Smoothness of Tunni curves
Linus Romer
Posts: 186
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.
8
Comments

Is there an "official" definition of what constitutes a Tunni curve anywhere?0

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_czOpw
Part 3) As a FL5 plugin
https://www.youtube.com/watch?v=BM0IlJ6jDO42 
Thanks, Pablo!0

Part 3) As a FL5 pluginThere 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 cursormove 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 editacrossglyphs mode, so I can shiftselect Tunni lines on related bowl shapes on multiple glyphs, and then use the cursor keys to adjust them all in unison.5

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 editacrossglyphs mode.
1 
In FontLab VI, 7, and 8 there is also the auto Tunni adjust: just doubleclick on the blue dotted line (the Tunni curve control). The line appears when your cursor is on its area.
2 
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 fourthorder and sixthorder curves, but others used cubic and quadratic curves, which would be applicable to font formats.0


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.0
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