Graphics Reference
In-Depth Information
Geodesics:
Assume that h(t) is a regular curve in
S
and g(s) is its arc-length
parameterization with Frenet frame (T(s),N(s),B(s)). If
n
S
(s) =
n
(g(s)) ¥ T(s),
then
g≤(s) = k(s)N(s) = k
n
(s)
n
(g(s)) + k
g
(s)n
S
(s)
k
2
= k
2
+ k
2
k
g
(s) = k(s)
n
(g(s)) • B(s) = k(s) cos a(s)
≤
()
∑
(
(
()
)
¥
()
)
h
t
nt
3
h
h
t
()
=
k
g
t
.
¢
()
h
t
D
dt
X
()
=¢
()
-¢
()
∑
(
(
()
)
)
(
( )
)
Covariant derivative:
t
XXnnt
t
t
h
t
h
Exponential map:
exp
p
:
U
(
p
) Æ
S
, where exp
p
(
v
) =g(1)
Parallel or offset surface: p
d
(u,v) = p(u,v) +
dn
(u,v)
(p
d
)
v
= p
v
+ d
n
v
and (p
d
)
u
= p
u
+ d
n
u
N
d
(u,v) = (1 - 2Hd + Kd
2
)N(u,v)
Define s by
n
d
=s
n
.
s
k
s
k
1
2
()
=
()
=
k
,
k
1
2
d
d
1
+
d
k
1
+
d
k
1
2
K
Hd
HKd
Hd
+
K
=
,
H
=
s
d
d
2
2
12
+
+
Kd
12
+
+
Kd
Ruled surface:
p(u,v) = q(u) + va(u)
(
)
∑¢
q
¢¥
aa
aa
If
l
=
,
then
¢∑
¢
2
2
M
EG
-
l
(
)
=
KKuv
=
,
=-
2
2
-
F
(
)
2
2
l
+
v
9.20
E
XERCISES
Section 9.2
9.2.1.
Find the length of the helix p(t) = (a cos t,a sin t,bt), t Π[0,p].
9.2.2.
Find the arc-length parameterization of the curve p(t) = ( cosh t,sinh t,t), t Π[0,3].
Find the arc-length parameterization of the curve p(t) = (e
t
cos t,e
t
sin t,e
t
), t Π[0,2].
9.2.3.
