Smoothness of Tunni curves

Linus Romer

While researching a curve optimization algorithm, I also examined Tunni curves for their smoothness. My findings may also be of interest to you. I have summarized them in the attached PDF.

Jens Kutilek

Is there an "official" definition of what constitutes a Tunni curve anywhere?

PabloImpallari

Yes! and video too (in Spanish)


Part 1) Eduardo's Introduction
https://www.youtube.com/watch?v=IVYX72rHemA

Part 2) Theory
https://www.youtube.com/watch?v=SWwr_czOp-w

Part 3) As a FL5 plugin
https://www.youtube.com/watch?v=BM0IlJ6jDO4

Jens Kutilek

Thanks, Pablo!

John Hudson

Part 3) As a FL5 plugin

There is native support for Tunni editing in FontLab 8 (introduced in FLVI). It should be noted that this works as an editing mode—you can drag and cursor-move Tunni lines—regardless of whether the curve follows Tunni smoothness. I really like it as a way to adjust a bowl in or out a bit while maintaining its tension. I also really like that Tunni line editing is accessible in FontLab’s edit-across-glyphs mode, so I can shift-select Tunni lines on related bowl shapes on multiple glyphs, and then use the cursor keys to adjust them all in unison.

Typedesigner

Tunni Lines are without doubt a great invention. I use Tunni Lines very often in FontLab 8. Incidentally, FontLab 8 has a similar function in Panel Node, which I also like very much. This function can also be applied to several letters at once in FontLab's edit-across-glyphs mode. 

Igor Freiberger

In FontLab VI, 7, and 8 there is also the auto Tunni adjust: just double-click on the blue dotted line (the Tunni curve control). The line appears when your cursor is on its area.

John Savard

This was interesting; a Tunni curve is just a special case of a Bezier curve, claimed to have superior aesthetic properties. And in looking that up, I found a reference to Hobby's curves; this refers to an algorithm for quickly producing a sequence of Bezier curves to pass through a series of points. However, continuity of curvature across those points is only approximate, not exact, although this would usually be invisible.

This inspired me to do a search for what I would be interested in, and I found that there were published mathematical papers on approximating conic sections with Bezier curves; some did so with fourth-order and sixth-order curves, but others used cubic and quadratic curves, which would be applicable to font formats.

Dave Crossland

https://github.com/raphlinus/spline-research 

Jens Kutilek

I think the normal “adjust” mode of my Curve EQ plugin (RoboFont/Glyphs) works with Tunni curves then, even if they are not called that in the UI.

Incidentally, John Hobby curves, as mentioned in Linus’ paper can also be used in Curve EQ.