GEOTRANS Kickoff Meeting

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3D Topology for Terrain Reasoning
Kevin Trott, kevin_trott_at_partech.com,
315-339-0491 x266 PAR Government Systems
Corporation Geocomp4 Conference 27 July 1999
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Why 3D Topology?

• There are a number of militarily significant
  features that simply cannot be adequately
  represented using 2D topology
• Bridges/Overpasses
• Tunnels
• Bodies of Water
• Caves/Overhanging Cliffs
• Building Interiors
• In 1996, a study performed by Dr. Nick Chrisman
  as part of the SEDRIS program recommended that
  partial and full 3D topology levels be defined.
• 3D topology is needed to provide improved support
  for advanced Computer Generated Forces (CGF)
  systems, which will perform 3D spatial reasoning
  within realistic, detailed synthetic battlefield
  environments.
• 3D topology also can aid in the reconstruction of
  full-scale feature representations from abstract
  vector data, using geometric attributes such as
  width and height.
• NIMA is preparing to produce Foundation Feature
  Data (FFD) on a worldwide basis. FFD prototypes
  contain 3D coordinates but only 2D topology.

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Example - A Simple Overpass
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FFD Line Bridge/Overpass
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3D Line Bridge/Overpass
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3D Topology Model

• Spatial Entities
• 0-Dimensional - Nodes
• 1-Dimensional - Edges
• 2-Dimensional - Faces
• 3-Dimensional - Volumes
• Topological Relationships
• Node-Edge Relationships
• Node-Face Relationships
• Node-Volume Relationships
• Edge-Face Relationships  Rings
• Edge-Volume Relationships
• Face-Volume Relationships  Shells

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Nodes

• Node  a zero-dimensional spatial entity that
  defines a location in 2D or 3D space
• Location of a node is defined by a single
  coordinate tuple
• Location of each node must be unique  multiple
  nodes cannot be colocated
• Cannot be located in the interior of an edge, but
  may be located within the interior of a face or
  within the interior of a volume (3D)

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Edges

• Edge  a one-dimensional spatial entity that
  defines a path through 2D or 3D space
• Geometry of an edge is defined by an ordered
  collection of two or more distinct coordinate
  tuples
• An edge is bounded by a node at each of its two
  endpoints (the endpoints are conceptually
  included in the edge)
• Orientation of an edge is defined by the order of
  its coordinate tuples
• An edge may not intersect with or overlap itself,
  or any other edges
• An edge may not intersect a node or a face
  without being broken into multiple edges
• Edges may meet only at nodes
• An edge may be completely contained within a face
  or within a volume

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Faces

• Face  a two-dimensional spatial entity that
  defines a closed area in 2D or 3D space
• Geometry of a face is defined by
• An ordered collection of one or more edges that
  bound the face
• A collection of zero or more nodes that are
  contained within the face
• A collection of zero or more interior points,
  like the interior points of an edge
• A face is bounded by one or more collections of
  edges, defining the outer boundary and zero or
  more inner boundaries
• Orientation of a face is defined by an explicit
  "up" vector
• The three-dimensional shape of a face must be
  monotone with respect to its up vector, forming a
  2D pseudomanifold
• A face may not intersect or overlap itself, or
  any other faces
• Faces may meet only along common edges, and/or at
  common nodes
• A face may be contained completely within a volume

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Volumes

• Volume  a three-dimensional spatial entity that
  defines a closed region of 3D space
• Geometry of a volume is defined by the unordered
  collection of faces that form its outer boundary
• A volume is bounded by one or more collections,
  each of two or more faces, defining the outer
  boundary and zero or more inner boundaries
• A volume may not intersect with or overlap
  itself, or any other volume
• Two or more volumes may meet only along common
  faces, along common edges, and/or at common nodes

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Node-Edge Relationships

• Each edge is associated with two connected nodes
  a start node and an end node
• Each node is associated with an unordered
  collection of zero or more connected edges
• The connected edges cannot, in general, be
  ordered, since they can connect to the node from
  any direction in 3D space
• The subset of a node's connected edges that are
  adjacent to a specified volume can be ordered

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Node-Face Node-Volume Relationships

• Node-Face Relationships
• Each node can be associated with zero or more
  containing faces (because multiple faces can meet
  at a common node that is not part of the boundary
  of any of the faces)
• Each face is associated with a collection of zero
  or more contained nodes
• Node-Volume Relationships
• Each node is associated with one containing
  volume
• Each volume is associated with a collection of
  zero or more contained nodes

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Edge-Face Relationships Rings

• Ring  a sequentially connected set of edges that
  bound a face
• A ring includes any edges contained within the
  face, but connected to the boundary
• An edge can appear twice in the same ring, once
  in each orientation
• Outer Ring  defines the outer boundary of a face
  
• Inner Ring  defines an inner boundary of a face
  (i.e., a "hole" in the face)
• A collection of one or more edges that are
  connected to one another, but that are not
  connected to the outer boundary of the face, form
  an inner ring even if they do not enclose an area
• An inner ring need not contain any faces - it may
  represent an actual hole in the face

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Edge-Face Relationships

• Each edge is associated with an ordered
  collection of zero or more bordered faces,
  ordered counterclockwise looking along the edge,
  starting with an arbitrary face
• Each face is associated with one outer ring, and
  an unordered collection of zero or more inner
  rings, each containing an ordered collection of
  one or more edges
• An edge may be completely contained within a face

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Edge-Volume Relationships

• Each edge that bounds one or more faces is also
  associated with a collection of one or more
  bordered volumes, ordered counterclockwise
  relative to the edge, starting with an arbitrary
  volume
• Each volume is associated with one or more
  collections of edges, each of which form the
  outer ring of one or more of the faces in the
  outer shell, or one of the inner shells, of the
  volume
• An edge may be completely contained within a
  volume

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Face-Volume Relationships Shells

• Shell  an unordered collection of two or more
  faces that bound a volume
• A shell includes any faces that are contained
  within the volume, but are connected to the
  boundary by a common edge
• A face can appear twice in the same shell, once
  in each orientation
• Outer Shell  defines the outer boundary of a
  volume
• Inner Shell  defines an inner boundary of a
  volume (i.e., a bubble in the volume)
• A collection of one or more faces that are
  connected to one another, but that are not
  connected to the outer boundary of the volume,
  form an inner ring even if they do not enclose a
  space

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Face-Volume Relationships

• Each face is associated with exactly two volumes,
  its top volume and its bottom volume
• Each volume is associated with one or more
  shells one outer shell, and zero or more inner
  shells, each containing an unordered collection
  of two or more faces
• A face may be contained completely within a volume

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Coordinates Topology
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Conclusions

• Full 3D topology (i.e., a 3D manifold) is a
  well-defined extension of full 2D topology,
  though it is more complex.
• Full 3D topology cannot be created by simply
  adding 3D spatial entities and relationships to
  full 2D topology
• One-to-many Node-Edge relationships are no longer
  ordered,
• One-to-two Edge-Face relationships become
  many-to-many, ordered in both directions.
• Partial 3D topology is a strange land where the
  rules of full 2D topology no longer hold, but the
  rules of full 3D topology do not yet hold either.
• NIMA is currently supporting the development of
  prototype demonstration 3D data sets with full 3D
  topology, and software that allows these data
  sets to be examined and interactively
  manipulated.