topology.xsd See ISO/DIS 19136 Clause 14. Topology is the branch of mathematics describing the properties of objects which are invariant under continuous deformation. For example, a circle is topologically equivalent to an ellipse because one can be transformed into the other by stretching. In geographic modelling, the foremost use of topology is in accelerating computational geometry. The constructs of topology allow characterisation of the spatial relationships between objects using simple combinatorial or algebraic algorithms. Topology, realised by the appropriate geometry, also allows a compact and unambiguous mechanism for expressing shared geometry among geographic features. There are four instantiable classes of primitive topology objects, one for each dimension up to 3D. In addition, topological complexes are supported, too. There is strong symmetry in the (topological boundary and coboundary) relationships between topology primitives of adjacent dimensions. Topology primitives are bounded by directed primitives of one lower dimension. The coboundary of each topology primitive is formed from directed topology primitives of one higher dimension. This abstract type supplies the root or base type for all topological elements including primitives and complexes. It inherits AbstractGMLType and hence can be identified using the gml:id attribute. gml:AbstractTopoPrimitive acts as the base type for all topological primitives. Topology primitives are the atomic (smallest possible) units of a topology complex. Each topology primitive may contain references to other topology primitives of codimension 2 or more (gml:isolated). Conversely, nodes may have faces as containers and nodes and edges may have solids as containers (gml:container). The gml:isolated property element implements the role of the same name of the ISO 19107 “Isolated In” association (see ISO 19107:2003, 7.3.10.4). The gml:container property element implements the role of the same name of the ISO 19107 “Isolated In” association (see ISO 19107:2003, 7.3.10.4). gml:Node represents the 0-dimensional primitive. The optional coboundary of a node (gml:directedEdge) is a sequence of directed edges which are incident on this node. Edges emanating from this node appear in the node coboundary with a negative orientation. If provided, the aggregationType attribute shall have the value “sequence”. A node may optionally be realised by a 0-dimensional geometric primitive (gml:pointProperty). A gml:directedNode property element describes the boundary of topology edges and is used in the support of topological point features via the gml:TopoPoint expression, see below. The orientation attribute of type gml:SignType expresses the sense in which the included node is used: start (“-“) or end (“+”) node. gml:Edge represents the 1-dimensional primitive. The topological boundary of an Edge (gml:directedNode) consists of a negatively directed start Node and a positively directed end Node. The optional coboundary of an edge (gml:directedFace) is a circular sequence of directed faces which are incident on this edge in document order. In the 2D case, the orientation of the face on the left of the edge is "+"; the orientation of the face on the right on its right is "-". If provided, the aggregationType attribute shall have the value “sequence”. An edge may optionally be realised by a 1-dimensional geometric primitive (gml:curveProperty). A gml:directedEdge property element describes the boundary of topology faces, the coBoundary of topology nodes and is used in the support of topological line features via the gml:TopoCurve expression, see below. The orientation attribute of type gml:SignType expresses the sense in which the included edge is used, i.e. forward or reverse. gml:Face represents the 2-dimensional topology primitive. The topological boundary of a face (gml:directedEdge) consists of a sequence of directed edges. If provided, the aggregationType attribute shall have the value “sequence”. The optional coboundary of a face (gml:directedTopoSolid) is a pair of directed solids which are bounded by this face. A positively directed solid corresponds to a solid which lies in the direction of the negatively directed normal to the face in any geometric realisation. A face may optionally be realised by a 2-dimensional geometric primitive (gml:surfaceProperty). The gml:directedFace property element describes the boundary of topology solids, in the coBoundary of topology edges and is used in the support of surface features via the gml:TopoSurface expression, see below. The orientation attribute of type gml:SignType expresses the sense in which the included face is used i.e. inward or outward with respect to the surface normal in any geometric realisation. gml:TopoSolid represents the 3-dimensional topology primitive. The topological boundary of a solid (gml:directedFace) consists of a set of directed faces. A solid may optionally be realised by a 3-dimensional geometric primitive (gml:solidProperty). The gml:directedSolid property element describes the coBoundary of topology faces and is used in the support of volume features via the gml:TopoVolume expression, see below. The orientation attribute of type gml:SignType expresses the sense in which the included solid appears in the face coboundary. In the context of a gml:TopoVolume the orientation attribute has no meaning. The intended use of gml:TopoPoint is to appear within a point feature to express the structural and possibly geometric relationships of this feature to other features via shared node definitions. The gml:topoPointProperty property element may be used in features to express their relationship to the referenced topology node. gml:TopoCurve represents a homogeneous topological expression, a sequence of directed edges, which if realised are isomorphic to a geometric curve primitive. The intended use of gml:TopoCurve is to appear within a line feature to express the structural and geometric relationships of this feature to other features via the shared edge definitions. If provided, the aggregationType attribute shall have the value “sequence”. The gml:topoCurveProperty property element may be used in features to express their relationship to the referenced topology edges. gml:TopoSurface represents a homogeneous topological expression, a set of directed faces, which if realised are isomorphic to a geometric surface primitive. The intended use of gml:TopoSurface is to appear within a surface feature to express the structural and possibly geometric relationships of this surface feature to other features via the shared face definitions. The gml:topoSurfaceProperty property element may be used in features to express their relationship to the referenced topology faces. gml:TopoVolume represents a homogeneous topological expression, a set of directed topologic solids, which if realised are isomorphic to a geometric solid primitive. The intended use of gml:TopoVolume is to appear within a solid feature to express the structural and geometric relationships of this solid feature to other features via the shared solid definitions. The gml:topoVolumeProperty element may be used in features to express their relationship to the referenced topology volume. gml:TopoComplex is a collection of topological primitives. Each complex holds a reference to its maximal complex (gml:maximalComplex) and optionally to sub- or super-complexes (gml:subComplex, gml:superComplex). A topology complex contains its primitive and sub-complex members. The property elements gml:subComplex, gml:superComplex and gml:maximalComplex provide an encoding for relationships between topology complexes as described for gml:TopoComplex above. The property elements gml:subComplex, gml:superComplex and gml:maximalComplex provide an encoding for relationships between topology complexes as described for gml:TopoComplex above. The property elements gml:subComplex, gml:superComplex and gml:maximalComplex provide an encoding for relationships between topology complexes as described for gml:TopoComplex above. The gml:topoPrimitiveMember property element encodes for the relationship between a topology complex and a single topology primitive. The gml:topoPrimitiveMembers property element encodes the relationship between a topology complex and an arbitrary number of topology primitives. The gml:topoComplexProperty property element encodes the relationship between a GML object and a topological complex.