1. Mesh Representation, part I
based on:
Data Structures for Graphics, chapter 12 in
Fundamentals of Computer Graphics, 3rd ed.
(Shirley & Marschner)

2. 3D objects
A single 3D object has a volume that is bounded by 2-dimensional patches (surfaces)
These 2-dimensional boundary patches are bounded by 1-dimensional features (curves or edges)
These 1-dimensional features are bounded by 0-dimensional things (points, called vertices)

3. 3D objects
facets
edges
vertices

4. 3D objects
We typically represent the boundary; the interior is implied to be the bounded subspace
We will assume linear boundaries here, so we have facets, edges, and vertices (and the interior)

5. Triangle meshes
Many freeform shapes consist of (many) triangles

6. Triangle meshes
How are triangles, edges, and vertices allowed to connect?
How do we represent (store) triangle meshes?
How efficient are such schemes?

7. Manifolds
A manifold is “watertight”: the interior is separated from the exterior everywhere
Very locally at every point on the manifold, it “looks like” the very local situation on a sphere

8. Manifolds
A manifold is “watertight”: the interior is separated from the exterior everywhere
Very locally at every point on the manifold, it “looks like” the very local situation on a sphere
These are non-manifolds, at the indicated places there are points whose very local neighborhood is not like on a sphere.

9. Manifolds Technically, we are discussing a 2-manifold in 3D
For any point on the 2-manifold, consider an arbitrarily small sphere centered at that point, and intersect the sphere boundary with the 2-manifold: this should be one closed loop
Left, there are two loops at the contact point. Right, at the contact edge there are two loops touching in two pinch points.

10. Manifolds

11. Manifolds
not a manifold
manifold

12. Triangle meshes for manifolds
A triangle mesh can be a manifold only if the following two local conditions are satisfied
Every edge is shared by exactly two triangles
Every vertex has a single, complete loop of triangles around it
A global condition is that the manifold does not self-intersect

13. Manifolds with boundary
A disk in 3D space is a manifold with boundary; the boundary is a circle
Any surface or surface patch is a manifold with boundary if
very locally at any point p, it is like how it is very locally at some point on a disk (boundary circle or interior)  the neighborhood of p is a single closed loop or one non-closed curve

14. Manifolds with boundary

15. Manifolds with boundary
Manifolds with boundary are not watertight
Self-intersecting surfaces cannot be manifolds with (nor without) boundary
Surfaces with parts that are thin tentacles cannot be manifolds with (nor without) boundary
Three things that are not a manifold, nor a manifold with boundary. Left, due to self-intersection of the surface patch. Middle and right, due to the existence of a point whose neighborhood is not a closed loop or single chain.

16. Triangle meshes for manifolds with boundary
A triangle mesh can be a manifold with boundary only if the following two local conditions are satisfied
Every edge is shared by one or two triangles
Every vertex connects to a single edge-connected set of triangles
A global condition is that the manifold does not self-intersect

17. Triangle meshes For manifolds with or without boundary Store triangles
Store coordinates of vertices
Possibly store edges, adjacencies of triangles, etc., depending on the scheme
9 vertices, 16 edges, and 8 triangles in a triangle mesh

18. Triangle meshes
Features of the same dimensionality are adjacent or not (two vertices, or two edges, or two triangles)
Features of different dimensionality are incident or not (vertex-edge, vertex-triangle, edge-triangle)
adjacent triangles
adjacent vertices
adjacent edges

19. Triangle meshes
Features of the same dimensionality are adjacent or not (two vertices, or two edges, or two triangles)
Features of different dimensionality are incident or not (vertex-edge, vertex-triangle, edge-triangle)
non-adjacent triangles
non-adjacent vertices
non-adjacent edges

20. Triangle meshes
Features of the same dimensionality are adjacent or not (two vertices, or two edges, or two triangles)
Features of different dimensionality are incident or not (vertex-edge, vertex-triangle, edge-triangle)
incident edge and triangle
incident edge and vertex
incident vertex and triangle

21. Triangle meshes
Features of the same dimensionality k>0 are adjacent if and only if their boundaries (or their closures) contain the same feature of dimensionality k – 1
Vertices are adjacent if they are endpoints of some edge (incident on a common edge)
Features of different dimensionality are incident if the feature with lower dimensionality is a subset of the boundary (or closure) of the feature with higher dimensionality

22. Orientation of triangles
By convention, triangles of a manifold are oriented so that from the outside, the triangle is counter-clockwise (and seen from the inside it is clockwise)
Orientation is not needed for representation, but it may be convenient for coding to use a consistent orientation

23. Orientation of triangles
For a manifold with boundary, we can define a front and a back where the front has triangles oriented counter-clockwise
This only works for orientable surfaces with boundary
Möbius strip

24. Non-orientable surfaces
Klein bottle
A Klein bottle is not realizable without intersections in 3D, so in 3D it is not a manifold.
Möbius strip
non-orientable surface without boundary

