Graphics Reference
In-Depth Information
Figure 3.37.
An inversion of a sphere maps
p
to
z
.
Z
z
X
r
q
p
c
|cp| |cz| = r
2
Every circle-preserving map of
R
•
is the composite of at most
3.10.6. Theorem.
three inversions.
Proof.
See [HilC99].
Finally, there is an interesting connection between the stereographic projection
and Poincaré's model of the hyperbolic plane. To learn about this we again refer the
reader to [HilC99]. Recall that one of the big developments in geometry in the 19
th
century was the discovery of non-Euclidean geometry. The big issue was whether the
axiom of parallels was a consequence of the other axioms of Euclidean geometry. The
axiom of parallels
asserts that given a line and a point not on the line, there is a unique
line through the point that is parallel to the line, that is, does not intersect it. This
axiom does not hold in other geometries. In the plane of elliptic geometry there is no
parallel line because all lines intersect. In hyperbolic geometry there are an infinite
number of lines through a point that are parallel to a given line.
3.11
E
XERCISES
Section 3.4
be an ordinary line in
R
2
. Carefully prove that the set
3.4.1.
Let
L
=»
{}
•
ll
in
P
2
is in fact a line in
P
2
.
Section 3.4.1
Find the equation of the line in
P
2
through the points [2,-3,1] and [1,0,1].
3.4.1.1.
3.4.1.2.
Find the intersection of the lines
-+
X
2
Y
-=
Z
0
and
2
X
+ =
Z
0
in
P
2
.
