DIFFERENTIAL GEOMETRY OF SURFACES 189

For a curve on a plane, we can form a surface of revolution by rotating it about a given line (axis).
A surface of revolution is expressed as

r(u,v) = λ (v) cos ui + λ (v) sin uj + μ (v)k≡ [λ (v) cos u,λ(v) sin u,μ (v)] (6.53)

for a curve, called the profile curve, lying on the x-z plane given by

r(v) = λ (v)i + μ (v)k = (λ (v), 0, μ (v)) (6.54)

This profile curve when rotated about the z-axis through an angle u, gives the equation of the surface
patch. Various positions of the profile curve around the axis are called meridians.Each point on this
curve creates a circle called parallels. The tangents to the surface of revolution are given by

rv ̇ ̇ ̇

vvv

= cos , sin , = [ cos , sin , ]

d

d

u

d

d

u

d

d

uu

λλμ

λλμ

⎡

⎣⎢

⎤

⎦⎥

(6.55)

ru = [–λ sin u, λ cos u, 0]

The normal at a point and the coefficients (G 11 ,G 12 ,G 22 ) and (L,M,N) of the first and second
fundamental forms of the surface can be determined as

Nr r = = [u× v λμ ̇ ̇ cos , uuλμ sin , – λλ ̇] = [ cos , sin , – ]λ μ ̇ ̇uuμ λ ̇

n

rr

rr

̇ ̇ ̇

̇ ̇

=

| |

=

[ cos , sin , – ]

(^22) +
u
u
× uu
×
v
v
μμ λ
μλ
G 11 = ru·ru = λ^2 , G 12 = ru·rv = 0, G 22 = = rrvv⋅ μλ ̇^22 + ̇ (6.56)
LMN = uu = u

• 
  
  
  
  • 
    
  
  
  
  

, = = 0, = =

+

(^2222) +
rn
̇
̇ ̇
rn rn
̇ ̇ ̇ ̇ ̇ ̇
̇ ̇
⋅⋅⋅
λμ
μλ
λμ λμ
μλ
vvv
SinceG 12 = 0, and from Eq. (6.14) cos θ = G 12 √(G 11 G 22 ) = 0, the meridians and parallels are
orthogonal as the angle θ between the tangents ru and rv is 90°. Since both G 12 = 0 and M = 0, the
conditions for the meridians and parallels to be the lines of curvature are also met. The Gaussian and
mean curvatures (K and H, respectively) in Eq. (6.36) for a surface of revoution are given by
K
LN M
GG G

• 
  
  
  
  • 
    
  
  
  
  

=

–( + )

( + )

2

11 22 122

2

222

̇ ̇ ̇ ̇ ̇ ̇ ̇

̇ ̇

λμ λμμ

λμ λ

H

GN G L GM

GG G

=

+ – 2

2( – )

=

(– ) + –

2( + )

11 22 12

11 22 12

2

23

223/2

λλ λ μ λλμ μ

λμ λ

̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇

̇ ̇

(6.57)

Example 6.6. A curve r(u) = ui + log uk lies in the x-z plane. It is rotated about the z-axis through
an angle v. Find the properties of the surface.
The equation of the surface, tangents, normal and coefficients of the first and second fundamental
forms are given by

r(u,v) = u cos vi + u sin vj + log uk = (u cos v,u sin v, log u)

⇒ rruuuu uu r
u

= cos , sin ,

1

, = (– sin , cos , 0), = – sin , cos , –

1

vvvvvvv 2

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