This program takes the parametric equations for arbitrary simple closed curves and finds the corresponding functional rotation that "deforms" a circle into a 3D curve for which the 2D projection is this simple closed curve.
The rotation is accomplished using a functional unit quaternion which can be shown to be a 1-parameter homeomorphism.
The problem is indeterminate, so there are multiple solutions; we randomly select for the purpose of plotting the resultant curves.
For this work, I colloborated with OpenAI's GPT-3.5 model via the OpenAI API
- Points on the curves (p) are expressed as "pure" quaternions (i.e. p=(0,x,y,z)).
- Rotations (q) are expressed as "unit" quaternions (i.e. norm(q) = 1).
- We perform a quaternion rotation using Hamilton products: p2 = q(p1)q'
More detailed theoretical foundations are available here:
P. T. Jardine and S. N. Givigi, "Flocks, Mobs, and Figure Eights: Swarming as a Lemniscatic Arch", IEEE Transactions on Network Science and Engineering, 2022.
Here are some results. Note that the Bernoulli curve does not make use of a pure quaternion and so is not strictly homeomorphic.
