On Date: Wed, 17 Oct 2012 09:53:59 +0200 Jasper van de Gronde wrote:

Now there's the real problem. I even asked a question related to this on

math.stackexchange, as it was driving me crazy. Essentially the problem is that curves with zero-length handles do tend to have infinite curvature at the endpoint(s) with a zero-length handle, but the curvature grows very quickly over a very short length of the segment. So visually, you hardly see anything. Put differently, the slope has a non-zero derivative at the endpoint, while the norm of the derivative is zero, so the slope changes by some finite amount over an infinitesimally small length, leading to infinite curvature, but a finite (and typically quite small) change in slope.

So essentially you have the extremely weird situation where a slope that looks very smooth and (almost) straight, can actually have infinite curvature at its endpoints (I think in principle this can also happen elsewhere, but that's less of a problem in this context).

The problem with taking an approximation of the path is that I'm not entirely sure how "stable" this is. So would taking a slightly different approximation give you a completely different curvature, or is there some "natural" curvature that can be defined? In fact, I've just tried fitting a quadratic curve through the first 1/3 of a B?zier curve with a zero-length handle (at the first node), and I can get a huge range of curvatures, without a big difference in fitting quality.

Of course this can always be "solved" by simply prescribing an approximation method. One thing you could try is to basically take a finite difference approximation to the curvature over a predefined range of the curve (if the SVG spec would do something like this, the range would probably have to be specified in the spec). For example, take the angle of the curve at t=0 and t=1/3, take the difference and divide by the arc-length. You could experiment with the range, or even try to do a "higher order" approximation. I'm not entirely sure what this would accomplish though.

I've not read the full conversation but in Synfig, when it is needed to evaluate

the tangent (and by perpendicularity the normal) at the start or end point of a bezier with a tangent small than a certain small length (zero-length tangent condition) we don't

evaluate the derivative of the bezier exactly on the start or end of the bezier but in a fixed small earlier or later bezier index. We use 0.025 as that value.

See CUPS_TANGENT_ADJUST usage in the code for the outline layer: https://github.com/synfig/synfig/blob/master/synfig-core/src/modules/mod_geo...

It works reasonably well in most of the cases.

I hope it helps. Cheers Carlos