Triangle meshes
Jarek Rossignac
GVU Center and College of Computing
Georgia Tech, Atlanta
http://www.gvu.gatech.edu/~jarek
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 1
Popular domain
• Surfaces decomposed into simple manifolds (with boundary)
Pseudo-manifold
Faces (flat or curved)
• Represent each manifold surface as a triangle mesh
– T-meshes are supported by optimized rendering systems
– Easily derived from polygons and parametric surfaces
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 2
Focus on explicit representations
• Samples: Location and attributes (color, mass)
• Connectivity: Triangle/vertex incidence
• Fit: Rule for bending triangles (subdivision surfaces, NURBS)
Samples
(vertices):
vertex 1 x y z c
vertex 2 x y z c
vertex 3 x y z c
V(3B+k) bits
Triangle/vertex
incidence:
v4
Triangle 1
Triangle 2
Triangle 3
Triangle 4
Triangle 5
T = 2V
Jarek Rossignac, GVU Center, Georgia Tech
V(6log2V) bits Triangle 6
1: T-meshes
1
3
4
7
6
8
2 3
24
5 2
5 6 v5
5 8
51
t3
v2
3D Compression, SM’02 3
Samples, connectivity, & attributes
• Samples (“vertices”)
– Location (x,y,z)
• Connectivity (“triangles”)
–
–
–
–
Define how surface interpolates samples
Specifies surface as a set of triangles
Associates each triangle with 3 samples (called corners)
Define how to interpolate corner attributes over triangle
• Attributes (parameters for color and texture calculations)
– One per corner of each triangle
• Could be the same for all 3 corners (flat triangle)
• Could be the same for two adjacent corners (smooth edge)
• Could be the same for all coincident corners (smooth surface)
– Linear interpolation of shape and attributes over triangle
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 4
Terminology
vertex
• Vertex:
corner
– Location of a sample
triangle
• Triangles:
– Decompose approximating surface
• Edge:
– Bounds one or more triangles
– Joins two vertices
border
edge
• Corner:
– Abstract association of a triangle with a vertex
– May have its own attributes (not shared by corners with same vertex)
• Used to capture surface discontinuities
• Border (half-edge, dart):
– Association of a triangles with a bounding edge.
– Defines an orientation of the border
• A triangle has 3 borders and 3 corners
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 5
Representation as independent triangles
• For each triangle:
– For each one of its 3 corners, store:
• Location
• Attributes (may be the same for neighboring corners)
• Each vertex location is repeated (6 times on average)
– geometry = 36 B/T (float coordinates: 9x4 B/T)
– Plus 3 attribute-sets per triangle (6 per vertex)
vertex 1 vertex 2 vertex 3
Triangle 1
Triangle 2
Triangle 3
xyzxyzxyz
xyzxyzxyz
xyzxyzxyz
Very verbose! Not good for traversal.
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 6
Connectivity = Incidence + adjacency
• Triangle/vertex incidence (“corner”)
– Associates each triangle with 3 vertices
• Defines Corners
• Triangle/triangle adjacency
– Associates triangle with neighboring triangles
• Neighboring triangles share a common edge
– Is completely defined by incidence!
– Convenient to accelerate traversal of triangulated surface
•
•
•
•
Walk from one triangle to an adjacent one (visit them all once)
Used to build triangle strips
Used to estimate surface normals at vertices
Used to compress triangulated surfaces
• Triangle/edge incidence (border)
– Associates triangles with their bounding edges
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 7
Triangle orientation
• Orient the plane supporting a triangle
– Pick one of 2 possible orientations for the normal
• List corners in counter clockwise order
– Cyclic order (equivalence under cyclic permutation)
• Orientation compatibility for adjacent triangles
– Common corners follow each other in reverse order
• Can try to propagate consistent orientation
– Pick orientation for first triangle
– Propagate to neighbors (edge-connected), needs not use geometry
– What if more than two triangles share same edge?
• Orientable set of triangles
– Can all triangles be oriented to be consistent with their neighbors?
