Graphics Reference
In-Depth Information
z
1
r
elliptic points
0 < f < p
k 1 = -
F (f,q)
hyperbolic points
p < f < 2p
r
C
q
y
1
R + r
maximum of k 2 =
f = p/2
R + r
R
R- r
q
elliptic points
0 < f < p
parabolic points
f = 0, p
hyperbolic points
p < f < 2p
x
-1
R- r
minimum of k 2 =
f = -p/2
Figure 9.24.
The torus.
fact that the formula for the derivative of the function n (F(u,v)) may be complicated,
one usually prefers to use the following formulas:
L
=∑
F
n
,
M
=∑
F
n
,
and
N
=∑
F
n
.
(9.56)
uu
uv
vv
9.9.18. Example. Consider the torus that is obtained by rotating the circle of radius
r with center (0,R,0), r < R, in the xy-plane about the x-axis. See Figure 9.24. It can
be parameterized by
(
) =
(
(
)
(
)
)
Ffq
,
r
cos ,
f
Rr
+
sin
f
cos ,
q
Rr
+
sin
f
sin
q
.
One can check that
(
) =-
(
)
F
F
fq
,
r
sin , cos
f
r
f
cos , cos
q
r
f
sin
q
f
(
) =-+
(
(
)
(
)
)
fq
,
0
,
Rr
sin
f
sin ,
q
Rr
+
sin
f
cos
q
q
(
) ¥
(
)
F
fq
,
F
fq
,
f
q
(
) =
= (
)
n
fq
,
cos ,sin
ffqfq
cos ,sin
sin
(
) ¥
(
)
F
fq
,
F
fq
,
f
q
(
) =-
(
)
F
fq
,
r
cos ,
f
-
r
sin
f
cos ,
q
-
r
sin
f
sin
q
ff
(
) =-
(
)
F
F
fq
,
0
0
,
r
cos
f
sin , cos
q
r
f
cos
q
fq
(
) =-+
(
(
)
(
)
)
fq
,
,
Rr
sin
f
cos ,
q
-+
Rr
sin
f
sin
q
.
qq
 
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