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I'm trying to parameterize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo:

orange peel

For example, the ideal curve would be inside the unit cube; have only one point of intersection with every horizontal plane $z=k$, when $k\in [-1,1]$; would start in $(0, 0, -1)$ and end in $(0, 0, 1)$, wrapping itself around them; and touch the boundary of the cube when $z=0$.

It's sort of a standard helix, compressed. I hope I was clear.

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up vote 10 down vote accepted

Well, you seem to have a lot of options; there are a number of spherical spirals that would do. The loxodrome is one (the spherical analogue of the equiangular spiral), and Seiffert's spiral is another.

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The first one was just what I was looking for. Thanks. – lentic catachresis Nov 6 '10 at 15:40
This answer was given a long time ago, but I'm curious... would a loxodrome really "start in (0,0,−1) and end in (0,0,1)"? Looking at the linked article on spherical spirals, it seems that those curves never quite reach the poles. – LarsH Jan 17 at 3:49

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