Graphics Reference
In-Depth Information
Inline Exercise 10.29: (a) Show that if T is any linear transformation on R 3 ,
then for any nonzero
R 3 , H ( T (
v )) = H ( T ( v )) .
(b) Show that if K is any matrix, then H ( T K ( v )) = H ( T α K ( v )) as well.
(c) Conclude that in a sequence of matrix operations in which there's an H at
the end, matrix scale doesn't matter, that is, you can multiply a matrix by any
nonzero constant without changing the end result.
α ∈
α
R and any vector v
Suppose we have a matrix transformation on 3-space given by T ( v )= Kv ,
and T is nondegenerate (i.e., T ( v )= 0 only when v = 0 ). Then T takes lines
through the origin to lines through the origin, because if v
= 0 is any nonzero
vector, then
v :
α ∈
R
}
is the line through the origin containing v , and when
we transform this, we get
, which is the line through the origin
containing T ( v ) . Thus, rather than thinking of the transformation T as moving
around points in R 3 , we can think of it as acting on the set of lines through the
origin. By intersecting each line through the origin with the w = 1 plane, we
can regard T as acting on the w = 1 plane, but with a slight problem: A line
through the origin in 3-space that meets the w = 1 plane may be transformed
to one that does not (i.e., a horizontal line), and vice versa. So using the w = 1
plane to “understand” the lines-to-lines version of the transformation T is a
little confusing.
The idea of considering linear transformations as transformations on the
set of lines through the origin is central to the field of projective geometry.
An understanding of projective geometry can lead to a deeper understand-
ing of the transformations we use in graphics, but is by no means essential.
Hartshorne [Har09] provides an excellent introduction for the student who has
studied abstract algebra.
T ( v ):
α ∈
R
}
Transformations of the w = 1 plane like the ones we've been looking at in this
section, consisting of an arbitrary matrix transformation on R 3 followed by H ,
are called projective transformations. The class of projective transformations
includes all the more basic operations like translation, rotation, and scaling of
the plane (i.e., affine transformations of the plane), but include many others as
well. Just as with linear and affine transformations, there's a uniqueness theorem:
If P , Q , R , and S are four points of the plane, no three collinear, then there's
exactly one projective transformation sending these points to ( 0, 0 ) , ( 1, 0 ) , ( 0, 1 ) ,
and ( 1, 1 ) , respectively. (Note that this one transformation might be described by
two different matrices. For example, if K is the matrix of a projective transforma-
tion S , then 2 K defines exactly the same transformation.)
For all the affine transformations we discussed in earlier sections, we've deter-
mined an associated transformation of vectors and of normal vectors. For projec-
tive transformations, this process is messier. Under the projective transformation
shown in Figures 10.24 and 10.25, we can consider the top and bottom edges of
the tan rectangle as vectors that point in the same direction. After the transforma-
tion, you can see that they have been transformed to point in different directions.
There's no single “vector” transformation to apply. If we have a vector v start-
ing at the point P , we have to apply “the vector transformation at P ”to v to find
out where it will go. The same idea applies to normal vectors: There's a different
 
 
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