• Only if the mesh is orientable
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 8
Incidence table
• Integer Ids for vertices (0, 1, 2… V-1) & triangles (0, 1, 2…T-1)
• Triangle orientation: cyclic order in which corners are listed
2
• Other connectivity info may be derived
– Borders: defined by 3 pairs of corners for each triangle
– Edges: Set of borders with same two vertices
– Adjacency: Triangles incident upon the same edge
• Incidence graph representation:
– List of corners (id for vertex & attribute)
– Corners of each triangle are consecutive
• Samples defined separately:
– List of vertex locations (x,y,z)
– List of attributes (color,normal, texture…)
• Not practical for traversing mesh
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
va
3
4
1
vertex 1 x y z
Triangle 0 1 a
vertex 2 x y z
Triangle 0 2 b
vertex 3 x y z
Triangle 0 3 c
vertex 4 x y z
Triangle 1 2 c
Triangle 1 1 d
Triangle 1 4 e
attribute a red
Triangle 2 1 a
attribute b blue
3D Compression, SM’02 9
T-meshes: manifold connectivity graph
• T-mesh = triangle set with a manifold connectivity graph
– The corners of each triangle refer to different vertices
– Each edge has exactly two incident triangles
– Manifold vertices: The star of each vertex is connected
• Star = union of edges and triangles incident upon the vertex
• Triangles form a surface that may be globally oriented
– All triangle orientations are consistent (No Klein bottle)
• All triangles form a connected set
– All pairs of triangles are connected
• Two adjacent triangles are connected
• Connectivity is a transitive relation
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 10
Connectivity/geometry discrepancy
• Connectivity of T-mesh may conflict with actual geometry
– Vertices with different names may be coincident
– Edges with different names may be coincident
– Triangles, edges, and vertices may intersect
• T-mesh with consistent geometry
– Triangles, edges, vertices are pairwise disjoint
• We consider edges and triangles to be open
– I.e., not containing their boundary
• Manifold graphs may be used with invalid geometry
– Coincident edges and vertices: Non-manifold singularities
– Self-intersecting surfaces
Manifold graph
Non-manifold shape
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 11
Handles in T-meshes
• Handles correspond to through-holes
– A sphere has zero handles, a torus has one
• The number of handles is well defined in a T-mesh
– A handle cannot be identified as a particular set of triangles
• An edge-loop is a cycle of oriented edges
– Each starts where the previous one ended, no repetition of vertices
• A T-mesh has k handles if and only if you can remove at
most 2k edge-loops without disconnecting the mesh
connected
• The genus of a T-mesh is the number of handles it has
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 12
Simple T-meshes (STM) and T-patches
• We first look at simple meshes (no handles)
– Homeomorphic to a sphere
– Incidence graph is a planar triangle graph
• We will also use the notion of a T-patch
– Connected portion of an STM
– Bounded by a single edge-loop
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 13
Simple mesh
• A simple mesh is homeomorphic to a triangulated sphere
–
–
–
–
Orientable
Manifold
No boundary (no holes)
No handles (no throu-holes)
• Properties
– Each edge has exactly 2 incident triangles
– Each vertex has a single cycle of incident triangles
– May be drawn as a planar graph
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 14
Dual graphs and spanning trees
From Bosen
• Dual graph:
– Nodes represent triangles
– Links represent edges
• Join centers of adjacent triangles
• Vertex spanning tree (VST)
– Edge-set connecting all vertices
– No cycles
– Cuts mesh into simply connected polygon with no interior vertices
• Triangle-spanning tree (TST)
– Graph of remaining vertices
– No loops
– Connects all triangles
