nal
the
hic
ces
an
| of
at-
the
ata
on-
)er-
] in
and
like
een
line
line
late
and
rec-
ject
the
ler-
iue
stly
nar
ion
attribute 3D
TIN
captured by
date
coverage
reverse
point
X,y,z
move(xt,yt,zt)
thematic
attribute
face
/planar
area
orientiation
Figure 1: 3D data model
is not advisable. On the lowest level of the model there
exist the geo-primitives point, edge and face. The points
contain the position as an attribute (either of a special 3D
coordinate type or as three attributes of the real or integer
type for x, y, z-coordinates). Point is the category for nodes,
e.g. edge-limiting points, and intermediate points, points be-
tween nodes. An edge is represented by initial or end nodes
as well as by intermediate points. A face again consists of
one or more clockwise ordered closed line segments. In this
way the inside and outside of faces and bodies is determined.
On a higher level there exist five geometrical attributes
which can be matched to a spatial object. A point-shaped
attribute is just composed of a node. A line-shaped attribute
is formed by one or more connected edges through joint
nodes. Face and body-shaped attributes consist of one or
more faces. In order to form a body the (planar) face have
to be tied over joint edges. An triangulated irregular net-
work (TIN) can also exist as an attribute of a geo-object
for representing the height model. Triangles are formed by
nodes and polygons.
As it appears from the data model a node can be referenced
by point-shaped, an edge by line-shaped and a face by face
and body-shaped attributes.
749
3 IMPLEMENTATION OF 3D
OPERATIONS BY 2D
OPERATIONS AND
TRANSFORMATIONS
A 2D-GIS is provided with comprehensive methods for 2D
calculations. In order to perform 3D calculations with a 2D-
GIS a macro language or better (like in Smallworld GIS)
programming language is necessary, in which additions and
multiplications and if possible trigonometrical functions on
coordinate values can be carried out. For the calculations
of 3D predicates the already existing 2D operations can be
used after transformations and the subsequent projections in
the two-dimensional space. By that planar faces are based
in one of the planes, which are defined by two of the three
coordinate axes and all objects (or their projections) in this
plane will be treated as ordinary in the planimetry of the
2D-GIS.
One possibility of transforming a face in one of the three
coordinate levels is the translation to the origin and 3 ro-
tations around the z-, y-, 2- axis. One can consider this as
a change of the coordinate system which mathematically is
the more perfect case. This will be exemplified by faces for
which intersections should be calculated.
In order to transform any face defined in the 3D Euclidean
space into the 2D xy-plane we take advantage of the well-
known characteristics of orthogonal matrices. The coordi-
nates of an object are transformed contragredient to the base
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
