Creating geomodels with Generalized Maps

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Creating geomodels with Generalized Maps
Marcus Apel, 2004based on works of the Gocad
Research Group, Nancyreference J-L Mallet
 Geomodeling , Oxford University Press 2002
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Reminders How are objects defined?

• cellular partition sub-division into cells in
  N-dimensions
• connectivity with other objects
• embedding
• geometry in 3d space
• numerical properties

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cellular partition and connectivity - incidence
graph

• describes all possible paths to go from a n-cell
  to a 0-cell, with only connecting k-cells to
  (k-1)-cells, ngtkgt0

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incidence graph

• nodes ? Cells, edges ? CellViews
• incidence cell C is an element of CellView w.

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3 relationship classes in a GMap

• a cell Ci is i-adjacent to another cell Ck, iff
• ki
• a (i-1)-cell incident to both can be found, or
  (i, k) are incident to at least one common
  (i1)-cell
• example 2 triangles (i,k2) sharing an edge
• a cell Ci is a sub-cell of Ck, iff
• iltk
• Ci can be reached by descending the incidence
  graph
• example polygon (k2) with sub-cells edges (i1)
  and vertices (i0)
• the k-adjacency of a i-cell Ci contains all
  k-cells Ck such that Ci is a subcell of Ck.
• example all polyhedra (k3) connected to a
  vertex (i0)

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adjacency
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adjacency
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adjacency
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adjacency classes
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darts, involution, GMap

• a dart corresponds to a path (CellView) in the
  incidence graph
• adjacency relations are translated in relations
  between pairs of darts
• a dart has N1 ai functions (maps), defined from
  Ai adjacency relations
• for example a0 connects each pair of darts
  (d1,d2) a0(d1)d2 a0(d2)d1 ? involution
• a N-GMap D, (a0, ..., aN) is a set D of darts
  with a set of N1 ai involution functions

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incidence graph
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Simplified data structure of a GMap

• struct GMap SetltDartgt d
• struct Dart Dart alphaN1 Embedding
  embN1

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GMap - Example

• GMapd1a0,d2a0,..., d3a1 // darts for 63
  involutions
• Dart d1a0d2 emb1,
• Dart d2a0d1 emb2, ...
• Embedding emb1coordxyz, p, ...

F1
E1
E2
E3
d2
d1
V1
V2
V3
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Functions acting on a GMap

• the basic function traversing the orbit of a
  dart d
• iterating the set of darts that can be reached by
  a given set of ai involutions

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Basic operations on GMaps

• recursive subdivision an GMap of dimension N may
  be thought of a set of GMaps of dimension N-1
  sewn along cells of dimension N-2
  (N-2)-cells are sewn (N-3)-cells etc. until
  vertices are reached
• example a volume can be represented by polyhedra
  sewn along their faces a polyhedron is
  represented by its boundary polygons sewn along
  their edges.

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Basic operations on GMaps

• create new GMap
• delete GMap
• create isolated dart and add it to a GMap,
  involutions are initialized as ai(d)d
• delete a dart from a GMap
• create an ai involution between 2 darts
• delete an ai involution between 2 darts

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Sew algorithm - create a N-GMap from (N-1)-cells

• 2 orbit traversals of the (N-1)-cells along which
  the N-cell will be sewn, and setting the
  involution
• sew(d1,d2,N) for (d'1 in lt gt(d1),
  d'2 in lt gt(d2) )
• dispatchCellEmbedding(d'1,d'2)
• d'1-gtaNd'2
• d'2-gtaNd'1
• how does unsew work?

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Representing Objects with G-Maps hierarchical
GMaps with Frame
1) Represents the objects boundaries
(Macro-topology)
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Representing Objects with G-Maps hierarchical
GMaps with Frame
2) Can be used to subdivide space in regions
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Representing Objects with G-Maps
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Representing 3d-Objects with G-Maps

• 3 glued surfaces along a radial line
• free edges a3(d) ? a2(d)
• glued edges a3(d) a2(d)

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Construction of a model

• create empty (N1) dimensional model M
• create a cellular GMap-partition L(W) of the
  N-dimensional dividing wall W
• create Frame F(W) such that L(W) is the Lattice
  of F(W) and add W to model M
• repeat (2) and (3) for a set of N-dividing walls
  ? MM(W1,..., Wi)
• if necessary, use the glue ("sew") operator to
  weld frames of the model M

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Building a Model with G-Maps (I)
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Building a Model with G-Maps (II)
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Building a Model with G-Maps (III)
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Building a Model with G-Maps (IV)
Horizon
Fault
Region
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Strong points of G-Maps

• Strong mathematical basis
• - data structure
• - operators
• only 2 concepts to understand darts and
  involution
• uncoupling of topology and geometry
• generically defined in N dimensions
• generic functions work for for N-cells

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more GMap model examples