Several people asked us the same two questions afterwards. First: how did you make such beautiful drawings? I can phrase it like that without fear of immodesty because Elena made the drawings, by hand with colored pencils, and then scanned them. Her colleagues at Eindhoven told me she also had some nice large 3d models made out of semitransparent plastic that would have made an interesting addition to the talk, but unfortunately she didn't bring them.
The second question we were asked is what applications we had in mind. The short answer is that this is a theory paper that isn't trying to be applied, but after the talk Yusu Wang discussed with Elena an application involving finding similarities between pairs of curves (one of Yusu's two SODA papers involved using Fréchet distance for this sort of comparison). Unfortunately I'm not sure of the details of the application. More generally, beyond the specific results in the paper itself, we think that one of its important contributions is the definition of a generalized terrain as a surface such that every point has a neighborhood within which the surface is a terrain. There should be many more problems in which these surfaces can be applied, and we tried to convey that in the talk slides by the images of real-world spiral ramps forming surfaces of this type that we found on Flickr. Sadly, we didn't end up using another photo we found, of the generalized terrain formed by a giant helicoid potato chip (or as Elena would call it a crisp), perhaps because it didn't have the proper salt-and-vinegar flavoring.
Beautiful drawings indeed! The Flickr images are nice too, and it is great they were CC-licensed by the respective photographers (although generally speaking it's somewhat unfortunate they're Non-Commercial).
Offtopic, but this seems the right opportunity to ask: how do you make such beautiful figures all the time? I know it's a matter of skill, not tools... but still, it would be greatly useful to many of us who produce terrible drawings if you did a screencast (say) showing how you drew some of the illustrations that have appeared on this blog (or at Wikipedia).
Thanks! My usual not-especially-helpful reply is that I make them in Adobe Illustrator. Sometimes I sketch what I want it to look like ahead of time on paper or my whiteboard, more often the sketch is only in my head. Perhaps some time when I'm planning a complicated figure I should take a sequence of screenshots showing how I put it together.