Graphics Reference
In-Depth Information
r
1
x
2
+ 2my = 0
k = 1:
(parabolic cylinder) , or
r
1
x
2
+ d = 0
(two planes)
r
1
x
2
+ r
2
y
2
+ d = 0
(If solutions exist, then we get an elliptic or hyperbolic
cylinder if d π 0 and two planes if d = 0.)
k = 2:
Proof.
See [Eise39].
3.7.3. Theorem.
The equation of the tangent plane to a quadric surface at a point
(x
0
,y
0
,z
0
) defined by equation (3.71) is
(
)
(
)
(
)
ax
+++
hy
fz
l x
+
by
+
hx
+
gz
+
m y
++++
cz
fx
gy
n z
0
0
0
0
0
0
0
0
0
++ + +=.
lx
my
nz
d
0
0
0
0
Proof.
See [Eise39]. (We again refer the reader to Chapter 8 for a precise definition
of a tangent plane).
For a classification of quadratic surfaces in
R
n
see [PetR98].
3.8
Generalized Central Projections
The standard central projections as defined in Section 3.2 have a center that is a point.
When dealing with higher-dimensional spaces it is sometimes convenient to allow the
“center” to be an arbitrary plane.
Let
O
n-k-1
be a fixed (n - k - 1)-dimensional plane in
R
n
. If
Y
k
Definition.
is a k-
dimensional plane in
R
n
, define a map
n
p
O
:
RY
Æ
by
()
=
(
)
«
(
)
p
O
p
aff
,
p
Y
, if aff
,
p
intersects
Y
in a single point,
=
undefined, otherwise.
The map p
O
is called the
generalized central projection with center
O
of
R
n
to the plane
Y
. If
X
k
is another k-dimensional plane in
R
n
, then the restriction of p
O
to
X
, p
O
|
X
:
X
Æ
Y
, is called the
generalized perspective transformation
or
generalized perspectivity
from
X
to
Y
with center
O
.
3.8.1. Theorem.
The generalized perspectivity p
O
|
X
from
X
to
Y
with center
O
is a
projective transformation.
Proof.
It is not hard to show that p
O
|
X
is a composition of ordinary central projec-
tions and parallel projections.
