Graphics Reference
In-Depth Information
Figure 3.36.
The stereographic
projection preserves
angles.
C
u
u
p
C
v
q
v
p¢
C
u
¢
u¢
q¢
C
v
¢
v¢
Proof.
See [HilC99].
All sphere-preserving maps of
R
•
are angle preserving.
3.10.4. Corollary.
Proof.
This is an immediate consequence of Theorems 3.10.2 and 3.10.3.
Now let
X
be a (n - 1)-sphere in
R
n
with center
c
and radius r. Let
X
¢ be the sphere
in
S
n
that is mapped onto
X
by the stereographic projection. Choose a point
p
on
X
and let
p
¢ be the point of
S
n
(in
X
¢) that maps onto
p
. Let s be the rotation of
S
n
around the great circle through
p
¢
and
e
n+1
that maps
p
¢
to
e
n+1
. Then
Y
¢
=s(
X
¢) is a
sphere through
e
n+1
and its projection to
R
n
is a plane
Y
. Let R be the reflection of
R
n
about
Y
.
Definition.
The map
(
)
-
1
()
n
n
m
=
p
s
p
s
:
RR
Æ
n
n
•
•
is called an
n-dimensional inversion
of
R
•
with respect to the sphere
X
, or simply an
inversion in a sphere
.
3.10.5. Theorem.
The map m is defined analytically as follows:
(1) m(
c
) = •.
(2) Let
p
Œ
R
n
,
p
π
c
. Let
q
be the point on
X
where the ray
Z
from
c
through
p
intersects
X
. Then m(
p
) is that unique point
z
on
Z
defined by the equation
2
r.
cp cz
=
cq
=
Proof.
See Figure 3.37. For a proof see [HilC99].
