snail shell, cut open view
snail shell 001
snail shell with varying parameter
snail shell circles 001
snail shell, showing how it is generated by moving a circle of increasing radius
snailshell st
snailshell st
About Snail Shell Surfaces

Traudel Karcher

These snail-like surfaces are included for their entertaining shapes;
try making one of your own.

Note that a snail is constructed as a one-parameter family
of curves C (u), depending on one of the surface parameter
u, where the curve C (u) pro jects onto the circle of radius
r := s (aa + bb cos(u)) about the origin in the x, z -plane; in
fact it is given parametrically by:

x := r cos(vv)
z := r sin(vv)
y := dd(1 − s) + s bb sin(u)

where the variable vv parameterizing C (u) is given in terms of the
second surface parameter v by

vv := v + (v + ee)^2/16.

The variable s is a function of v and the parameters cc and ee:
s := exp( −cc vv).

It scales the diameter of the opening of C (u), while the parameter dd
stretches in the z-direction. (The snail may not close nicely at the
top after parameter changes. The quadratic term in the definition of
vv can be used to close the snail if one also adjusts vmax by

The parameter ee controls the size of the opening — the default is
ee = −2.

Make only small changes to cc, and keep bb ≥ aa.

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