25. Triangle mesh schemes (simple)
Separate triangles mesh
Shared vertex mesh
Indexed triangle mesh
....

26. Separate triangles mesh
Each triangle is an object that stores the coordinates of its vertices  the same coordinates are stored multiple times v6 T1
(x1,y1,z1)
(x5,y5,z5)
(x6,y6,z6) v1 v7 T2
(x1,y1,z1)
(x2,y2,z2)
(x5,y5,z5) T1 T6 T5 T2 v5 T3
(x2,y2,z2)
(x3,y3,z3)
(x5,y5,z5) v2 T7 T3 T4 v8 v4   v3

27. Shared vertex mesh
Each vertex is an object that stores its coordinates
Each triangle is an object that stores references to its vertices v1
(x1,y1,z1) v2
(x2,y2,z2) v6 v3
(x3,y3,z3) v1 v7  T1 T6 T1 v1 v5 v6 T5 T2 v5 v2 T7 T2 v1 v2 v5 T3 T4 v8 v4 T3 v2 v3 v5 v3  

28. Indexed triangle mesh
Same as shared vertex mesh but the j-th vertex of the i-th triangle is addressed as Array[ i ][ j ], j = 1,2,3 and i = 1,2, ..., no. of triangles v1
(x1,y1,z1) v2
(x2,y2,z2)
A[1][1] = 1
A[1][2] = 5
A[1][3] = 6
A[2][1] = 1
A[2][2] = 2
A[2][3] = 5 v6 v3
(x3,y3,z3) v1 v7  T1 T6 T5 T2 v5 v2 T7 T3 T4 v8 v4  v3

29. Comparison of simple meshes
Assume same storage for floats, ints, and pointers:
Separate triangles mesh: 9 nt units for nt triangles
Other two: 3 nv + 3 nt units for nv vertices and nt triangles
In meshes we roughly have: nt  2 nv Why?
All triangles have 3 edges, and most of the edges are incident to 2 triangles. So 3 nt  2 ne
Euler’s formula for connected planar graphs states nv – ne + nt = 2

30. Comparison of simple meshes
Assume same storage for floats, ints, and pointers:
Separate triangles mesh: 9 nt units for nt triangles
Other two: 3 nv + 3 nt units for nv vertices and nt triangles
In meshes we roughly have: nt  2 nv
Separate triangles mesh:  18 nv units of storage
Shared or indexed mesh:  9 nv units of storage

31. In-memory versus transfer
Indexed triangle meshes are the most common in-memory representation of triangle meshes
For transferring meshes, triangle fans and triangle strips can save bandwidth
v12 v6
T11 v1 v7 T1 T6 T5
v11 T2
T12 v5 v4 v2 T7 T3 T4 v8
T10 T9 T8 v3
v10 v9

32. In-memory versus transfer
Indexed triangle meshes are the most common in-memory representation of triangle meshes
For transferring meshes, triangle fans and triangle strips can save bandwidth
v12 v6
indexed mesh:
T11 v1 v7
345, 465, 647, 487 T1 T6 T5
v11 T2
T12 v5 v4 v2 T7
triangle fan T3 T4 v8
T10
means triangles with fan center at 4 and sequence 35678 T9 T8 v3
v10 v9

33. In-memory versus transfer
Indexed triangle meshes are the most common in-memory representation of triangle meshes
For transferring meshes, triangle fans and triangle strips can save bandwidth
v12 v6
indexed mesh:
T11 v1 v7
345, 465, 647, 487 T1 T6 T5
v11 T2
T12 v5 v4 v2 T7
triangle strip T3 T4 v8
T10
235467(12) means first triangle 235, then 35 with 4, then 54 with 6, then 46 … T9 T8 v3
v10 v9

34. In-memory versus transfer
Triangle fans/strips use k+2 indices for a fan/strip with k triangles; an indexed mesh would use 3k indices
We need a decomposition of the mesh into fans or strips
Strips are often long, fans seldom
v12 v6
T11 v1 v7
v11 T1 T6 T5 T2
T12 v5 v4 v2 T7 T3 T4 v8
T10 T9 T8 v3
v10 v9

35. Model with triangle strips shown

36. Efficiency of triangle strips
Suppose a surface with n triangles can be represented using m triangle strips, then we need n + 2m indices avg strip length indices per triangle without strips
Minimizing the number of strips is basically a puzzle
To get few triangle strips, one uses heuristics to decompose a triangle mesh into strips.

37. Connectivity structures for meshes
Let triangles have direct access to the adjacent triangles
Allows efficient editing of the mesh
Allows neighborhoods on the mesh to be explored efficiently
Allows paths on the mesh to be followed efficiently
v12 v6
T11 v1 v7 T1 T6 T5 T2 v5 v4 v2 T7 T3 T4 v8
T10 T9 T8 v3
v10 v9

38. Connectivity structures for meshes
Let triangles have direct access to the adjacent triangles
Allows efficient editing of the mesh
Allows neighborhoods on the mesh to be explored efficiently
Allows paths on the mesh to be followed efficiently
In an indexed mesh, how do you find the three vertices “opposite” a triangle? (v1, v3, and v7 for the shown triangle)
v12 v6
T11 v1 v7 T1 T6 T5
Answer: You have to go through al triangles in sequence, and for each, test whether it has exactly two vertices in common with the given triangle. This will be the case for at most three triangles. For these, we report their third vertex (the one they do not have in common with the given triangle). T2 v5 v4 v2 T7 T3 T4 v8
T10 T9 T8 v3
v10 v9

