DIFFERENTIAL GEOMETRY OF SURFACES 171
6.2 Curves on a Surface
A curve c on a parametric surface r(u,v) may be expressed in terms of an additional parameter t as
c(t) = [u(t)v(t)]T by letting the parameters u and v as functions of t (Figure 6.7).
Figure 6.7 A curve c(t) on a parametric surface
The tangent to the curve is given by
T
cr r
= rr A
(, )
=
(, )
+
(, )
= [ ] = =
du
dt
u
u
du
dt
ud
dt
du
dt
d
dt
du
dt
d
dt
x
u
x
y
u
y
u
vvv
v
v
vv
v
v
∂
∂
∂
∂
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
∂
∂
∂
∂
∂
∂
∂
∂∂
∂
∂
∂
∂
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
v ⎥
v
v
z
u
z
du
dt
d
dt
(6.8a)
The differential arc length ds of the curve is given by
ds
du
dt
dt
du
dt
d
dt
dt
du
dt
d
dt
du
dt
d
dt
= uuudt
(, )
= + = + +
c
rr rr rr
vvvv
vvv
⎛
⎝
⎞
⎠
⋅⎛
⎝
⎞
⎠
= [ ] =
du
dt
d
dt
du
dt
d
dt
dt
du
dt
d
dt
u
u
uu u
u
v
v
v
v
v
v
vvv
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢
⎤
⎦
⎥⋅
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣⎢
⎤
⎦⎥
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
r
r
rr
rr rr
rr rr⎥⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
du
dt
d
dt
dt
v
=
du
dt
d
dt
du
dt
d
dt
dt
v
v
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
G (6.8b)
where G
rr rr
rr rr
= = AA = =
11 12
21 22
uu u
u
T GG
GG
EF
FG
⋅⋅
⋅⋅
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
v
vvv
(b)
c(t)
rv
T
ds
rudu
rvdv
ru
c(t)
P
T
v
rv
ru
(a)
u
P
n