@tpfto He's right that it's annoying, but to me the more annoying part is not so much the trolly deliberate misidentification as the non-monotonic curvature of the red spiral. I wonder if some other shape than quarter-ellipses would work better for this while still (unlike an actual log spiral) staying axis-parallel at the points where it crosses the rectangle corners.

@11011110 That's actually a good question. Of course, Randall's joke is effectively what's called a "conic spline" with at least \(G^1\) continuity, but one could then ask if there are other splines that could be devised, such that they satisfy the required conditions at those rectangle points.

@tpfto @11011110 One only needs a curve with specified endpoints and tangents whose curvature decreases monotonically from some κ at the small end to κ/√2 at the large end, yes? I imagine such a curve could be found quickly by adding a couple more sinusoids to the parametric form of the ellipse.

@narain @11011110 I should confess that I haven't even attempted to modify the usual (signed) curvature function for an ellipse to get the curvature function for Randall's spiral. Regarding Rahul's suggestion, I had been idly thinking about adding control points to the spline, but did not consider the "epicycle" approach. (I wonder if they might even be found to be equivalent, since a rational parametrization can be easily converted to a trigonometric one...)

@tpfto @11011110 After some numerical experimentation I've found that it's easy to get specified curvatures at the endpoints, but it's very much not easy to get monotonic curvature. In fact now I'm pretty sure it's impossible: if the curvature decreases monotonically from κ to κ/√2 then the curve has to lie "outside" the osculating circle of radius 1/κ and "inside" the circle of radius √2/κ, but it seems there's no way to choose κ so that both endpoints lie outside/inside each other's circles.

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