258 Annexes
NOTE
Closed under the boundary operations means that if a topological
primitive is in the topological complex, then its boundary objects are also in the
topological complex.
topological dimension
minimum number of free variables needed to distinguish nearby direct positions
within a geometric object from one another (lS039 2003)
NOTE The free variables mentioned above can usually be thought of as a local coordinate system. In a 3D coordinate space, a plane can be written as P(u, v) = A + u
X + v Y, where u and v are real numbers and A is any point on the plane, and X and
Y are two vectors tangent to the plane. since the locations on the plane can be distinguished by u and v (here universally), the plane is 2D and (u, v) is a coordinate system for the points on the plane. On generic surfaces, this cannot, in general, be done
universally. If we take a plane tangent to the surface, and project points on the surface onto this plane, we will normally get a local isomorphism for small neighbourhoods of the point of tangency. This "local coordinate" system for the underlying
surface is sufficient to establish the surface as a 2D topological object.
Since ISO 19107 deals only with spatial coordinates, any 3D object can rely on coordinates to establish its topological dimension. In a 4D model (spatio-temporal),
tangent spaces also play an important role in establishing topological dimension for
objects up to 3D.
topological expression
collection of oriented topological primitives which is operated upon like a multivariate polynomial (lS039 2003)
NOTE
Topological expressions are used for many calculations in computational topology.
topological object
spatial object representing spatial characteristics that are invariant under continuous
transformations (lS039 2003)
NOTE
A topological object is a topological primitive, a collection oftopological primitives, or a topological complex.
topological primitive
topological object that represents a single, non-decomposable element (lS039 2003)
NOTE
A topological primitive corresponds to the interior of a geometric
primitive of the same dimension in a geometric realization.
topological solid
3-dimensional topological primitive (IS039 2003)
NOTE
The boundary of a topological solid consists of a set of directed
faces.
trans action time
time when a fact is current in a database and may be retrieved (IS040 2002)
NOTE
Closed under the boundary operations means that if a topological
primitive is in the topological complex, then its boundary objects are also in the
topological complex.
topological dimension
minimum number of free variables needed to distinguish nearby direct positions
within a geometric object from one another (lS039 2003)
NOTE The free variables mentioned above can usually be thought of as a local coordinate system. In a 3D coordinate space, a plane can be written as P(u, v) = A + u
X + v Y, where u and v are real numbers and A is any point on the plane, and X and
Y are two vectors tangent to the plane. since the locations on the plane can be distinguished by u and v (here universally), the plane is 2D and (u, v) is a coordinate system for the points on the plane. On generic surfaces, this cannot, in general, be done
universally. If we take a plane tangent to the surface, and project points on the surface onto this plane, we will normally get a local isomorphism for small neighbourhoods of the point of tangency. This "local coordinate" system for the underlying
surface is sufficient to establish the surface as a 2D topological object.
Since ISO 19107 deals only with spatial coordinates, any 3D object can rely on coordinates to establish its topological dimension. In a 4D model (spatio-temporal),
tangent spaces also play an important role in establishing topological dimension for
objects up to 3D.
topological expression
collection of oriented topological primitives which is operated upon like a multivariate polynomial (lS039 2003)
NOTE
Topological expressions are used for many calculations in computational topology.
topological object
spatial object representing spatial characteristics that are invariant under continuous
transformations (lS039 2003)
NOTE
A topological object is a topological primitive, a collection oftopological primitives, or a topological complex.
topological primitive
topological object that represents a single, non-decomposable element (lS039 2003)
NOTE
A topological primitive corresponds to the interior of a geometric
primitive of the same dimension in a geometric realization.
topological solid
3-dimensional topological primitive (IS039 2003)
NOTE
The boundary of a topological solid consists of a set of directed
faces.
trans action time
time when a fact is current in a database and may be retrieved (IS040 2002)