Computer Aided Engineering Design

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