Graphics Reference
In-Depth Information
n n
:
SR
p
n
Æ
•
by mapping
e
n+1
to •. This map is also called the
stereographic projection
of the
n-sphere.
The identification of the two space
S
n
and
R
•
using p
n
gives us a one-to-one cor-
respondence between maps of
S
n
and
R
•
. Specifically, for any map
n
n
h
:
SS
Æ
define
()
n
n
ph
:
RR
Æ
n
•
•
by
()
=
-
1
.
ph p hp
oo
n
n
n
Alternatively, p
n
(h) is the unique map that makes the following diagram
commutative:
h
n
n
S
ææ
S
p
Ø
Ø
æÆ
p
.
n
n
n
n
R
ææ
R
•
•
(
)
ph
n
3.10.1.
Theorem.
(1) If
X
is an k-dimensional sphere in
S
n
that misses the point
e
n+1
, then
X
¢
=
p
n
(
X
) is a k-dimensional sphere in
R
n
.
(2) If
X
is an k-dimensional sphere in
S
n
through the point
e
n+1
, then
X
¢
= p
n
(
X
)
is a k-dimensional plane in
R
n
.
Proof.
Consider the case of circles and n = 2. The argument for part (1) proceeds as
follows. Let
X
be a circle in
S
2
that does not pass through
e
3
. Figure 3.35 shows a ver-
tical slice of the three-dimensional picture. The points
A
and
B
are points of
X
and
A
¢ and
B
¢ are the image of
A
and
B
, respectively, under the stereographic projection.
The tangent planes at the points of
X
envelop a cone with vertex
C
. One can show
that the image
C
¢ of
C
under the stereographic projection is then the center of the
circle
X
¢. See [HilC99]. Part (2) follows from the fact that a circle through
e
3
is the
intersection of a plane with the sphere. The general case of spheres and arbitrary k-
dimensional spheres is proved in a similar fashion.
If we consider a plane as a “sphere through infinity,” then Theorem 3.10.1 can be
interpreted as saying that the stereographic projection takes spheres to spheres. With
this terminology we can now also talk about
sphere-preserving transformations
of both
S
n
and
R
•
. (Note that in
R
n
these would be the maps that send a sphere to a sphere
or
a plane and a plane to a plane
or
a sphere. We would have had a problem talking
about such “sphere-preserving” transformations in
R
n
because we would have to allow
