Helicoid to Catenoid

Definition of surface, in parametric equations:

x(u,v) = cos() sinh(v) sin(u) + sin() cosh(v) cos(u)
y(u,v) = -cos() sinh(v) cos(u) + sin() cosh(v) sin(u)
z(u,v) = u cos() + v sin()


  • is a parameter ranging from 0 to 2, giving the 80 frames shown here with a step of /40
  • u ranges from 0 to 3, resulting in the three folds of the surface
  • v ranges from -/2 to /2, defining the diameter of the "cylinder" within which the surface lies.

Note: The values /2 and 3/2 for result in a catenoid (a 2-D rendering of which is produced when we hold a flexible string, or chain, from its two ends, and keep it loose). All other values for result in a helicoid.


Weisstein, Eric W. CRC Concise Encyclopedia of Mathematics, pp. 205-206

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