Computer Aided Engineering Design

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