back

by raphlinus·2y ago·view on hn ↗
I can perhaps refine (1) and provide some insight in support of (2).

It does make intuitive sense for handles to be 1/3 of the length of the curve, as that makes the "speed" close to constant, but this is not the smoothest possible choice in general. In particular, it's not the best approximation to a circular arc, which by most reasonable definitions is the smoothest possible curve. I did the math[1] and found that there's a parabola around each endpoint that does result in an accurate arc approximation, and in fact may be easier to apply as it doesn't move around as the length of the curve changes. Rather, each of the parabolas can be applied independently.

Evidence in favor of this is O(n^5) accuracy scaling to approximate Euler spirals (which are also a good candidate for "smoothest possible curve" depending on the way you define that). I suspect, but haven't yet demonstrated, that there's some variational sense in which this choice is optimum, or at least to a first order approximation.

You can think of cubic Béziers as forming a four-dimensional parameter space (the parameters can be accounted as the lengths and angles of the control handles; in this scheme, the positions of the endpoints are not counted). By applying a rule such as 1/3 arc length, or falling on a parabola, that reduces it to a two-dimensional space (same dimensionality as quadratic Beziers, Euler spirals, and sections of rectangular elastica). I think two dimensions is too few, as it doesn't give you the ability to easily form superellipse shapes, a particular strength of cubic Béziers and quite handy for font design, as superellipse is extremely common in fonts. I think there may be some value in exploring a three-dimensional parameter space (which, among other things, is the count for general elastica) and have some ideas.

As for (2), the problem is that for s-shaped curves, cubic Béziers tend to put curvature maxima near but not exactly on endpoints. Since the curvature maximum is visually salient, you do better subdividing your curves so you can place them at joins. Other curve families, notably those based on elastica and Euler spirals, don't have this problem.

[1]: https://raphlinus.github.io/curves/2021/02/19/parallel-curve...

1 comments
The ideal dimensionality of the parameter space depends on the jaggedness of the curve. Derivatives are useful when they stay constant over a stretch of time; when derivatives vary too much, it becomes more efficient to use more points of lower degree. This is the same calculation in other domains of approximation: numerical integration, DiffEq, and Taylor series.

For example, to draw rough surfaces, point-to-point lines are the most efficient way, with a ton of points. For industrial geometric shapes, lines and ellipses become efficient. As the smoothness of the curve goes up, more derivatives become valid. But sometimes those higher derivatives are not useful.

This is why the parent comment likes the Pencil tool; it's optimal for high variation, because it is a local (smoothed) control of position, the zeroth derivative. The cost is some loss of smoothness, which likely doesn't matter for her domain, since her hands move smoothly enough.

I think the Pen tool would be a more competitive alternative with a better control scheme. For example, a Pen tool with local control would unify the Pencil and Pen: letting you draw jagged curves by holding your mouse down, and letting you draw straight lines and smooth curves by clicking. There would be no interruption of flow, as you can use either method freely to continue a curve. They should inherently be the same tool anyway, just with different numbers of derivatives.

For example, to draw rough surfaces, point-to-point lines are the most efficient way, with a ton of points.

I usually just draw a quick simple line and tell Illustrator to rough it up for me. There's a lot of ways to do this; "use a custom brush" is a common one, so's "apply the Roughen effect". These will dynamically generate a buttload of virtual points in the vicinity of the basic path when it gets rendered; the editing view is still just a handful of points. Much easier to edit, much faster to draw. If I need precise, human-defined jagginess to a path, then I whip it out with the Pencil, and push it around with stuff like the Puppet Warp tool, which lets me drop a few pins into a complex set of paths, and distort it based on how I move those around.

Illustrator's Pencil tool actually performs a certain amount of smoothing on the raw positional input from the drawing tablet. There's a slider for how much it does this. I have it right in the middle of the range and it makes for a pretty nice compromise between catching every deliberate motion of my fingers/hand/arm across the tablet, and throwing away the little irregularities that I would be working hard to eliminate if I was working in pen and ink.