Beyond Bézier Curves (2013)

Sadly the author died years ago (at a fairly young age...)

It does look amazing. I tried using it for a drawing app, though, and found it too slow. Maybe I was using it wrongly, but long or large numbers of strokes caused issues as it seemed to redraw the whole image more than necessary. I had better luck with just plain canvas, but I'd like to investigate CanvasKit (Skia in WASM) as well, as their drawing demo seems very performant.

It is amazing, I made https://boolean.method.ac with it.

From Shewchuk’s thesis:

> From the beginning of the implementation of Triangle, and well into the development of Pyramid, floating-point roundoff problems plagued me. Each program would sometimes crash, sometimes find itself stuck in an endless loop, and sometimes produce garbled output. At first I believed that I would be able to fix the problems by understanding how the algorithms went wrong when roundoff error produced incorrect answers, and writing special-case code to handle each potential problem. Some of the robustness problems yielded to this approach, but others did not. Fortunately, Steven Fortune of AT&T Bell Laboratories convinced me, in a few brief but well-worded email messages (and in several longer and equally well-worded technical papers), to choose the alternative path to robustness, which led to the research described in this chapter. For reasons that will become apparent, exact arithmetic is the better approach to solving many, if not all, of the robustness worries associated with triangulation.

Of course, what can be done with polygons is often not feasible for more general kids of shapes like parametric cubics.

* * *

What is a “clever vector path algorithm”, and how do you avoid them?

> The other issue that comes up again and again is self-intersecting paths. Most published algorithms you'll find don't bother to support them, which also makes them not very useful in reality because self-intersecting paths are everywhere.

Exactly this. This is the biggest challenge when doing useful stuff with curves in graphics design/3D.

Curve offsetting is a common tool to produce paths (2D), extrusions & bevels (3D) etc. Whenever the offset distance exceeds the local radius of curvature intersections occur.

When you have some clever algorithm to deal with those, e.g. clipping them out, you're then faced with offset curves that have different parameterization than the original. Try to create a patch mesh from those then.

This is a huge brain fuck and every time you think you got it worked out another edge case you didn't think about will creep up from out of nowhere.

Ah, thanks—I didn't see that mentioned in the article at all. It definitely seems like the most important property for a curve that someone will be drawing should have.

That's right the Hobby splines are a strict subset of the polynomial cubic Béziers.

The polynomial cubic Béziers are defined by 4 control points. In 2D that's 4 x 2 = 8 degrees of freedom. In the terminology of HN raphlinus's PhD thesis, we fix the 2 endpoints, which removes 4 degrees of freedom, and they're referred to as a 4-parameter family.

In addition to the 2 endpoints, Hobby splines are defined by the start and end tangent angles Θ0 and Θ1, plus a tension parameter τ.

If we fix the 2 endpoints and the tension parameter τ, and allow Θ0 and Θ1 to vary, then we have a 2-parameter subspace of the 4-parameter space of polynomial cubic Béziers for 2 fixed endpoints.

If we allow the tension parameter τ to vary we have a 3-parameter subspace.

So we have lost 1 degree of freedom, but in terms of aesthetics we haven't lost so much because the majority of the splines that we've thrown away are likely uglier than the splines we retained.

One TL/DR from HN raphlinus's thesis: Euler spiral splines are about the best you can do for a 2-parameter family that cover all reasonable possibilities for Θ0 and Θ1 with good aesthetics.

That's an unrelated spline algorithm to the article discussed here. Interesting on its own though.

Indeed. The biquadratics aren't particularly smooth, and don't support inflections naturally. See https://raphlinus.github.io/curves/2018/12/21/new-spline.htm... for more on this.

Hesitate to link anything from Elsevier, but IIRC the two standard industry 3D CAD/CAM journals are:

● Computer-Aided Design - more applications and commercial stuff https://www.journals.elsevier.com/computer-aided-design

● Computer Aided Geometric Design - mainly geometric algorithms for curves and surfaces https://www.journals.elsevier.com/computer-aided-geometric-d...