Graphics Reference
In-Depth Information
(10.10)
2
2
2
2
uvzw
-+- =
0
and
uv
z+=0.
(10.11)
The three-dimensional slice w = 0 is defined by
2
2
2
uvz
-+=
0
(10.12)
and
uv = 0.
(10.13)
Equation (10.12) defines a cone. Equation (10.13) defines two planes as before. See
Figure 10.3(a). What basically is happening is that if we were to consider the family
of hypersurfaces V(X
2
+ Y
2
- c), c Π[0,1], and let c approach 0, then we would get
pictures like in Figure 10.1 except that the circle
C
0
would shrink to
0
and the lines
H
would “straighten” out. The final analog of Figure 10.2(b) would be Figure 10.3(b).
In other words, thought of as living in
P
2
(
C
) the space V(X
2
+ Y
2
) is topologically the
union of two spheres which meet at a point.
V (X
2
+ Y
2
10.2.3. Example.
+ c), 0 π c Œ
C
Analysis.
One can show that the closure of this space in
P
2
(
C
) is topologically a
sphere (Exercise 10.2.1).
V (Y
2
- X (X
2
10.2.4. Example.
- 1))
Analysis.
This time, replacing X by u +
i
v and Y by z +
i
w leads to the equations
p
•
z
u
v
p¢
•
(a)
(b)
Figure 10.3.
Example 10.2.2.
