On the surface - The Visualization of Spatial Social Structure - page 208

Box 7.2 The perspective projection

The orthographic projection on to image space ( u, v ) of a point ( x,y,z ) with
the viewpoint at an angle (

θ ϕ

, )is

u = x cos θ − z sin θ

v = x sin θ sin φ + y cos φ + z cos θ sin φ

The perspective projection at a distance ( r) and with a particular focal length

( f )isgivenby

θ)
r − (x sin θ sin ϕ − y sin ϕ + z cos θ cos ϕ)

f(x cos θ − z sin

u =

f(x sin θ sin ϕ + y cos ϕ + z cos θ sin ϕ)
r − (x

v =

sin θ sin ϕ − y sin ϕ + z cos θ cos ϕ)

For derivation, extension and a full discussion see Plantinga (1988).

7.4 Landscape painting

Applying 2-dimensional tools to 3-dimensional problems has been only
moderately successful at best. As the new 3-dimensional geoprocessing
tools get into the hands of the users, answers will be discovered to
the questions that we currently don't understand or even realize we
can ask.

(Smith and Paradis, 1989, pp. 153-154)

As should be realised from the difficulty of visualizing nonflat two-
dimensional surfaces on paper, the variability of their structure can be nowhere
near as great as that of graphs. Only the simplest surfaces are susceptible to the
depth cue method. Most surfaces in our real world are of this simple form.
What is seen in an image containing simple surfaces is not truly three-
dimensional, but something just beyond the plane, just the outsides. 10 To visualize
true three-dimensional complexity we would have to be able to unravel a ball
of wool in our mind, to see all facets and aspects of an object at once, to
understand how features would intersect from all around, above and below, and

stereo maps compared with the mono maps. However the quality of the answers to the “stereo-
questions” does not differ significantly from the “mono-questions”. Viewing a Spatial Map Image in
stereo means a faster, but not necessarily better, understanding of the map' (Kraak, 1989, p. 112).
10 Surfaces show two-dimensional elevation, not three-dimensional structure: 'The definition of
three dimensional mapping has been incorrectly pre-empted in many cases, by the advertising of
so-called 3-D computer programs and video displays that are nothing more than 2-D representations
of perspective or similar type projections' (Hardy, 1988).