its intersection with another irregular shape, or the volume
of an isosurface of a variable value, and the ability to ana-
lyze the contents of any volume or intersection in terms of
contained variable values. Most progress realizing these
tasks have been reached in geology, demonstrated by S.W.
Houlding (1994), G. F. Bonham-Carter, (1994), T. Bode
et al. (1994), M. Breunig (1996), and H. Kasper et al.
(1995).
3.1 Raster data models
To create a solid model for fully three-dimensional geo-
metry hexahedral volume elements, or vowels are intro-
duced, particularly to represent geobodies (S.W. Hould-
ing, 1994). In the simplest voxel representation, a cube,
each face is a square. In a more complex voxel, each face
can have a different size and shape. This flexibility ensures
that the output of practically any finite difference model-
ling application can be readily accepted as model input.
Voxels are therefore defined using any of several grid struc-
tures. In fig. 6 four common structures are sketched that
are also implemented in available 3D GIS products (for
instance the MGE Voxel Analyst, INTERGRAPH, 1993).
Figure 6: Geobody modelling - (a) uniform, (b) regular,
(c) irregular, and (d) structured (Copyright/Courtesy: IN-
TERGRAPH Co., Huntsville).
In the uniform approach the grid spacing along all ortho-
gonal axes is constant and identical; all edges are the same
length (cubic voxels). The regular modelling uses differ-
ent grid spacing but constant along each orthogonal axis;
edges are constant along each axis. In the contrary, if grid
spacing varies along each orthogonal axis, and the edges
vary in length along each axis, then the irregular voxel
representation is used. Last, a most deformed voxel rep-
resentation that is often referred to in geological modelling
is the structured one, in which grid spacing varies along
each orthogonal axis; each edge of each voxel can be of
different length. The data results from modelling tech-
niques to conform to geophysical or geological formations
and shapes.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
3.2 Tetrahedral tessellations
A generalization of the planar Delaunay tringulation leads
to Delaunay tetrahedral tessellation (DTT). This is one
kind of a simple data structure for spatial solid modelling,
particularly if irregular distributed 3D data are captured.
Due to its many advantages there are some proposals to
use the DTT for 3D GIS (X. Chen et al., 1994). The DTT
is derived by the 3D Voroni diagram that is quite similar
to the definition of section 2.
Figure 7: The 3D-Voronoi diagram (a) and the
Delaunay tetrahedral tesselation (b) (Copyright/Courtesy:
X.Chen/K.Ikeda, 1994)
The Delaunay tetrahedral tessellation is the straight-line
dual of the 3D Voroni daigram and is constructed by
connecting the points whose associated Voroni influence
volumes share a common boundary. The Delaunay tetra-
hedral tessellation is thus formed from four adjacent points
whose Voroni influence volumes meet a vertex, which is the
center of the circumsphered sphere of the Delaunay tetra-
hedral.
The integration of the Delaunay tetrahedral tessellation
(DTT) in 2D GIS data structures is proposed in a fur-
ther contribution given by X. Chen et al. (1994). It is
shown, that not only DTT representations but also voxel
definitions can be linked with geometric-topological data
structures.
4 STRUCTURED QUERY
LANGUAGE
The integration of height information extends the query
space for spatial queries considerably. New spatial op-
erators can be defined and the existing structured query
language, the standard of relational databases, has to be
redefined. In general, three different classes of queries can
be defined
e Numeric (e.g. the altitude of a mountain), also called
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