Sea Shells


Seashell around a helix.
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Introduction:

Given the parameterization of a tube, it is simple enough to alter it so that the tube can change radius as the value of t changes. Such a surface is called a sea shell, because of the effect when the variable-radius tube is plotted around a helix (see above).

Definition:

,

where x(t) is a curve, N and B are the normal and binormal vectors to that curve, and r is a scaling factor for the radius of the shell.

Properties:

As with tubes, the seashell has the property that its volume depends only on the length of the curve x, around which the tube is constructed, and the radius of the shell at the beginning and end of the curve.