Greg Egan on Nostr: But to make a Möbius strip with the geometry of part of the hyperbolic plane ...
But to make a Möbius strip with the geometry of part of the hyperbolic plane requires even more care with the choice of centreline.
In general, we can take a curve and extend it into a surface of constant Gaussian curvature, by means of a partial differential equation that describes the way the surface continues away from the original curve. But to make a Möbius strip this way, the centreline must have an inflection point, but the derivatives for the curves that cross it have terms with the curvature of the centreline in their denominator.
At the inflection point, where that curvature is zero, those derivatives will become infinite, unless the centreline meets further conditions that ensure the numerators of those terms also go to zero in the same way.
There are an infinite number of such terms, and there is no systematic way to deal with all of them.
But for the image at the top of the thread, enough terms were made finite for a 6th-order Taylor series to have finite coefficients, and that’s enough to give a good approximation to the idealised hyperbolic Möbius strip that we want.
More details at:
https://www.gregegan.net/SCIENCE/PSP/PSP.html#CYG
In general, we can take a curve and extend it into a surface of constant Gaussian curvature, by means of a partial differential equation that describes the way the surface continues away from the original curve. But to make a Möbius strip this way, the centreline must have an inflection point, but the derivatives for the curves that cross it have terms with the curvature of the centreline in their denominator.
At the inflection point, where that curvature is zero, those derivatives will become infinite, unless the centreline meets further conditions that ensure the numerators of those terms also go to zero in the same way.
There are an infinite number of such terms, and there is no systematic way to deal with all of them.
But for the image at the top of the thread, enough terms were made finite for a 6th-order Taylor series to have finite coefficients, and that’s enough to give a good approximation to the idealised hyperbolic Möbius strip that we want.
More details at:
https://www.gregegan.net/SCIENCE/PSP/PSP.html#CYG