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Most of the applications of Hobby's spline would be better served by using Euler spirals instead. These are G2 continuous and have less curvature varation (in fact, Euler spirals are solutions of the Minimum Variation Curvature problem). I think it's fair to characterize Hobby's spline as an approximation to Euler spiral splines, just less expensive to compute.

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).

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I was just going to comment to ask if Hobby spines were the minimal solution to any mathematical loss. Natural cubic splines are the solution to minimum L2 penalty on "roughness" (integrated squared second derivative).

No. Among other things, they'd have a higher degree of continuity if they were.

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.

Hi Raph! I'm the author of the blog post. I actually read your PhD thesis when I was working on this project and trying to wrap my head around splines. It was a huge help to me in understanding the landscape of the field and how to think about and compare different classes of splines. Just wanted to say thanks!

> then arguably the piecewise Euler spiral spline [...] is the 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.

This is a bit of a tangent but your expertise makes you the best person to ask, so I'll give it a try:

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?

It's a good question. I touch on it a little in my thesis (mostly chapter 2). First, there is psychology literature from the 50's [1] that indicates that curvature minima and maxima are salient, so having superfluous curvature extrema makes a curve less aesthetically pleasing. Hobby splines suffer from this, if you arrange the control points in a zigzag to make a serpentine, it places curvature extrema just on either side of the control point.

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...

[3]: https://levien.com/phd/LevienSequinCAD09_014.pdf

Sure. Hobby splines minimize the supremal difference between your favorite curve and a Hobby spline.

That's a joke answer, as you meant it to be trivially true for any curve family, but there is an important sense in which it is not true - Hobby splines are not closed under subdivision. So if you put an additional point down that lies exactly on the spline, then you won't in fact minimize the supremal difference (I assume you mean Hausdorff or Fréchet distance) to the original Hobby spline.

Fun! Thank you :)

> 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.