Kevin Trott, kevin_trott@partech.com, 315-339-0491 x266PAR Government Systems Corporation

Geocomp4 Conference

27 July 1999

3D Topology for Terrain Reasoning

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.

Example - A Simple Overpass

FFD Line Bridge/Overpass

Node is not associated with any feature(s)

Edges associated with both Bridge/Overpass and upper Road/Railroad line feature

Edges associated with lower Road/Railroad

line feature

Bridge TUC value matches upper

line feature

Endpoint Zs on lower edges do not

match node Z

Face 1Face 2Face 3Face 4

3D Line Bridge/Overpass

Edges associated with both Bridge/Overpass and upper Road/Railroad line features come up out of base surface

Edges associated withlower Road/Railroad

line feature

Bridge/Overpass TUC value matchesupper line feature

Node associated with Underpass/Tunnell point

feature (TUC value matcheslower line feature)

Face 1Face 2Face 3Face 4Optional Upper node not associated with

any feature

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

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)

Connectedto Edges

Containedin a Face

Contained in a Volume

Multiple Relationships

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

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

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

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

StartNodeEnd

Node

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

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

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

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

BorderedVolumes

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

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

FaceTopVolumeBottomVolumeOrientation

Coordinates & Topology

2D Coordinates3D Coordinates3D Topology2D TopologySomeFaces

UniversalFacesNo

FacesSome

VolumesUniversalVolumes

NoVolumesSomeFacesNo

FacesUniversal

Faces2.5D TopologySomeFaces

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.