〈 Berger | Dillon 〉

When I was making my u-sub animation a while ago, i was thinking about something similar.. how even an innocent non-linear u-sub seems to deform the region in a way that’s not obviously equal in area to its ‘parent’ region so i was wondering if there were .. 1/2

ways to quantify the notion of ‘intuitive’ equal area .. started by fixing 2 points in the x-axis then started deforming curves connecting them .. it started to smell like topology so i dropped it and went back to research

Maybe deform one curve into the other along a geodesic defined by the earth move distance, which preserves the integral

hmm.. i think I understand .. so if I have a curve γ(t) on some manifold, then moving along a geodesic to some γ’(t) will have equal area?

It'll be the path that preserves area but moves the smallest amount of "stuff" over time to do it. I'm sure @gabrielpeyre tweeted an animation along these lines a while back.

Ah...it's this. https://twitter.com/gabrielpeyre/status/1137585164886978562?s=21 … I think it's just saying linearly interpolate the "cumulative" density. I don't know if that's interesting in this case.

Interesting.. reminds me of using fractional derivatives to interpolate between functions in an animation, though those aren’t area preserving in general

That's sounds close to something I've played with: using fractional "functional" powers of functions to interpolate. Eg. f is "functional" square root of sin if f(f(x))=sin(x)