8
$\begingroup$

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.

1 Answers 1

11

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.

  • 1
    The first one was just what I was looking for. Thanks.2010-11-06
  • 0
    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.2016-01-17