1. CS418 Computer Graphics John C. Hart
Meshes
CS418 Computer Graphics
John C. Hart

2. Simple Meshes Cylinder (x,y,z) = (cos q, sin q, z) Cone
(x,y,z) = (|z| cos q, |z| sin q, z)
Sphere
(x,y,z) = (cos f cos q, cos f sin q, sin f)
Torus
(x,y,z) = ((R + cos f) cos q,(R + cos f) sin q, sin f)

3. Good Meshes
Manifold: 1. Every edge connects exactly two faces 2. Vertex neighborhood is “disk-like”
Orientable: Consistent normals
Watertight: Orientable + Manifold
Boundary: Some edges bound only one face
Ordering: Vertices in CCW order when viewed from normal

4. Indexed Face Set Popular file format VRML, Wavefront “.obj”, etc.
(x4,y4,z4)
(x3,y3,z3)
Popular file format
VRML, Wavefront “.obj”, etc.
Ordered list of vertices
Prefaced by “v” (Wavefront)
Spatial coordinates x,y,z
Index given by order
List of polygons
Prefaced by “f” (Wavefront)
Ordered list of vertex indices
Length = # of sides
Orientation given by order
(x2,y2,z2)
(x1,y1,z1)
v x1 y1 z1
v x2 y2 z2
v x3 y3 z3
v x4 y4 z4
f 1 2 3
f 2 4 3

5. Other Attributes Vertex normals Prefixed w/ “vn” (Wavefront)
v x0 y0 z0
v x1 y1 z1
v x2 y2 z2
vn a0 b0 c0
vn a1 b1 c1
vn a2 b2 c2
vt u0 v0
vt u1 v1
vt u2 v2
(x2,y2,z2) (a2,b2,c2)
(u2,v2)
Vertex normals
Prefixed w/ “vn” (Wavefront)
Contains x,y,z of normal
Not necessarily unit length
Not necessarily in vertex order
Indexed as with vertices
Texture coordinates
Prefixed with “vt” (Wavefront)
Contains u,v surface parameters
Faces
Uses “/” to separate indices
Vertex “/” normal “/” texture
Normal and texture optional
Can eliminate normal with “//”
(x0,y0,z0) (a0,b0,c0) (u0,v0)
(x1,y1,z1) (a1,b1,c1)
(u1,v1)
f 0/0/0 1/1/1 2/2/2
f 0/0/0 1/0/1 2/0/2

6. Catmull Clark Subdivision
First subdivision generates quad mesh
Generates B-spline patch when applied to 4x4 network of vertices
Some vertices extraordinary (valence  4)
Rules:
Face vertex = average of face’s vertices
Edge vertex = average of edge’s two vertices & adjacent face’s two vertices
New vertex position = (1/valence) x sum of…
Average of neighboring face points
2 x average of neighboring edge points
(valence – 3) x original vertex position
(boundary edge points set to edge midpoints, boundary vertices stay put)

7. Implementation Face vertex For each face add vertex at its centroid
Face vertex = average of face’s vertices
Edge vertex = average of edge’s two vertices & adjacent face’s two vertices
New vertex position = (1/valence) x sum of…
Average of neighboring face points
2 x average of neighboring edge points
(valence – 3) x original vertex position
Face vertex
For each face add vertex at its centroid
Edge vertex
How do we find each edge?
New vertex position
For a given vertex how do we find neighboring faces and edges?
v x1 y1 z1
v x2 y2 z2
v x3 y3 z3
v x4 y4 z4 … f
...

8. Half Edge class HalfEdge { HalfEdge *opp; Vertex *end; Face *left;
HalfEdge *next; };
HalfEdge e;

9. Half Edge e e->opp() class HalfEdge { HalfEdge *opp; Vertex *end;
Face *left;
HalfEdge *next; };
HalfEdge e; e
e->opp()

10. Half Edge e->end() e e->opp()
class HalfEdge {
HalfEdge *opp;
Vertex *end;
Face *left;
HalfEdge *next; };
HalfEdge e;
e->end() e
e->opp()
e->start() = e->opp()->end();
e->start()

11. Half Edge e->left() e->right() e e->opp()
class HalfEdge {
HalfEdge *opp;
Vertex *end;
Face *left;
HalfEdge *next; };
HalfEdge e;
e->left()
e->right() e
e->opp()
e->right() = e->opp()->left();

12. Half Edge Can walk around left face until e(->next)n = e
class HalfEdge {
HalfEdge *opp;
Vertex *end;
Face *left;
HalfEdge *next; };
HalfEdge e;
e->next() e
e->opp()
e->opp()->next()
e->next()->next()
Can walk around left face until e(->next)n = e

13. Vertex Star Query e e->next() e->next()->opp()7 8 6 … 5 e
e->next()
e->next()->opp()
e->next()->opp()->next()
e->next()->opp()->next()->opp()
e->next()->opp()->next()->opp()->next()
e->next()->opp()->next()->opp()->next()->opp()
e->next()->opp()->next()->opp()->next()->opp()->next()
… until e(->next()->opp())n == e 4 3 2 1

14. Digital Michelangelo
In 1998 Marc Levoy takes a sabbatical year in Florence to scan a bunch of Michelangelo sculptures
David at 1mm resolution
St. Matthew at 290m resolution

15. Edge-Face-Vertex Half Edges Faces Vertices HalfEdge *opp; Vertex *end;
Face *left;
HalfEdge *next;
f <he>
f <he>
f <he> …
v <x1> <y1> <z1> <he>
v <x2> <y2> <z2> <he>
v <x3> <y3> <z3> <he> …
HalfEdge *opp;
Vertex *end;
Face *left;
HalfEdge *next;

16. Mesh Simplification
Meshes often contain more triangles than are necessary for visual fidelity
Some surface generation methods run at a fixed resolution regardless of surface detail
Meshes might be used on different resolution devices, e.g. cellphone
Need a method to reduce the number of triangles in a mesh
Must figure out which triangles to remove while preserving visual shape and appearance
424,376 triangles
60,000 triangles

17. Edge Collapse Removing a vertex turns triangle mesh into polygon mesh
Removing an edge…
Merges two vertices into one vertex
Removes two faces
Mesh still consists of triangles
Which edges should be removed first?
Where should the new vertices go?

18. QEM(v) = S (Ni ∙ v + Di)2 (for adjacent polygons i)
Vertex Importance
Plane equation
Pi(x,y,z) = Aix + Biy + Ciz + D
Pi(x) = Ni ∙ x + D
Pi(x) returns signed distance from x to plane passing through polygon i
Best position of new vertex position minimizes squared distance to original planes of original polygons
QEM(v) = S (Ni ∙ v + Di)2 (for adjacent polygons i) 2 1 1 3 6 4 5

19. Matrix Representation
Squared distance from point x to plane P
P2(x) = (Ax + By + Cz + Dz)2
= xT Q x
Sum of squared distances from point x to planes P1 and P2
P12(x) + P22(x) = xT (Q1 + Q2) x

20. Q(v) = S Qi (for adjacent polygons i)
Quadric Error Metric
Find Q for each vertex v
Q(v) = S Qi (for adjacent polygons i)
Edge collapse edges whose vertices v1,v2 have least Q(v1) + Q(v2)
New vertex v12 can be at v1 or v2 or optimally at …

21. Examples