DIFFERENTIAL GEOMETRY OF SURFACES 183

Example 6.4. A toothpaste tube is a ruled surface. For p(u) = acosui + asinuj + 0k with aas the
radius of the circle and q(u) = (a– 2au/π)i + 0j + bk, the straight line parallel to the plane of the circle
and at a distance b from it, the half section r(u,v) of the tube with u in [0, π] is given by

r(u,v) = [(1 –v)acosu + v(a– 2au/π)]i + asinu(1 –v)j + vbk

To show that the toothpaste tube is not developable, the Gaussian curvature is determined.

ru= [–(1 –v)asinu– 2av/π]i + acosu (1 –v)j

rv= [–acosu + (a– 2 au/π)]i–asinuj + bk

condition conveys that the tangent plane touches the surface along a straight line (in u and v) at
parabolic points. If both the curvatures κminandκmaxare nonzero, the surface is called a doubly
curved surface.
Consider two curves p(u) and q(u) in Figure 6.14(b). If a straight line moves uniformly such that
its one end is always on the curve p(u) and the other is always on q(u), a ruled surface is generated.
The equation of the resulting surface is

r(u,v) = (1 –v)p(u) + vq(u) = p(u) + v[q(u)–p(u)] (6.38a)

or r(u,v) = p(u) + vd(u) (6.38b)

withd(u) = q(u)–p(u). For ruled surfaces rvv = 0 which makes N= 0 and the Gaussian curvature in
Eq. (6.36) becomes

K

M

GG G

=

2

(^112212)
2 (6.39a)
Further, for a ruled surface to be developable, it is required that K and thus M is zero, that is,
ruv·n = 0 at every point on the surface. The mean curvature for developable ruled surfaces is then
given by
H
GL
GG G

2( – )
22
11 22 122
(6.39b)
Figure 6.14 (a) Developable and (b) ruled surfaces
q(u)
d(u)
p(u)
(a) (b)