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

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Keenan Crane

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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

shiv

What a terrific service you have done by offering these classes on YouTube! 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!

scottpet

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.

ObsessiveClarity

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

erinzhang

Terrific! Even more terrific is that it is free and beneficial for so many people like me.

jiongwang

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

alfredoarroyog.

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

saturdaysequalsyouth

Great lecture! Thank you! absolute blessing to have found your course.

sakarapu

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

xanthirudha

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.

zeyonaut

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

bryanbischof

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

joshuaclavel

7:25 Can someone please tell me where was “link” introduced? Thanks.

maxwang

Oh hello! I have a twin. And that twin,

I S M E

columbusmyhw

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 $\rho \circ \eta$ so the halfedges are coming outwards from a vertex?

spghet

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?

felipekersting

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

abenedict

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"

KaliFissure

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?

daniellesman

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

ЗакировМарат-вщ

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