From your thesis:
> Intuitively, curvature is the position of the steering wheel when driving a car along the curve. Therefore, G2 continuity is equivalent to the lack of jerks of the steering wheel
I particularly like this analogy because I've worked simulating NPC cars in computer games and I found lack of G2 continuity of 3 Beziers annoying - the jerk of the steering wheel clearly put a sudden change of cornering forces through the suspension simulation which looked very unnatural.
I'm curious, because I haven't seen much work built on top of Knuth and Hobby's "Most Pleasing Curve" as defined by Mathematical Typography  and Hobby's follow-up papers   . Has anybody investigated these? As far as I know they're just not very convenient to rasterize.
There has been some work on this. Apple's Pages uses an implementation of the Hobby spline, demonstrating that it can be used in real consumer products. I think if you put these two side-by-side though, mine comes out quite a bit better; when you push the Hobby spline you get curvature extrema quite a distance from the control points, and mine doesn't do that.
Hobby's work has of course been an inspiration for mine for a long time, and it holds up academically quite well too - it has just the right blend of practical and aesthetic considerations and theoretical analysis; his solver runs in linear time and there's a proof it always converges (as long as tension values are within a reasonable range).
And of course there's the k-Curve work from Adobe. I think it is a step forward from Hobby in some ways (more robust) but a step back in others (it's not as smooth and most especially doesn't form true inflections).
Higher degree Béziers illustrate the "curse of too many parameters." Yes, they're very expressive, and yes you can have very high degrees of continuity, but it's also easy to make very wiggly or lumpy curves. I don't know of anyone who uses them, certainly not for font design.
It’s certainly tricky to figure out how to manage Béziers, but with the right tools and some knowledge of the math, you can do amazing things. Using the sketcher constraint tools in the SolidWorks it’s possible to join higher degree Béziers with G3 tangency (or even higher). I also build approximate offset curves of the same degree using a modified version of Tiller and Hansen’s method (I keep the control polygon legs parallel, but I add constraints to control the offset distance of the curve at n-2 places along the curve).
If I was doing font design, I’d love to have higher degree Béziers. Being able to make a sans-serif ‘S’ with single span curves would be great. Or lining up 3 control points on either end of the top of a lower case ‘n’ with vertical straight lines to have it be G3 would be nice.
> weights of the tangent vectors
Is this a synonym for curvature? In theory you could directly specify the curvature at a knot.
Or do you mean something more explicitly like weights in NURBS? Those could also conceivably be added, but IMO the hit to human intuition / intelligible UIs makes it not worth the trouble.
If I wanted to try Raph's prior work on spiro splines for comparison, is there some way to do that that's easier than learning to use Fontforge? (When I last tried Fontforge, it didn't seem to handle hi-DPI screens well...)
You should add a link to the demo in the word "demo" for the people like me that can't follow instructions :). I tried to drag the points in the screenshot instead of clicking it.
Thinking about "fairness" of a spline: Intuitively I'd say a curve is "most fair" as compared to other curves when it minimizes the sum of squared centrifugal forces of a body moving along the path. In other words: The faster I can drive a car along the path the fairer the path :)
Would such a definition make any sense?
The problem with MEC is that it tends to optimize for low tension; for inputs that are not well behaved, it sometimes optimizes for infinitely low tension, ie an infinitely long road. So there are other metrics, like Minimum Variation of Curvature, that work better for actual spline use, even though they have an energy metric that's "worse."
These curves are absolutely useful for path planning. A lot of the literature on "clothoids" from a hundred years or so ago is on transition curves for railroads. The G2 continuity condition basically means that you never have to suddenly turn the steering wheel to follow the curve, it's always moving continuously.
My favorite spline editor is the Bezigon tool in Macromedia Freehand of yore. It sprang to mind as I read the passages you wrote.
Probably not the use it was intended for, but I wonder if this could be used to improve imagemap polylines. I use Wikipedia's imagemap tool  and I wish the interface was like this (actually, I use a separate site  to generate the polyline values).
