 # Seashell surface

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In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.

## Parametrization

The following is a parameterization of one seashell surface:

x = 5 4 ( 1 v 2 π ) cos ( 2 v ) ( 1 + cos u ) + cos 2 v y = 5 4 ( 1 v 2 π ) sin ( 2 v ) ( 1 + cos u ) + sin 2 v z = 10 v 2 π + 5 4 ( 1 v 2 π ) sin ( u ) + 15

where 0 u < 2 π and 2 π v < 2 π \\

Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like

F ( θ , φ ) = e α φ ( cos ( φ ) , sin ( φ ) , 0 sin ( φ ) , cos ( φ ) , 0 0 , 0 , 1 ) F ( θ , 0 )

which starts with an initial generating curve F ( θ , 0 ) and applies a rotation and exponential magnification.

## References

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