Triangle meshes

1. Triangle meshes Jarek Rossignac GVU Center and College of Computing Georgia Tech, Atlanta http://www.gvu.gatech.edu/~jarek

2. Pseudo-manifold Faces (flat or curved) Popular domain • Surfaces decomposed into simple manifolds (with boundary) • Represent each manifold surface as a triangle mesh • T-meshes are supported by optimized rendering systems • Easily derived from polygons and parametric surfaces

3. x y z c vertex 1 Samples (vertices): x y z c vertex 2 x y z c vertex 3 Triangle/vertex incidence: v4 1 2 3 Triangle 1 3 24 Triangle 2 4 5 2 Triangle 3 t3 7 56 Triangle 4 v5 6 5 8 Triangle 5 v2 8 51 Triangle 6 Focus on explicit representations • Samples: Location and attributes (color, mass) • Connectivity: Triangle/vertex incidence • Fit: Rule for bending triangles (subdivision surfaces, NURBS) V(3B+k) bits T = 2V V(6log2V) bits

4. 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

5. vertex corner triangle border edge Terminology • Vertex: • Location of a sample • Triangles: • Decompose approximating surface • Edge: • Bounds one or more triangles • Joins two vertices • 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

6. Triangle 1 vertex 1 vertex 3 vertex 2 Triangle 2 x y z x y z x y z Triangle 3 x y z x y z x y z x y z x y z x y z 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) Very verbose! Not good for traversal.

7. 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 byincidence! • 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

8. 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

9. v a Triangle 0 1 a Triangle 0 2 b Triangle 0 3 c Triangle 1 2 c Triangle 1 1 d Triangle 1 4 e Triangle 2 1 a vertex 1 x y z vertex 2 x y z vertex 3 x y z vertex 4 x y z 2 attribute a red attribute b blue 3 4 1 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 • 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

10. 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 • Manifoldvertices: 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

11. Manifold graph Non-manifold shape 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

12. 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 2kedge-loops without disconnecting the mesh • The genus of a T-mesh is the number of handles it has connected

13. 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

14. 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

15. 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

16. 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

17. vertices Add 1 vertex, 2 triangles, and 3 edges triangles edges 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 • Three times more edges than vertices: E=3V-6+6H • 2E/3-E+V=2-2H

18. c.v c 2 c.n 3 1 3 2 4 5 0 4 c.o 1 v o a Triangle 0 corner 0 1 7 a Triangle 0 corner 1 2 8 b Triangle 0 corner 235 c Triangle 1 corner 3 2 9 c Triangle 1 corner 4 1 6 d Triangle 1 corner 542 e vertex 1 x y z vertex 2 x y z vertex 3 x y z vertex 4 x y z attribute a red attribute b blue Corner table:data structure for T-meshes • Table of corners, for each corner c store: • c.v : integer reference to vertex table • c.o : integer reference to opposite corner • c.a : index to table of corner attributes • Make the corners of each triangle consecutive • 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 c.n.n c.t

19. v o a 2 Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 23 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 54 e 3 1 3 2 4 5 0 4 1 v o a Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 235 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 542 e 2 3 1 3 2 4 5 0 4 1 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} • 230, 131, 122, 143, 244, 125, … • Sort: • 122, 125 ...131... 143 ...230...244 … • Pair-up consecutive entries 2k and 2k+1 • (122, 125)...131... 143...230...244… • Their corners are opposite • (122,125)...131...143...230...244…

20. c.v c.l = c.n.n.o c.r = c.n.o c c.p=c.n.n c.n c.o 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

21. 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);

22. 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