I'm mostly familiar with b-splines and NURBS and such, and there's some behaviour in this demo that I don't understand. If I create a round control point, and spin that point's handles around, sometimes the curve coming in and out of the point is a beautifully smooth tangent, coming in one side and out the other, and sometimes it forms a spike, doubling back on itself. Is there a term for that? Why does it happen? I find it fascinating and would like to learn more.
This is one of those "deep UX" questions I find fascinating.
That’s actually the whole point of this type of curve (and also Adobe’s recent “k-curve” tool) – knots are more or less at the curvature extremes; pulling a knot outward will make first a ruffle shape, then a sharp cusp, then a loop; the curve in general doesn’t extend far outside the rough shape of a polygon through the knots; and the solver is robust.
There exist plenty of curve primitives more like what you are suggesting here (see Raph’s PhD thesis), with a full-turn periodicity. But they end up creating shapes which loop out past the knots, and they also tend to not have unique solutions all the time, with the result that they sometimes will discretely jump from one local minimum to another as you make slight changes in the inputs.
The only thing difference I suggest is the control. Its the difference between:
effect = control_angle mod PI ,or
effect = control_angle / 2
When a full turn of the control produces a full cycle of effect a better ergonomic relationship might result.
If you instead had a full-turn periodicity, you would have substantially different properties for the curve shape. Several such curve primitives exist, you can use one of those if you want. This is one of the trade-offs involved in the choice of tool.
With this particular type of curve primitive, the primary way to change the spline is by adding and moving control points, not by trying to manually specify tangents. The latter should not be used for general shape design but should be seen as a specialty tool to support niche technical requirements (such as straight segment to curve interfaces), and tangent adjustments should be sparing because large tangent adjustments will create lumpy shapes.
> If you instead had a full-turn periodicity, you would have substantially different properties for the curve shape
We would have precisely the same effect on curve shape from a 360 degree control than with a wrapped 180 degree control, but have twice as much travel to adjust the outcome more finely.
For example, the guy who is writing Prosemirror (an editor), started such a project, and was able to secure a source of income.
You could start a bunch of these projects, see which one is funded first, and then focus all your energy on that project.
So whether this spline is better for, say, animation, is what I consider a fascinating research question. Hopefully putting it out there will start to get answers.
YMMV, but the problem I have is that it generally makes ugly shapes unless you are very skilled/practiced with it and very careful about twiddling parameters, because the control parameters are based on the implementation details of the mathematical tool rather than directly on features of human perception.
There’s basically a mismatch between human-centered goals and the mathematical model of those.
As an analogy, you might think of it like trying to learn how to dance vs. learning how to make a puppet dance by manipulating a control bar with your fingers to pull on its strings. The latter involves more indirection, which makes it harder for novices to get approximately their desired result.
For anyone unfamiliar with this tool, see the image in https://en.wikipedia.org/wiki/Kochanek%E2%80%93Bartels_splin...
Topology optimization is the broader field that allows topological changes to the design (e.g. the addition or removal of nodes). These optimization’s often have a solution that first have to be interpreted by an engineer to turn it into a manufacturable design.
A shape optimization performs optimization only of certain variables of a predefined design. The splines as described here seems to fit well for describing for instance sheet metal parts or cross-sections of beams.
This seems like the sort of thing that LGPL is really well-suited for. Any reason for going with GPL? Not many people are going to want to GPL-infect an entire project just to adopt a new spline solver.
If you want them to be widely adopted outside of research circles, you should avoid the GPL. Many companies would be interested in a curve like this but would avoid GPL just avoid any possible litigation. It's not great, and it's probably not a popular opinion I have but it's what I've seen from experience.
What is it with pushing people to unpaid labour to make megacorps happy with some fuzzy notion of potential "wide adoption" as the only reward?
I'm assuming the idea behind making such a comment is the realization that many people, especially in academia, do not realize how unusable GPL is.
Nonsense. Raph Levien's spiro spline code already is used in Inkscape and Fontforge, as you could easily have found out yourself. How exactly is releasing code under GPL going to to prevent its adoption in widely used GPL'ed software?
And if the approach is any good, why would a commercial software vendor not be convinced by the GPL'ed prototype to reach out to Raph to find a licensing deal or commission work from him rather than doing it themselves?
