TOPOLOGY

1
TOPOLOGY

• Edge based data structures for solid modeling
• wireframe modeling, surface modeling, solid
  modeling
• mass properties, integrity and physical
  realizability
• Topology and Geometry separation

2
Solid modeling representations

• Boundary based vs. Volume based
• Object based vs. Spatially based
• Evaluated vs. Unevaluted
• Focus on boundary based, object based, evaluated
  form
• Various data structures for topology for planar
  and curved surface forms

3
Topology Data Structures

• V-E, F-E structures focus on use of edges
• Easy access to which side or end of edge is
  intended
• Simpler access procedures which are fast
• Requires additional data model elements
• New structures proposed by Weiler

4
Graphs and Adjacency

• Solid model has many kinds of information
  including surfaces and adjacency relationships
• Adjacencies can be derived using numerical
  techniques with high computational expense
• Evaluated representations have adjacency
  information explicitly

5
Graphs and Adjacency(contd)

• Adjacency information is referred to as the
  topology of solid model
• geometry descriptions (point, curve, surface
  definitions) are then geometry of solid model
• Topology serves as the framework to glue various
  pieces of information

6
Graphs and Adjacency(contd)

• Topology is saved in graph format
• Embedded graphs of boundaries and the topology
  relationships
• Vertex - a unique point
• Edge - an unordered set of 2 vertices
• Graph - a set of vertices and set of edges that
  use the vertices

7
Graphs and Adjacency(contd)

• Multigraph - a graph where multiple edges can
  join the same 2 vertices
• Self Loop - an edge joins a vertex to itself
• Pseudograph - a graph that allows both self-loop
  and multiple edges
• Labeled graph - a graph in which each vertex and
  edge is uniquely identified

8
Topology Concepts

• Topology is study of properties invariant under
  homeomorphisms
• Homeomorphism is a one-to-one, onto, topological
  transformation that is continuous and has a
  continuous inverse
• Example open-disk - 2d
• open sphere 3d analog of the open disk

9
Topology Concepts

• A manifold, in particular, a two-manifold is a
  2-d connected surface where each point on the
  surface has a neighborhood topologically
  equivalent to an OPEN DISK
• Embedded graph Drawn on a surface so that no
  two edges intersect, except at their incident
  vertices

10
Topology Concepts

• 9 element adjacency relationships (see figure 1
  page 23 in class notes) - VV, EV, FV, VE, EE, FE,
  VF, EF, FF
• Ordering terminology group - unordered list,
  (group) - linear ordered list, ltgroupgt - circular
  ordered list
• ltVgt - group of 0 or more circular vertices

11
Graph Data Structures

• Based on edge as the reference
• WE - winged edge, MWE - modified winged edge ,
  use of edge structures
• Shell structure list of faces
• face structure next, last, one vertex, edgelist
• vertex structure x, y, z coordinates

12
Topology data structures

• Inference analysis tasks need only check the
  higher level shell extents
• Backpointers are also common for efficiency
  purposes
• Winged Edge (WE structure) Provides boundary
  graph of faces, edges, vertices. Also provides
  for euler operators

13
Winged Edge (WE) structure

• Adjacency of given edge with other edges
• edge pointers to vert1, vert2, cwe1, cwe2,
  ccwe1, ccwe2, face1, face2 (Figure 3, page 26)
• graphical appearance w.r.t. reference edge
• cw and ccw looking from outside the volume
  toward surface.

14
WE, MWE structures

• E(V)-E((E)(E))-E(F) is available in WE
• Modified Winged Edge Structure Has additional
  data. Each cwe, ccwe pointers have edge-side
  fields cwe - cwehalf, ccwe- ccwehalf (figure 4,
  page 26)
• Reduces algorithm complexity - in curved surfaces

15
Additional structures

• V-E structure edges around a vertex forms a
  circular- ordered list (cwe field)
• F-E structure edges around a face forms a
  circular-ordered list (cwe field) - each side of
  edge is used only once as boundary of face
• many variations with back-pointers are possible

16
Topological Sufficiency

• Representation has data operators
• complete specification of the domain, proof of
  sufficiency, operators that cover entire domain
  yet cannot violate domain
• compact, orientable two-manifolds, embedded graph
  adjacency topology, pseudographs (figure 7, page
  28)

17
Topology sufficiency

• Labeled graphs to attach attributes, uniqueness,
  derivable relationships with other entities
• sufficiency is to recreate all 9 topology
  relations without error/ambiguity
• VltEgt, FltEgt, some EE sufficient individually over
  the selected domain

18
Conclusions

• WE, MWE, V-E, F-E structures
• 3 of four sufficient topologically
• WE needs additional data/algorithms to deal with
  curved surface environments
• needs to be aware of domain and occurrence of
  elements (use)
• WE, MWE have better record-accessing

19
Conclusions

• MWE has better efficiency w.r.t. WE
• V-E, F-E better performance for complex adjacency
  querries
• V-E, F-E require 20 more storage than W-E, MWE
  structures
• F-E is particularly well suited for hierarchical,
  edge-oriented, evaluated, object based, boundary
  graph solid models

20
STEP STANDARD

• ISO-10303 is an international standard for
  product data exchange (Standard for Exchange of
  Product Model Data)
• Modules AP203, AP209, AP224 etc.,
• geometry, product identification, product
  structure(assembly), configuration, manufacturing
  features, tooling, processes etc.,