Graphics Reference
In-Depth Information
Figure 3.35.
The stereo-
graphic projection maps
circles to circles.
e
3
A
C
B
B¢
C¢
A¢
these transformations to not be defined at a point and not onto a point. We had a
similar problem with affine projective transformations.) The sphere-preserving trans-
formations of
R
•
are easy to characterize. First, any similarity of
R
n
extends to a map
of
R
•
to itself by sending • to •. Call such a map of
R
•
an
extended similarity
.
3.10.2.
Theorem.
(1) An extended similarity of
R
•
is a sphere-preserving map. Conversely, every
sphere-preserving maps of
R
•
that leave • fixed is an extended similarity.
(2) An arbitrary sphere-preserving maps of
R
•
is a composition of an extended
similarity and/or a map p
n
(h), where h is a rotation of
S
n
around a great circle
through
e
n+1
.
Proof.
See [HilC99].
Another interesting and important property of the stereographic projection is that
there is a sense in which it preserves angles. Let
p
be any point of
S
n
other than
e
n+1
.
Let
u
and
v
be linearly independent tangent vector to
S
n
at
p
. See Figure 3.36. Tangent
vectors will be defined in Chapter 8. For now, aside from the intuitive meaning, take
this to mean that
u
and
v
are tangent to circles
C
u
and
C
v
, respectively, in
S
n
through
p
and that “tangent at a point
p
of a circle with center
c
” means a vector in the plane
containing the circle that is orthogonal to the vector
cp
. Let
p
¢,
C
u
¢, and
C
v
¢ in
R
n
be
the images of
p
,
C
u
, and
C
v
, respectively, under the stereographic projection. The
vectors
u
and
v
induce an orientation of the circles
C
u
and
C
v
(think of
u
and
v
as
velocity vectors of someone walking along the circles) and these orientations induce
orientations of the circles
C
u
¢ and
C
v
¢ via the stereographic projection. Choose tangent
vectors
u
¢ and
v
¢ to the circles
C
u
¢ and
C
v
¢ at
p
¢ that match their orientation. Let q be
the angle between the vectors
u
and
v
and q¢ the angle between
u
¢ and
v
¢.
3.10.3. Theorem.
The stereographic projection is an
angle-preserving
or
conformal
map, that is, q=q¢.
