Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)

Full playlist: • Discrete Differential ...
For more information see geometry.cs.cmu.edu/ddg

[@shiv093]

0:36 Manifold - First Glimpse 5:05 Simplicial Manifold - Visualized 6:50 Simplicial Manifold - Definition 11:49 Manifold Triangle Mesh 14:02 Manifold Meshes - Motivation 16:21 Topological Data Structures 16:41 Adjacency List 19:21 Incidence Matrix 25:03 Aside: Sparse Matrix Data Structures 28:44 Data Structures - Signed Incidence Matrix 31:18 Half Edge Mesh 33:41 Half Edge - Algebraic Definition 39:31 Half Edge - Smallest Example 42:20 Other Data Structures - Quad Edge 43:08 Dual Complex 44:00 Primal vs. Dual 45:20 Poincare Duality 46:16 Poincare Duality in Nature

[@scottpet100]

What a terrific service you have done by offering these classes on KZbin! You are a terrific teacher that explains the concepts very thoroughly in plain english without assuming we know the math jargon. I've always wanted to understand these concepts better, and this class bridges a lot of that gap in my knowledge. Thank you!

[@ObsessiveClarity]

22:09 I think this just shows how great these lectures are. You present the ideas so naturally that it all feels obvious, but I know other lecturers would fall short, and I would feel so lost reading a technical definition like that.

[@alfredoarroyog.5705]

"If the Tilapia can do it, then so can you. " - Keenan Crane. Words to live by! Thanks for these lectures!

[@utof]

47:16 Yeah 😂

[@erinzhang8664]

simplicial manifold 6:50 manifold triangle mesh 11:49 manifold mesh motivation 15:25 adjacency list 16:40 incidence matrix 19:21 sparse matrix data structure 25:03 signed incidence matrix 28:45 half edge mesh 31:18

[@saturdaysequalsyouth]

Wow, this is the first time I feel like I'm starting to understand this stuff. This is amazing. Thank you.

[@xanthirudha]

This is a very nice geometry course, can't believe it's 2021 lecture, it seems this can be great for programmers as well

[@bryanbischof4351]

I really liked the slide on “how hard is it to check for manifold by value of k”.

[@zeyonaut]

I followed along with the C++ exercises for this lecture, and there appears to be a bug (I think it’s a memory leak caused by accessing coefficients from an Eigen::DenseCoeffsBase) that can cause an implementation of the boundary operator to silently crash when running the test suite. If anyone else has this issue, just destroy and rebuild the guilty dense vector as often as necessary. I ended up rebuilding it on every coefficient access to avoid crashing, which didn’t seem to affect performance significantly.

[@joshuaclavel9666]

I love this my goodness something that my mind needs to know

[@columbus8myhw]

Oh hello! I have a twin. And that twin, I S M E

[@columbus8myhw]

35:41

[@frittenpeter]

highly underrated channel

[@sp4ghet]

Hi, been following these lectures and they're super helpful! I noticed your algebraic definition of a vertex in a half-edge seems to disagree with the course notes and your previous description of the vertex struct with pseudo code. Is it perhaps meant to be $ ho \circ \eta$ so the halfedges are coming outwards from a vertex?

[@keenancrane]

Ah, you're right! Yes, just a typo (or, as you say, a different convention for whether vertices are at the "head" vs. "tail" of the halfedge).

[@felipekersting7065]

Hi professor Crane, amazing lectures. I have a question: Given that you defined manifold considering only the topology of the mesh, you didn't account for self-intersections (i.e. two faces intersecting each other but without an edge in the "middle"). I have seen manifold definitions that had the additional restriction of not having such intersections. This makes sense to me, but at the same time it would include geometry information. What is your take on this?

[@JAYMOAP]

Well done

[@abenedict85]

Do you expand on the signed incidence matrix's connection with discrete exterior calculus in another video in any more detail?

[@Iamfafafel]

Professor Crane, I have a question on the definition of the incidence matrix. If we think of the free vector space generated by the n-cells, then your incidence matrices correspond to the linear transformation induced by inclusion of an n-cell into an (n+1) -cell. However, there's an equally natural map from n-cells to (n-1) -cells induced from the boundary operator. The corresponding matrices are exactly the transpose of your incidence matrices. Is there any reason to take one over the other?

[@keenancrane]

Yes, the transpose is important. We will see later on that in discrete exterior calculus these matrices (the boundary and coboundary operators) correspond to a discrete notion of exterior derivative (for the dual and primal mesh, respectively).

[@forheuristiclifeksh7836]

14:11

[@KaliFissure]

Surface(cos(u/2)cos(v/2),cos(u/2)sin (v/2),sin(u)/2) 0>u>4π 0>v>2π. A single sided closed surface. The missing Klein. "Shirley's Surface"

[@hamedzahmati4530]

The best language to describe mathematics is mathematics itself.

[@daniellesman6836]

Quick question. If the adjacency list is the top dimensional simplex, why wouldn't the adjacency list for the tetrahedron be (0,1,2,3) given that the tetrahedron itself is a 3-simplex?

[@keenancrane]

In this example the mesh we want to describe is a "hollow tetrahedron," i.e., the four triangles that bound a tetrahedron, but no actual tetrahedron.

[@forheuristiclifeksh7836]

6:42

[@user-sv5vb1mj1q]

As a third party person where can I find homework assignments for this course to do it myself?

[@forheuristiclifeksh7836]

42:42

[@utof]

47:16 if tilapia can do it then so can you 😂😂