Graphics Reference
In-Depth Information
3.5.1. Theorem.
(The Fundamental Theorem of Projective Geometry for
P
n
) Given
two sets {
P
i
} and {
P
i
¢} of n + 2 points in
P
n
with the property that no n + 1 points of
either set lies in a hyperplane, then there is a unique projective transformation T of
P
n
that sends
P
i
to
P
i
¢.
The proof is analogous to what we did for
P
2
.
Proof.
3.5.2. Corollary.
Given two k-dimensional projective planes
X
and
Y
in
P
n
, there is
a projective transformation T of
P
n
that sends
X
onto
Y
.
The reader should remember two fundamental ideas: One is that, as was pointed
out earlier, whenever one works with homogeneous coordinates one is really dealing
with projective space, whether one is consciously thinking about that or not, because
those coordinates are the natural coordinates for projective space. Second, if one has
to deal with (projective) transformations of
R
n
, then it is often simpler to translate
the problem into one involving
P
n
, to solve the corresponding problem in that space,
and finally to map the answer back down to
R
n
(equivalently, solve the problem using
homogeneous coordinates first). This idea can be expressed very compactly by the
commutative diagram
F
P
n
P
n
Æ
i ≠ »
» Ø p
Æ
R
n
R
n
f
If one needs to deal with a transformation f, then deal with its lift F to
P
n
instead,
where F is defined by the equation f =pFi with i and p the standard inclusion and
projection, respectively. (Recall that p is actually not defined on all of
P
n
.)
3.5.1
Homogeneous Coordinates and Maps in 3-Space
The homogeneous coordinates of a point (x,y,z) in
R
3
are a 4-tuple
(
)
X Y Z W
,,, ,
where W
π
0
and x
=
X W
,
y
=
Y W
,
and
z
=
Z W
.
Every projective transformation of
R
3
is an affine transformation or a composite of
an affine transformation and a single perspectivity. The affine part can in turn be
decomposed into a composition of translations, rotations, reflections, shearings, or
local scaling. Using homogeneous coordinates, we can express such a map either as
a product of 4 ¥ 4 matrices that correspond to the maps in that composition or via a
single 4 ¥ 4 matrix whose parts can be described as shown below:
