opposite direction, coherent edge B C D

6. 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

7. 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

8. 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

9. 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.

10. 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

11. Example: OFF format
# vertices
# faces
# edges
LetterL.off
OFF
first face:
4 vertices:
with indices
3, 2, 1 and 0
index 0
index 3
index 2
index 1
x,y,z
2nd
vertex

12. 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

13. 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

14. More difficult tasks Compute a Bounding sphere Stripification
Parametrize
Automated simplification
Detail recovery
...

15. More difficult tasks Stripification
Organize the triangles in triangle strips
As long as possible

16. 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

17. More difficult tasks Automated simplification
Parameters: A maximal error or a given number of faces
automatically
mesh originale
500K triangles
mesh semplificata
2K triangles

18. Automated simplification
p e r f o r m a n c e
q u a l i t y

19. 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

20. Detail preservation (or texture for geometry)
Idea:
Synthesize a texture
To compensate for the detail lost during simplification

21. 500k 2k Ad-hoc created TEXTURE detail recovery rendering triangles
Automated
simplification
2k triangles, with texture mapping
rendering 2k
triangles

22. simplified but textured
2K triangles
originale
500K triangles
simplified
2K triangles