Graphics Reference
In-Depth Information
(
)
-
{}
Æ
Æ
(
AD
,
-
C
:
L
D R
B D
)
,
C
is a one-to-one, onto, and continuous function. This fact is easily deduced from
equation (3.1).
Now, it is easy to check that the formula on the right-hand side of equation (3.1)
is invariant under affine maps. Although the x-axis may seem like a very special line,
all lines look like it once they are parameterized. We prove the following fundamental
theorem of projective geometry.
3.2.1. Theorem.
Both parallel projections and perspectivities between hyperplanes
preserve the cross-ratio.
Proof.
By Theorem 2.4.1.2 parallel projections preserve the ratio of division. There-
fore, parallel projections preserve the cross-ratio. Before we get to a proof of the
theorem in the case of perspectivities, we derive a formula for the ratio of division.
Assume that we have points
A
,
B
, and
D
on an oriented line
L
and that
O
is a point
not on the line. See Figure 3.5(a). If
n
A
and
n
D
are unit vectors that are orthogonal to
the vectors
OA
and
OD
, respectively, then
AB
•
n
=
OB
•
n
,
(3.2)
A
A
and
BD
•
n
=
BO
•
n
,
(3.3)
D
D
because the quantities in equations (3.2) and (3.3) are just the signed lengths of the
perpendiculars dropped from
B
to the lines through
O
,
A
, and
O
,
D
, respectively.
Next, let
u
be a
unit
direction vector for
L
that induces its orientation. Define real
numbers a and b by
AB
= a,
u
(3.4)
O
O
v
C
n
D
n
D
v
B
v
B
L
n
B
L
D
n
A
n
A
C
D
B
L¢
u
B
A
D¢
C¢
B¢
A
A¢
(a)
(b)
Figure 3.5.
Perspectivities preserve the cross-ratio.