39. Triangle neighbor structure
Take the shared vertex mesh and add a pointer from each triangle to the three adjacent triangles
Optional: let each vertex have a pointer to one incident triangle
nbr[2]
Triangle {
Triangle nbr[3]
Vertex v[3] … }
v[0]
Note that edges are not represented at all in a triangle neighbor structure. They are in the mesh, but not in the mesh representation. Their existence is implied by the triangles and vertices.
v[2]
nbr[0]
nbr[1]
v[1]

40. Triangle neighbor structure
For manifolds with boundary, a triangle at the boundary will have one of its nbr[.] point to NIL, or use index –1 to reference a neighbor (a triangle can also have two edges that are on the boundary of the manifold)

41. Triangle neighbor structure
How do you find the three vertices opposite a triangle t?
The adjacent triangles are directly accessed as t.nbr[0], t.nbr[1], and t.nbr[2]
For t.nbr[0], we determine the vertex that is not equal to t.v[0] or t.v[1]
For t.nbr[1] and t.nbr[2] it is analogous
v12 v6
T11 v1 v7 T1 T6 T5
Code optimization question: how many vertex-vertex comparisons are needed to find these three vertices? You can do it with at most six in total, if you use the consistent orientation of triangles having their vertices enumerated counterclockwise in v[]! This shows that making structuring choices and being consistent in these structuring choices can be more efficient in running time (and program code length). T2 v5 v4 v2 T7 T3 T4 v8
T10 T9 T8 v3
v10 v9

42. Triangle neighbor structure
Expressed in algorithmic efficiency notation:
In an indexed mesh, finding opposite vertices for a triangle takes linear time in the mesh size, O(n)
In a triangle neighbor structure, finding opposite vertices takes constant time (independent of mesh size), O(1)

43. Triangle neighbor structure
Storage usage expressed in number of vertices nv:
An indexed mesh uses 9 nv units of storage
A triangle neighbor structure uses 15 nv units of storage (a triangle uses 6 units and a vertex uses 3 units, and there are roughly twice as many triangles as vertices)
Explain why 15 nv is the right estimate for a triangle neighbor mesh

44. Questions: height profile
A triangle mesh can be used to represent a terrain, where the third coordinate is height above sea level. How efficiently can you compute a height profile, a cross-section with a given vertical plane
using an indexed mesh?
using a triangle neighbor structure?
Assume there are proportional to sqrt(n) triangles on the outer sides of the terrain.
We intersect each of the n triangles separately with the vertical plane, leading to O(n) time.
We find the starting triangle on a side in O(sqrt(n)) time, and then we traverse the terrain in O(1) time per step to the next triangle. So we get O(sqrt(n)+ size of profile) time. The size of the profile is typically O(sqrt(n)).
So method b is usually much more efficient.

45. Winged-edge structure
Stores connectivity at edges instead of triangles
For one edge, say, e1:
Two vertices are important: v4 and v6
Two triangles are important: T5 and T6
Four edges are important: e2, e14, e5, and e8 v6 e7 e8 v1 v7
e14 T1 T6 e3
e12 e1 T5
e10 T2 v5 e5 v2 T7 e9 e2 e4 T3 T4 e6 v8
e11 v4
e13 v3

46. Winged-edge structure
Give e1 a direction, then
v4 is the tail and v6 is the head
T5 is to the left and T6 is to the right
e2 is previous on the left side, e14 is next on the left side, e5 is previous on the right side, and e8 is next on the right side v6 e7 e8 v1 v7
e14 T1 T6 e3
e12 e1 T5
e10 T2 v5 e5 v2 T7 e9 e2 e4 T3 T4 e6 v8
e11 v4
e13 v3

47. Winged-edge structure
Give e1 a direction, then
v4 is the tail and v6 is the head
T5 is to the left and T6 is to the right
e2 is previous on the left side, e14 is next on the left side, e5 is previous on the right side, and e8 is next on the right side v6
head e8
rnext
e14
lnext T6
right e1 T5
left e5
rprev e2
lprev v4
tail

48. Winged-edge structure
Edge lprev, lnext, rprev, rnext;
Vertex head, tail;
Triangle left, right }
head
rnext
lnext
Vertex {
double x, y, z;
Edge e; // any incident }
right
left
rprev
lprev
tail
Triangle {
Edge e; // any incident }

49. Questions
The triangle strip savings table shows only the average number of indices when triangle strips have an average length. We must also transfer the vertices themselves. Assuming that indices and coordinates use the same amount of storage, make a table with the total transfer savings when using triangle strips
Analyze how much storage the winged-edge structure needs for a mesh with nv vertices
For a given triangle t, write code for reporting the coordinates of its vertices (winged-edge)
For a given triangle t, write code for reporting the coordinates of the opposite vertices (winged-edge)
For a given vertex v, write code for reporting the coordinates of all the adjacent vertices (winged-edge)