final draft of curves PhD thesis

raph's picture

After years of work, I finally have a final draft of my thesis done. There's still time to make some more tweaks, but for the most part this is the version I plan to submit formally.

My defense is scheduled for Sept 3, on the UC Berkeley campus. If you're in the area and fascinated by the technical aspects of curves and font design, you're welcome.

Now that the PhD is done, I'm hoping to have more time to actually draw fonts, but we'll see - there are a bunch of last minute things, as well as other important stuff I've pushed aside while I've been crunch mode on the thesis.

Many thanks to all of you for the encouragement and feedback over the years. It's amazing to finally feel done, but it hasn't quite sunk in yet.

joeclark's picture

Maybe you could actually divulge the topic of your thesis. Or maybe you’d like us to guess.

Joe Clark

raph's picture

Sure, sorry. Here's the abstract, and of course I'm happy to answer questions. Chapter 10 is specifically about applications to fonts, so people here who would want to pick and choose might look for that one.

A basic technique for designing curved shapes in the plane is interpolating splines. The designer inputs a sequence of control points, and the computer fits a smooth curve that goes through these points. The literature of interpolating splines is rich, much of it based on the mathematical idealization of a thin elastic strip constrained to pass through the points. Until now there is little consensus on which, if any, of these splines is ideal. This thesis explores the properties of an ideal interpolating spline. The most important property is fairness, a property often in tension with locality, meaning that perturbations to the input points do not affect sections of the curve at a distance. The idealized elastic strip has two serious problems. A sequence of co-circular input points results in a curve deviating from a circular arc. For some other inputs, no solution (with finite extent) exists at all.

The idealized elastic strip has two properties worth preserving. First, any ideal spline must be extensional, meaning that the insertion of a new point on the curve shouldn't change its shape. Second, curve segments between any two adjacent control points are drawn from a two-parameter family (and this property is closely related to good locality properties). A central result of this thesis is that any spline sharing these properties also has the property that all segments between two control points are cut from a single, fixed generating curve. Thus, the problem of choosing an ideal spline is reduced to that of choosing the ideal generating curve. The Euler spiral has excellent all-around properties, and, for some applications, a log-aesthetic curve may be even better.

Shapes in applications such as font outlines contain extra features such as corners and transitions between straight lines and smooth curves. Attaching additional constraints to control points expresses these features, and, carefully applied, give the designer a richer palette of curve types.

The splines presented in this thesis are entirely practical as well, especially for designing fonts. Sophisticated new numerical techniques compute the splines at interactive speeds, as well as convert to optimized cubic Bézier representation.

William Berkson's picture

Congratulations! Look forward to this being an option in FontLab :)

eliason's picture


Frode Bo Helland's picture

The amount of time spent on this is breathtaking.

blank's picture

Congratulations, Raph. I hope that Spiro gets integrated into a professional font-editing tool.

hrant's picture

Very nice to see this completed Raph.
Go get 'em.


abattis's picture

Well done Raph!

sgh's picture

Good luck with the defense, and congratulations on the thesis! I've enjoyed reading it. The result about a 2-parameter family being defined by a single curve I think is particularly pleasing.

raph's picture

Thanks everybody! The defense went brilliantly, and (most importantly) I got the title page signed. There's a bit more bureaucracy before I actually get the degree issued, but I've met all the requirements, so it's feeling done.

Now to rest and relax a bit :)

blank's picture

Congratulations! I look forward to designing type with Spiro in the future.

hrant's picture

Raph, we are not surprised. :-)

Here's an idea for a vacation for you: go to Béziers, France,
and when you leave don't throw a coin into the fountain.


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