Discrete Differential Geometry and Developability

Keynote talk given by Keenan Crane at the third Symposium on Geometry and Computational Design, hosted at TU Wien on November 18, 2016.

Developable surfaces are those that can be made out of flat sheet materials like paper, thin plywood, or sheet metal, by pure bending, i.e., without shearing or stretching the material itself. This talk takes a look at the geometry of developable surfaces through the dual perspectives of traditional smooth differential geometry, as well as discretization, i.e., descriptions involving only finitely many pieces of information. Such descriptions are valuable both for computational modeling and simulation, as well as modeling phenomena in nature that are inherently discrete. In particular, it looks at (i) a new notion of discrete developability for triangle meshes that can be used to approximate arbitrary curved surfaces, and (ii) a connection between physical developable surfaces comprised of discrete elements and a smooth model based on conformal geometry. Applications to design, fabrication, and manufacturing are discussed.

[albertchern]

Beautiful talk. I'm intrigued by how gaussian curvature concentrates along curves. They are like the shockwaves of the normal vector field.

[leromiRos]

Awesome talk! Did you guys publish or plan to publish any of this? I couldn't find any papers.

[nachobev]

Absolutely amazing. All your work is mind blowing. Will this code be eventually uploaded on your page as you've done with other applications you developed?
Thank you, great work!

[Kram]

Nice talk as always. Too bad there was a lot of clipping on your mic. Should have been turned down just a little.
I visually like how these developable surfaces turn out. They look sort of "low poly" except smooth.
There is this idea of "facial planes" meant for sculptors and 2D artists, to get down the volume and shape of faces more clearly. That technique tends to result in takes on faces that look very similar to these developable surfaces. So I wonder if you could use that technique on 3D scans of real faces to then auto-generate such a concept.
The only problem I saw is that small features, such as ears of that horse statue, tended to be abstracted away, which probably wouldn't be useful for this scenario. If that same think could be done while preserving such details, it'd be perfect. I'm guessing that's what that curvature directed thing is gonna be all about.

Though this was literally a year ago. So... what's new? Did this already work out by now?

[VicenteCuellar]

Amazing talk, so many applications! :-)