Triangle meshes

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Triangle meshes

• Jarek Rossignac
• GVU Center and College of Computing
• Georgia Tech, Atlanta
• http//www.gvu.gatech.edu/jarek

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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

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Focus on explicit representations

• Samples Location and attributes (color, mass)
• Connectivity Triangle/vertex incidence
• Fit Rule for bending triangles (subdivision
  surfaces, NURBS)

V(3Bk) bits
T 2V
V(6log2V) bits
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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

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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

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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.
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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

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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

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Incidence table

• Integer Ids for vertices (0, 1, 2 V-1)
  triangles (0, 1, 2T-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

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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

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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

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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
• The genus of a T-mesh is the number of handles it
  has

connected
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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

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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

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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

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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
• T2V-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
• C3T6V-12
• On average, a vertex is used 6 times

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Properties of manifold meshes

• T triangles, E edges, V vertices, H handles, S
  shells
• Euler T-EV2S -2H
• Example a tetrahedron has T4, E6, V4, S1,
  H04-642-0
• Number of handles HS-(T-EV)/2
• Shared edges E3T/2
• 3 borders per triangle, 2 borders per edge
• Twice more triangles than vertices T2V4(H-S)
• T-3T/2V2S-2H
• Assume H and S are much smaller than V
• Three times more edges than vertices E3V-66H
• 2E/3-EV2-2H

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Corner tabledata 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 (c1)MOD 3

c.n.n
c.t
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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 2k1
• (122, 125)...131... 143...230...244
• Their corners are opposite
• (122,125)...131...143...230...244

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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

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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)

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Summary

• Samplesincidence 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, T2V-4, H0, S1