Graphics Reference
In-Depth Information
aa a
bb b
cc c
1
2
3
π
0
1
2
3
1
2
3
is called a fractional transformation .
3.2.8. Theorem.
Every projective transformation is a fractional transformation and
conversely.
Proof.
See [Gans69].
3.2.9. Theorem. The set of all projective transformations is a group containing the
planar affine transformations as a subgroup.
Proof. This is obvious. The planar affine transformations correspond to the frac-
tional transformations above where c 1 = c 2 = 0 and c 3 = 1.
This is as far as we shall take things in affine space. Clearly, one of the unpleasant
aspects of perspectivities in this context is that they are not defined everywhere, nor are
they onto. Basically, there are “missing” points. We shall have more to say about that
shortly, but a few comments are appropriate now because it prepares the reader for the
concept of “ideal point” and we also want to relate perspectivities to parallel projec-
tions. We know what the missing points are in the case of a perspectivity of the type
shown in Figure 3.3. We need to add some points, both to the view plane and the object
plane. Suppose we add a new point to a plane for each of its families of parallel lines.
If L is a line, then let L denote this new point associated to the family of lines paral-
lel to L . L will be called an ideal point . With these new points, we could extend our
definition of the perspectivity by saying that the point J in Figure 3.3(b) should map to
L and L 1 should map to E ¢ in Figure 3.3(a). This would give us a one-to-one and onto
map between these extended planes , which are ordinary planes together with their
ideal points. In addition, with these new points we could consider parallel projections
between hyperplanes as a special case of perspectivities if we allow the center of
projection to be an ideal point.
3.3
Homogeneous Coordinates
One of the key ideas in the study of analytic projective geometry is that of homoge-
neous coordinates . The standard Cartesian coordinates are sometimes referred to as
“nonhomogeneous” coordinates and are simply one of many ways to specify points
in space with real numbers. Other ways are polar coordinates in the plane and cylin-
drical and spherical coordinates in 3-space. Barycentric coordinates are a type of
“homogeneous” coordinates. They specify points relative to a fixed set of points.
Out of the many ways that one can coordinatize points, which is the most con-
venient depends completely on the type of problem we are trying to solve. Homoge-
neous coordinates are just another way of coordinatizing points. Historically they find
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