> people [...] do not realize how unusable GPL is.
The GPL is so unusable that almost every computer or device you touch is full of GPL'ed software.
GPLed code can be used by other researchers (good for popularity), but you can't use it for proprietery software.So if Adobe wants to use this curve, they have to license it.
If it actually works out, I don't know, but that's been the intention of some researchers that I asked why they used GPL.
The intent of releasing something under a MIT or BSD license, maybe.
The GP actually quoted the article "and solves most of the problems that held that back from wider popularity."
Which imply the author wanted people to adopt it. I'm assuming it wasn't about other academics adoptions but of hobbyists and commercial ventures.
There are so many examples of GPL software being blacklisted by commercial companies and dying out as a result. Linux, Bash, gcc, Blender, Ansible, MySQL, git… It really is quite a graveyard.
In all seriousness, the licensing choice seems very reasonable for someone wishing to partner with commercial companies rather than have their work lifted without royalties.
I use lots of GPL software for building my product (like git), but I can't ship any as a part of it.
I presume that code for a curve is meant to be included in a product, rather than being used on its own.
Companies find various ways to ship GPL software, depending on the situation. Sometimes it requires discussing it with the authors. Overall, it does ensure greater protection on the authors and on the propagation of the software.
Shipping bash as a part of macOS is not what I had in mind. (and it's funny, because macOS ships an ancient version of bash, because they don't want to touch GPL3...)
As for Amazon RDS, I thought that was SaaS? So they don't "ship" it either...
I mean the whole purpose of GPL is to make using it with closed source difficult / impossible.
And in practice, apart from the big Open Source projects, GPL is used a lot by people who don't want to give away their code for free, they just want to get people to pay for a commercial license. Which is odd, because it feels like it is totally against the spirit of the free software movement.
And yet, it is completely within the original intended spirit. It is not about "giving source code away for free", it is about the user (who may very well have paid for the right to use the software) being "free" to do whatever they want with it...up to and including modifying the source code for their own benefit. With traditional proprietary (aka "non-free") software, even if you had access to the source code, you are not legally allowed to modify it. And then GPL goes a step further and allows you to redistribute your modifications! That's freedom.
The idea that "open source" means "give everything away" is imho one of the primary mistakes of the "free software" movement. and why RMS now encourages the term "libre software" instead.
GPL is perfect, we get to play around with it and the author may make money from it. And if it doesn't work, the author can still switch to a more permissive license (MIT, BSD,...), the other way around won't work.
Also worth noting that it is presumably not patented, so nothing prevent companies from rewriting the algorithm.
Surely the curve function is worth something. Take that something and invest in a non GPL license.
GPL = Share and share alike
A company using the curve function for profit is not sharing. And that is OK, but they should also be paying for that, due to the fact that they are unwilling to pay by contributing more open code.
The open code comes from people who have to eat. It makes perfect sense for closed projects to pay for a suitable license. The author makes mention of all this, indicating commercial license plans.
That's exactly as it should be.
Most companies won't touch GPL code with a barge-pole.
It looks great but you should add some basic usage howto. It is obvious that you know how to use it, you created it, but other people do not.
The first time I used it it was very frustrating because I only created circle handlers, and said: This is sh*t!! Then I close the window and looked back at the sample image and saw square pointers, so I said: There must be some key that creates square pointers. After trying again I discover it creates square handlers by default.
This is the typical engineer mindset, after having spent years doing something they consider obvious what it is not.
Read "The Design of everyday things" for more design concepts if you are interested.
More charitably, this is the way all research prototypes work in any field.
Production vector illustration packages hire a team of people who spend man-years of effort canvassing customers, collecting bug reports, adding features and affordances and refining the user interface, writing documentation, and so on. Someone making a research prototype where the UI per se was built in a few weeks is not going to have the same feature richness, level of polish, or clarity of outward-facing documentation.
Give it some time.
Part of why I put it up with no instructions is to get feedback on how intuitive it is to use without any explicit instruction. I've gotten mixed response; some people seem to get it, others don't. So thanks for this bit of feedback. There's a basic help panel now.