Geometry Processing with Intrinsic Triangulations (Day II)

This video is the second in a series of two lectures given by Keenan Crane at the Harvard FRG Workshop on Geometric Methods for Analyzing Discrete Shapes: cmsa.fas.harvard.edu/frg-2021/
Day I: • Geometry Processing wi...
Abstract: The intrinsic viewpoint was a hallmark of 19th century geometry, enabling one to reason about shapes without needing to consider an embedding in space---and leading to major developments in the 20th century such as Einstein's theory of general relativity. Yet 21st century digital geometry processing still largely adopts an extrinsic mindset, where the geometry of a polyhedral surface is expressed via vertex positions in n-dimensional space. This talk explores how the intrinsic view of polyhedral surfaces helps relax some standard assumptions in geometric computing, leading to algorithms that are more flexible and numerically more robust. In particular we will examine fundamental data structures for intrinsic triangulations, extensions of important triangulation algorithms to curved surfaces, as well as methods for finite element problems, finding geodesics, and computing discrete conformal maps.

[@omerkose8735]

These are gold. Finding sources about Geometry Processing is really hard. You are doing a beatiful job Dr. Crane. Very appreciated.

[@tinkeringengr]

Non-linear ripple effects in our civilization. He is altering the course of humanity with his next level understanding coupled with the ability to convey it. Generations will honor him. Keep it coming please (with demo code -- the transition from theory to practice is steep :)

[@keenancrane]

See 53:16 for demo code. :-)

[@eelcohoogendoorn8044]

Such a great series of papers; definitely the most inspirational in geometry i've read in a long time. Simplices in higher dimensions are so aweful though. So seducingly simple to define, but you cant even properly regularly subdivide a tetrahedron. The day I found out about that came as such a blow to me; as if everything id learned about triangles was a complete dead end. I suppose surfaces are still a joy in their own right though.

[@jacobchateau6191]

Such an awesome talk! I'd love to use this kind of thing to do fast worldvolume meshes of meshes in motion. I imagine a GPU could incorporate these methods to optimize ray tracing and numerical integrators for wave phenomena. Now I'm extremely curious about Delaunay in higher dimensions.

[@siegfriedstow]

Hey isn't it guaranteed that you can't find a unique extrinsic geometry for a surface without some prior knowledge? Eg, volume or some notion of curvature distribution? Because I can always take any 'hill' and invert it, preserving edge lengths and angles (sort of), to get a new surface. So you can only hope to produce a surface under some conditions eg minimise the sum of edge bending or signed binormals... You can't even do it with a plane except for the trivial case

[@keenancrane]

Yes, but you can still ask for an algorithm that computes *one* of the solutions. Or all the solutions. Or tells you with 100% certainty that no embedding exists. Lots of ways to state the problem. Also, your "hill" example works out only if the neighbors of the vertex sit in a common plane. So, if a vertex has degree 3, then it will always be possible to reflect it in the plane of its neighbors and preserve the metric. But if the vertex has degree 4 or greater, then generically its four neighbors will not sit in a common plane. Empirically (i.e., based on numerical experiments I and others have done) isometric embedding of polyhedral metrics seems to be pretty darn rigid, even in the nonconvex case. Check out this beautiful paper for instance: cseweb.ucsd.edu/~alchern/projects/ShapeFromMetric/. The question is: how rigid exactly, and what can you do about it algorithmically?

[@numoru]

Road to hexaflexagon here we go mwhahahaha