Graphics Programs Reference
In-Depth Information
Remove the string temporarily, close the hinged leaf, and mark the intersection point
of the wires on the paper. This is repeated for many points on the object, which later
permits the artist to interpolate the points and complete the drawing.
In contrast with Renaissance and classical artists, who mostly tried to create works
true to nature, many impressionist and modern artists consider the use of color and
technique more important than accurate perspective. Figure C.3 (page 234) is a classic
example of this approach. It shows the famous yellow chair painted by Vincent van
Gogh several times during his short stay in Arles. Even a quick glance at it creates the
impression that something is wrong. However, van Gogh fans (this author not numbered
among them) claim that his mastery of color, combined with his technique and style,
resulted in paintings full of appeal and charm, in spite of the crude perspective (or even
because of it). Another example that some may call divergent perspective is The Chair
by David Hockney (1985).
3.4 The Mathematics of Perspective
The mathematics of linear perspective is easy to derive and to apply to various situa-
tions. The mathematical problem involves three entities, a (three-dimensional) object
to be projected, a projection plane, and a viewer watching the projection on this plane.
The object and the viewer are located on different sides of the projection plane, and
the problem is to determine what the viewer will see on the plane. It is like having a
transparent plane and looking through it at an object. Specifically, given an arbitrary
point P =( x, y, z ) on the object, we want to compute the two-dimensional coordinates
( x ,y ) of its projection P ontheprojectionplane. Oncethisisdoneforallthepoints
of the object, the perspective projection of the object appears on the projection plane.
Thus, the problem is to find a transformation T that will transform P to P .Weuse
the notation P = PT from Chapter 1.
Often, there is no need to compute the projections of all the points of the object.
If P 1 and P 2 are the two endpoints of a straight line on the object, then only their
projections P 1 and P 2 need be computed and a straight line is then drawn between
them on the plane. In the case of a curve, it is enough to compute the projections
of several points on the curve and either interpolate them on the projection plane or
simply connect them with short, straight segments.
It is obvious that what the viewer will see on the projection plane depends on the
position and orientation of the viewer. The viewer and the object have to be located on
different sides of the plane, and the viewer should look at the plane. If the viewer moves,
turns, or tilts his head, he will see something else on the projection plane and may not
even see this plane at all. Similarly, if the object is moved or if the projection plane is
moved or is rotated, the projection will change. Thus, the mathematical expressions for
perspective must depend on the location and orientation of the viewer and the projection
plane, as well as on the location of each point P on the object.
We start with a special case—where the viewer is positioned at a special location,
looking in a special direction at a specially placed projection plane—and show how to
project any three-dimensional point to a two-dimensional point on the projection plane.
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