DIFFERENTIAL GEOMETRY OF SURFACES 167
Figure 6.4 Catenoid and pseudosphere
x
z
y
x
(a) Catenoid (b) Pseudosphere
of constant length a. Rotation of a tractrix about its asymptote results in a pseudosphere. If
the asymptote is the x-axis, the equation of a tractrix for uvarying between 0 and π/2 is given
by
xu( ) = cos + ln tan a u a u
2
, y(u) = asinu
and for the rotation angle v varying from 0 to 2π, the equation of the pseudosphere is
xu a u a
u
( , ) = cos + ln tan
2
v , y(u,v) = asinu sin v, z(u,v) = asinu cos v
Figure 6.3 A sphere
z
P
B
P 1 y
0
u
A
x
a
v
(e) A Helicoid: This is the surface formed by the perpendiculars dropped from a circular helix to its
axis. The parametric equation of a helicoid for vvarying from 0 to 2π is represented by