Hobby's algorithm for aesthetic Bézier splines (2020)
jakelow.com
jakelow.com
I'm saying all this, because it seems Hobby's algorithm is something designed precisely to construct splines that maximize this aesthetically pleasing C¹ continuity... However it tends to explode splines by favoring arc connections, so I think some optimization is needed to limit that behavior: https://i.imgur.com/W9ssWTu.png
https://blog.mikeswanson.com/photoshop-script-for-ios-7-roun...
Your visual system can actually see those discontinuities.
Because it's also the biggest difference of curvature (actually the radius of curvature, the inverse of the curvature value, which is what is geometrically significant, "what we see") possible, the one between infinity (a line is a circle with an infinite radius) and a "normal" (finite) value.
I wrote a post about this on Blender Stack Exchange: https://blender.stackexchange.com/a/306616/60486
Which means that the radius of curvature of the "seams" of the spline is the same for both curves in the common point - the spline is "smoother", there is no jump in the curvature. Like C1 means, that each point (except for start and end) of the spline has a tangent (the samé tangent from the left and the right side), no spikes. And C3 is about the torsion of (3D, as in "not contained in a plane") curves.
So, the curve should really be a clothoid, sometimes called Euler spiral, but a cubic Bézier can be a good enough approximation.
I think there's even more to be done on this topic. In particular, I think the tension should increase as angles go up, which can solve the "flipping" behavior you see with both Hobby and Euler spiral splines. This is true for the Adobe κ-curves paper (which considerably overstates the extent to which it's the basis for the "curvature" tool in Adobe Illustrator; the latter has not to my knowledge been adequately documented).
I personally find this a fascinating topic, so much so that I did a PhD on it. The most natural loss function to optimize for is minimizing bending energy, and there are solutions to that going back about 300 years. However, in practice that's unlikely to be what you want - the problem with it is that its scaling properties give an advantage to curves with a longer arclength even if they have more curvature variation. Intuitively, the smoothest curve through a set of co-circular points should be a circular arc, but that's not what you get from the minimum energy curve, at least unless you impose an additional arc length constraint.
The long story is of course much more complicated, but the short version is that the Euler spiral fixes these scaling issues and has some other really nice properties. If your problem is "what is the smoothest curve that goes through this sequence of points" then arguably the piecewise Euler spiral G2 continuous spline is the distinguished answer.
It is the easiest "not wrong" answer, as it doesn't for example work with points on a circle either, where the (not only "intuitive") solution would be a circle. An Euler spiral is the answer on how to ("easily") "sufficient smoothly" connect 2 points with a given radius of curvature (and therefore tangent) at both points, and no constraints between those two points.
The OP talks about aesthetic curves, but most comments are about continuity and smoothness. Intuitively that makes sense. Do you happen to know if there is any serious research that looked into which 2D curve properties do people find aesthetically pleasing?
Second, there's all the log-aesthetic stuff from Japan, [2] has links to a lot of papers. Honestly, I find their claims of aesthetic beauty to be fluffy, not backed up by hard empirical evidence. I did do a small amount of human evaluation research in [3], which suggests that the log-aesthetic curves are visually pleasing, but the math for their use in an interpolating spline (which is the subject of TFA) is less stable than Hobby splines, Euler spiral splines, or Minimum Energy Curves (all of which are fairly similar).
I think there's more that could be done here. Thanks for your question.
[1]: Fred Attneave. Some informational aspects of visual perception. Psychological Review, 61(3):183–193, 1954.
[2]: https://www.yoshida-lab.net/english/research-e/log-aesthetic...
> supremal ==? something else
Yeah, I couldn't come up with the name as I was writing the comment and didn't want to splay a bunch of definitions across HN, hoping that "supremal" would evoke something kind of like what was in my mind.
In particular, I was imagining something closer to the Fréchet distance (allowing arbitrary indexing sets and not requiring continuity or anything nice in the reparameterization -- the infimum across all possible bijections (inf if there are none) of the suprememum of the set of distances between each bijected pair of points), but Fréchet, Hausdorff, and whatever that idea is called all have the same unique solution for this problem.