Lecture 2A: What is a 'Mesh?' (Discrete Differential Geometry)

[shiv]

0:21 Today: What is a "Mesh?"
1:54 Connection to Differential Geometry
3:02 Convex Set
6:00 Convex Hull
7:34 Example
9:25 Simplex
10:19 Linear Independence
11:38 Affine Independence
13:49 Simplex - Geometric Definition
18:28 Barycentric Coordinates - 1-Simplex
19:27 Barycentric Coordinates - k-Simplex
20:50 Simplex - Example
22:02 Simplicial Complex
23:04 Face of a Simplex
25:03 Simplicial Complex - Geometric Definition
27:00 Simplicial Complex - Example
28:50 Abstract Simplicial Complex
30:34 Abstract Simplicial Complex - Graphs
31:01 Abstract Simplicial Complex - Example
33:10 Application: Topological Data Analysis
38:04 Example: Material Characterization via Persistence
39:23 Persistent Homology - More Applications
41:19 Anatomy of Simplicial Complex
44:23 Vertices, Edges, and Faces
45:23 Oriented Simplicial Complex
45:38 Orientation - Visualized
46:52 Orientation of 1-Simplex
47:58 Orientation of 2-Simplex
49:17 Oriented k-simple
50:36 Oriented O-Simplex?
51:17 Orientation of 3-Simplex
52:25 Oriented Simplicial Complex
54:54 Relative Orientation

[SliversRebuilt]

I think this might be the most genuinely, *generally* useful mathematical knowledge condensed into the shortest amount of time that I have ever come across. It's just...overwhelming.

[erinzhang]

linear Independence 10:22
affine Independence 11:38
k-simplex 14:06
Barycentric Coordinate, convex combination 18:30
standard n-simplex, probability-simplex 20:50
face of a simplex 23:07
simplicial complex 25:03
abstract simplicial complex 28:51
persistent homology 34:39
closure 41:35
star 42:22
link 43:02
oriented 1-simplex 46:55
oriented 2-simplex 48:00
oriented k-simplex 49:20
oriented 0-simplex 50:39
oriented simplicial complex 52:26
relative orientation 55:00

[AkamiChannel]

He really did just go and say "grow some balls" without laughing. What a lad

[ahmaddarawshi]

Hello professor Keenan Crane, thank you very much for these pretty useful lectures. I have a question which is not a scientific one. What drawing tool did you use to draw the nice surfaces such as the two shaded surfaces at the upper- left corner of the slide 46:08 ?

[maxwang]

18:15 this is very important and I think it’s the essence of the concept of simplex. Nice!

[maxwang]

21:42 this concept is beautiful! I think its usefulness in probability is not just that all of the coordinates of any point in the standard simplex is between 0-1, but much more importantly, is its use in the whole probability concept due to the nice attribute that the sum of the coordinates of any point is always 1.

But I think I have a question. What’s the k for this standard k-simplex? 2?

[vinitsingh]

My question pertains to understanding k-simplicies from a geometric point of view.
In particular, what are the dimension"s" a simplex can exist in?

1. Geometrically is natural to think 1-simplex lies in R^2.
A) But can a 1-simplex exist in R^1?
I think the answer to this is probably a yes.
For example:
A linearly independent vector OR two affinely independent points (p1, p2) can exist as:
a. p1: {0} and p2: {1}. Here both points can be thought of as numbers lying on the number line.
b. p2: {0, 0} and {1, 1}. Here the points can be thought of a line y=x, where 0 <= x <= 1.

B) Can a 1-simplex exist in any space from R^1, R^2, ... to R^N? Or is it constrained to be only in R^1 and R^2?


2. Similarly it is natural to think 2-simplex lies in R^2.

A) But can a 2-simplex exist in R^3?
a. I think the answer to this is also probably a yes.
Eg: In R^2, (column vector notation)
[ 0 1 2
1 0 2 ]
describes a triangle or a 2-simplex.
Eg: In R^3,
[ 0 0 1
0 1 0
1 0 0 ]
also describes a triangle or a 2-simplex.

B) Can a 2-simplex exist in any space from R^2, R^3, ... to R^N? Or is it constrained to be only in R^2 and R^3?

[maxwang]

Hi professor, is DDG just another name of computational geometry? If not, what’s the difference? Thank you.

[sarvagyagupta]

This is a great lecture. Learning so much from these videos. I have one question about affine independence. I was thinking if it could also be called as "relative" or "positional" independence since it depends on the location of points instead of an absolute position (origin). Let me know.

[kksh]

亮点:
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06:03



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18:07
-在讨论单纯形时，使用重心坐标（barycentric

-单纯形复合体（simplicial
24:11



30:15



36:19



42:24



48:29



54:32




由sider.ai生成

[maxwang]

16:58 question: in the case of 1-simplex, where there are two points (vertices), from intuition I know if these two vertices coincide, they are not affinely independent. The problem is, how can we say so based on the definition of affine independence? It seems like the two coincidental points are also affinely independent because this is already a degraded case where there is only one vector to work with, and their coincidence does not make any difference in this regard.

[maxwang]

11:28 question: what’s the relationship between linear independence and orthogonality? Obviously the latter fits in the former but not vice versa. But is there anything deeper between the two?

[성이름-mec]

Dear Prof. Crane. I'm really appreciate to your lectures :) I have a question. In 9:20, you said that the convex hull of S={(1, 1, 1), (-1, -1, -1)} in R^3 is the (unit)-cube. But, I think that the set of line segment connecting two points (1, 1, 1) and (-1, -1, -1) is the smallest convex set containing S, so that the line segment is the convex hull of S. Where is the missing link?

[johnginos]

This is so high quality that for it to be free and for us to be listening to this for no price seems like stealing!

[AdrianBoyko]

At 21:30, I think the sum needs to run from 0 to n, not 1 to n.

[quanghungle]

Dear Prof. Crane, i still don't know what is the mesh. Is it simplicial surface? If yes, is it always is simplicial surface?

[JerrysMathematicsChannel]

For the slide about abstract simplicial complex, do you mean that a set of size k+1 is called an (abstract) k-simplex? (but not just simplex)

[Jason-sqcc]

A quick question, does 2-simplex contain vertices and line segments of the triangle (i.e. its faces)? It seems so from the convex hull definition. But we have simplices doesn't contain its faces when talking about star and closure. Thank you!

[Tannzrz]

Observations:

- A k-simplex has $2^{k + 1}$ faces. (23:04)

- The possible permutations of orientations of a k-simplex are $(k + 1)!$. (51:17)

- An orientation of a k-simplex $A$ may be considered to be negative the orientation of another k-simplex $B$ if the number of swaps of 0-simplices in the tuple to get from $A$ to $B$ is odd. (51:17)