Surfaces/Meshes Well stick to triangles Working with Meshes CS 176 - PDF document

Surfaces/Meshes We’ll stick to triangles Working with Meshes CS 176 Winter 2011 CS 176 Winter 2011 1 2 Discrete Surfaces What’s a Mesh? Setup Formally “pointers” “floats”  topology & geometry  abstract simplicial complex K  singletons, pairs, triples,… of integers  simplicial complex: “triangle mesh”  2-manifold if ld abstract simplices  containment property  Euler characteristic  partial order ¹ , face, coface, ∅ CS 176 Winter 2011 CS 176 Winter 2011 3 4 Simplicial Complex Topological Structure Topological realization 2-manifold (with boundary)  identify V with unit vectors in R N  every point has an open, (half-) disklike subset surrounding it complex convex hull of vertex images make subc  subset topology of ambient space b l b  closure, star, and link incidence  |K| 2-manifold iff |St v| ≈ R 2 “1-ring” CS 176 Winter 2011 CS 176 Winter 2011 5 6

Topological Invariants Simplicial Complex Euler characteristic Geometric realization  for surfaces: F-E+V=  =2(g-1)  the concrete embedding  v (|K|)  not required to be simplicial  more generally for simplicial ll f i li i l  vertex images specify everything t i if thi complexes  piecewise linear approximation  proof by  presumably approximation of induction (shelling) underlying smooth surface CS 176 Winter 2011 CS 176 Winter 2011 7 8 Mesh Structure Building the Mesh Input What do we need? x1 y1 z1 … x2 y2 z2 …  typically  array of pointers to vertices x3 y3 z3 … …  list of vertices (how long?) i j k  choices for basic topology primitive j k l  list of triangles (until EOF) li t f t i l ( til EOF) …  (half-)edges (h lf ) d  need to build mesh structure  different variants  triangles  infer topology  check topology  we’ll use triangles  oriented (orientable?) CS 176 Winter 2011 CS 176 Winter 2011 9 10 Types of Operations Types of Operations What do we need to support? What do we need to support?  iterate over all vertices (easy)  for a vertex visit  iterate over all triangles (easy)  star  link li k  for a triangle visit l  different flavors  incident vertices (easy)  need back pointer  incident triangles (easy)  vertex points to one incident triangle  careful at boundary! CS 176 Winter 2011 CS 176 Winter 2011 11 12

Types of Operations Operations to Support What about edges? For later (think about it now…)  visit all edges  edge collapse  not explicitly represented…  legality?  do we need edges? Yes! d d d ? Y !  edge flip d fli  discover triangle adjacencies  map pairs of integers to triangles CS 176 Winter 2011 CS 176 Winter 2011 13 14 Data Structures What Data Where? Triangles Attributes  consistent ordering of vertex and  normal, color, texture coordinates triangle incidences  later: forces, velocities, mass  why not just lay everything out in h t j t l thi t i Triangle{ Vertex *v[3]; arrays? Triangle *t[3];  changes in structure! }  very hard to debug…  triangles across from vertices CS 176 Winter 2011 CS 176 Winter 2011 15 16 Examples Example Vertex normals Gaussian curvature  gradient of volume CS 176 Winter 2011 CS 176 Winter 2011 17 18

Principles Other Tricks As you write code… As you write code  assumptions are ok, but you must  use two sided lighting assert them explicitly  abstract the iterators!  orientability  orientability  what about boundary vertices? h b b d i  2-manifold property  keep iterators sorted  avoid storing the same information  interior then boundary vertices multiple times  interior then boundary triangles  nasty to keep current under changes CS 176 Winter 2011 CS 176 Winter 2011 19 20