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a natural principle.
It can be noted here that the digital-to-digital transformation is
the only step required if no graphic presentation is concerned.
However, if graphics is considered, one needs to take into
account the geographical requirements, multi-purpose
requirements and cartographic requirements. It is now clear
that cartographic requirements should be considered in the
digital-to-graphic transformation after the scale-driven digital-
to-digital transformation. Of course, one can also use some of
the cartographic requirements as constraints for the digital-to-
digital transformation. Some of the multi-purpose and
geographical requirements may also be used as constraints for
this scale-driven transformation and for selecting data layers
for generalization.
3. TRANSFORMATION OF SPATIAL REPRESENTATION
IN SCALE DIMENSION
There are many mathematical models available for the
transformation of spatial objects, such as conformal, affine,
projective, etc. After such a transformation, the shape, size,
orientation and even the topology of an object can be altered.
However, these are transformations in space dimension. What
will be discussed in the next two sub-sections are about the
transformation in scale dimension, a concept introduced by Li
(19942).
3.1 The concept of scale dimension
It has been noted by researchers that what is supposed to be a
reality is dependent on scale and time. ^ After many
illustrations, Li (1994a) introduced the concept of scale-
dimension and time-scale systems, which can be illustrated in
Fig.3.
Y
Oo A
Y
X
(a) In 3-D space, a point represented in a 2-D plan by orthogonal
projection;
4 Time
Space ^ 3
Time O A spatial
representation
— A
Scale Scale
(b) In new 3-D system, a point in time-scale plan is a representation of
spatial variations
Fig.3 A new 3-dimensional system
Just as a point of 3-D space can be represented in the X-Y
systems, a spatial object in the new 3-D system can also be
represented in the time-scale systems. In other words, a spatial
representation is a record of spatial variations in the time-scale
systems.
3.2 Transformation in scale dimension
Now comes the question: *why do we need to introduce the
concept of scale dimension?" or “Is there any difference
between scale and scale dimension". Fig.4 illustrates some
examples to show the difference between scale and scale
dimension.
MN
Scale 2 Scale 3 Scale 4
PS
za X
(a) Simple scale reduction in space: complexity not reduced
Time
Scale 2 Scale 3 Scale 4
En
(b) Transformed in scale dimension: complexity reduced
Fig.4 Difference between simple scale reduction and
transformation in scale dimension
» Scale
It can be noted that by scale reduction in space dimension it is
meant a simple reduction in size. In this case, the complexity
of spatial representation is not reduced. On the opposite side,
by transformation in scale dimension it is meant that the
representation is simplified to suit the representation at another
(smaller) scale. It might be better to call the term scale in space
dimension size.
The transformation in scale dimension is a transformation in
time-scale systems when the time is fixed. The transformation
of spatial representation in time dimension is a transformation
in the same systems when scale fixed. This transformation is
called temporal modelling and lies outside the scope of this
article.
4. THEORETICAL BASIS FOR TRANSFORMATION IN SCALE
DIMENSION: THE NATURAL PRINCIPLE
After the introducing the concept transformation in scale
dimension, it is the time to examine the theoretical basis for
such a transformation.
4.1 The natural phenomena
In order to understand the underlining problem better, some
practical examples are desirable to illustrate such a
transformation implied in natural phenomena. Li and
Openshaw (1993) have used the Earth being viewed from
various distance as an example. When a person is nearer an
object, s/he can see more detail. When one gets further away
from the object, less detailed information can be seen but the
main characteristics of the object can be better observed, thus
better overview being gained. Surveyors all have such
experiences: When one stands somewhere near the peak of a
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