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 15
Euler formula for Simple Meshes
• Mesh has V vertices, E edges, and T triangles
• E = (V-1)+(T-1)
– VST has V nodes and thus V-1 links
– TST has T nodes and thus T-1 links
• E = 3T/2
– There are 3 borders (edge-uses) per triangle
– There are twice more edge-uses then edges
•
T=2V-4
– Because (V-1)+(T-1) = 3T/2
– And hence V-2 = 3T/2-T = T/2
• There are twice more triangles than vertices
• The number C of corners (vertex-uses) is about 6V
– C=3T=6V-12
• On average, a vertex is used 6 times
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 16
Properties of manifold meshes
• T triangles, E edges, V vertices, H handles, S shells
• Euler: T-E+V=2S -2H
– Example: a tetrahedron has T=4, E=6, V=4, S=1, H=0…4-6+4=2-0
• Number of handles: H=S-(T-E+V)/2
• Shared edges: E=3T/2
– 3 borders per triangle, 2 borders per edge
• Twice more triangles than vertices: T=2V+4(H-S)
– T-3T/2+V=2S-2H
– Assume H and S are much smaller than V
Add 1 vertex,
2 triangles,
and 3 edges
vertices
triangles
edges
• Three times more edges than vertices: E=3V-6+6H
– 2E/3-E+V=2-2H
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 17
Corner table:data structure for T-meshes
c.v
• Table of corners, for each corner c store:
c
– c.v : integer reference to vertex table
– c.o : integer reference to opposite corner
– c.a : index to table of corner attributes
c.n
• Make the corners of each triangle consecutive
c.o
– List them according to consistent orientation of triangles
– Can retrieve triangle number: c.t = c DIV 3
– Can retrieve next corner around triangle: c.n = 3t + (c+1)MOD 3
voa
Triangle 0 corner 2 3 5 c
Triangle 1 corner 3 2 9 c
Triangle 1 corner 4 1 6 d
c.t
2
Triangle 0 corner 0 1 7 a
Triangle 0 corner 1 2 8 b
c.n.n
3
2
1 3
0
4
1
5
4
vertex 1 x y z
attribute a red
vertex 2 x y z
attribute b blue
vertex 3 x y z
vertex 4 x y z
Triangle 1 corner 5 4 2 e
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 18
Computing adjacency from incidence
• c.o can be derived from c.v (needs not be transmitted):
• Build table of triplets {min(c.n.v, c.n.n.v), max(c.n.v, c.n.n.v), c}
voa
– 230, 131, 122, 143, 244, 125, …
3
2
2
Triangle 1 corner 0 1
a
1 3
Triangle 1 corner 1 2
b
Triangle 1 corner 2 3
c
Triangle 2 corner 3 2
c
Triangle 2 corner 4 1
d
Triangle 2 corner 5 4
e
0
• Sort:
4
5
4
– 122, 125 ...131... 143 ...230...244 … 1
• Pair-up consecutive entries 2k and 2k+1
voa
– (122, 125)...131... 143...230...244…
• Their corners are opposite
2
– (122,125)...131...143...230...244…
3
2
4
1
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
a
Triangle 1 corner 1 2
b
Triangle 1 corner 2 3 5 c
1 3
0
Triangle 1 corner 0 1
5
4
Triangle 2 corner 3 2
c
Triangle 2 corner 4 1
d
Triangle 2 corner 5 4 2 e
3D Compression, SM’02 19
Accessing left and right neighbors
• Can identify opposite corners of right and left neighbors
– c.r = c.n.o
– c.l = c.n.n.o
c.v
c.l = c.n.n.o
c
c.r = c.n.o
c.p=c.n.n
c.n
c.o
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 20
Using adjacency table for T-mesh traversal
• Visit T-mesh (triangle-spanning tree)
– Mark triangles as you visit
– Start with any corner c and call Visit(c)
– Visit(c)
• mark c.t;
• IF NOT marked(c.r.t) THEN visit(c.r);
• IF NOT marked(c.l.t) THEN visit(c.l);
• Label vertices
– Label vertices with consecutive integers
– Label(c.n.v); Label(c.n.n.v); Visit(c);
– Visit(c)
•
•
•
•
IF NOT labeled(c.v) THEN Label(c.v);
mark c.t;
IF NOT marked(c.r.t) THEN visit(c.r);
IF NOT marked(c.l.t) THEN visit(c.l);
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 21
Summary
•
•
•
•
•
Samples+incidence graph define triangles and corners (c.v)
Attributes attached to corners (may be same for neighbors)
T-mesh: oriented manifold incidence (consistent geometry?)
Adjacency (c.o) supports T-mesh traversal (derived from c.v)
Simple meshes, T=2V-4, H=0, S=1
Jarek Rossignac, GVU Center, Georgia Tech
1: T-meshes
3D Compression, SM’02 22