Graphics Reference
In-Depth Information
z
z
2
- v
2
= 1
+1
H
H
+1
z = -v
z = v
C
u
C
0
v
p¢
•
p
•
-1
H
v = 0
-1
u = 0
(a)
(b)
Figure 10.1.
Example 10.2.1.
z
H
e
p
•
u
2
+ z
2
= 1 + e
2
+ v
2
v ≥ e
C
v
(e)
v ≤ e
C
0
u
1
z = -(v/e)u
p¢
•
H
e
(a)
(b)
Figure 10.2.
Example 10.2.1 continued.
Equation (10.8) by itself defines hyperboloids
H
e
, which are the union of the circles
C
v
(e), of which
H
in Figure 10.1(a) is a special case. Equation (10.9) defines a twisted
plane. For fixed v it defines a line. These lines start with slope 0 and then rotate to
almost vertical lines as v gets much bigger than 1. Figure 10.2(a) shows the circles
C
v
(e) and hyperbolas
H
e
defined by equations (10.8) and (10.9) projected to the u-z
plane. One can show that the curves defined by the intersection of the hyperboloids
and twisted planes (the simultaneous solutions to equations (10.8) and (10.9)) “fill
out” the skeleton of the sphere shown in Figure 10.1(b). See Figure 10.2(b). In other
words, thought of as living in
P
2
(
C
) the space V(X
2
+ Y
2
- 1) is topologically a sphere.
V (X
2
+ Y
2
)
10.2.2. Example.
Analysis.
Again replacing X and Y by u +
i
v and z +
i
w, respectively, leads to the
equations
