You’ve probably seen a pattern like this on Christmas decorations or chocolates, but it’s actually a demonstration of a technique I just learned from @zenorogue for creating a coordinate system (u,v) on a surface such that moving along a grid line of constant v from (u,v) to (u+s,v), or along a line of constant u from (u,v) to (u,v+s), means travelling precisely a distance s.
Suppose you start out with coords (x,y) on the surface with the property that the metric g depends only on y, not x. For example, with the standard geographical coordinates on a unit sphere, the metric only depends on latitude, not longitude, so we would set x=longitude and y=latitude, and g would be the matrix
cos(y)^2 0
0 1
Consider the curve C(t) = (t,f(t)) in (x,y) coordinates, and solve the differential equation for f(t) that ensures that the rate of motion along the curve wrt t is a constant, c:
C'(t) g C'(t) = c^2
For (x,y) = (longitude, latitude) on a unit sphere, this becomes:
cos(f(t))^2 + f'(t)^2 = c^2
which is solved by:
f(t) = am(t √(c^2-1) | 1/(1-c^2))
where am() is the Jacobi amplitude function.
We now define new coordinates (u,v) so that:
(x,y) = ((u+v)/c, f((u-v)/c))
Because of the way we have chosen f(t), and because g depends only on the second coordinate, y, the rate of motion along a curve of constant u wrt v, or vice versa, is guaranteed to be 1.
The animation uses c ≈ 1.4866, which was chosen to make the curves meet up neatly along the international date line.
But as a bonus for wondering about the connection, I just discovered that Erdős[*] wrote about Seiffert’s spirals as a way of teaching about Jacobi functions!
@gregeganSF Thanks for the shoutout! I have obtained that by analyzing your earlier toot https://mathstodon.xyz/@gregeganSF/110774975354528301, which could be obtained by using this technique for the horocyclic coordinates on the hyperbolic plane, covering a horodisk (I attach an animation where more of the horodisk is covered).
As I have already mentioned on 𝕏, we can also use Lobachevsky coordinates in the hyperbolic plane, covering an equidistant band.
This leaves one more natural case: azimuthal (e.g. polar) coordinates in hyperbolic or Euclidean geometry, to cover a disk.
(In spherical, we would get the same result as with longitude/latitude, although possibly not covering the whole sphere if c<1; and the Euclidean analog of longitude/latitude is boring.)
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