61–70 of 167 posts
http://ddg.cs.columbia.edu/SGP2014/LaplaceBeltrami.pdf
“The Swiss Army knife of geometric operators.”
I always thought that was cool since I usually think of diffusion in the context of fluid flow.
Earlier quoted context omitted.
It's the common way to solve a linear system in octave, matlab and julia. You have an invertible square matrix A, a vector b of the same dimension, and you want to find a vector x such that "A*x=b". Then you write "x=A\b", which is like "x=A^(-1)*b" but does not get to compute the full inverse matrix (which is useless).
Sorry, yes I know about the syntax. I'm just struggling with what exactly you would be plugging in. Like with respect to any of the example problems given, what would A and x and B be?
A is the five-point laplacian (a symmetric matrix with five non-zero diagonals), as implemented in the "step" function in the article, let's call it L
b is the datum h
If you set-up the sparse matrix L and the vector b (with the same code that performs the iterations in the article), then you can solve Poisson equation "L*f=h" by doing "f=L\h".
I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…
> you're not supposed to learn everything in college, you're supposed to learn how to learn
But to learn how to learn, you gotta learn some things to a somewhat decent degree. I think at some point you need to have these linalg/divgradcurl things down, if only briefly. You might forget any particular topic, but if you've indexed it you should be able to pick it up again, particularly in the modern learning environment.
Just imagine coding without access to StackOverflow.
A friend of mine broke a badminton racket during a match, and I was struck by how the sharply bent and twisted metal rim was transformed into a smooth, continuously double-curved surface by the racket weave. I looked closely at the balance of tension in the woven cord, thought of how it resembles Poisson's equation, and suddenly it all made sense. Edit - it looked something like this: https://thumbs.dreamstime.com/b/…
Hmmm, this looks more like a minimal surface, i.e. a solution to the minimal-surface equation[0], than a solution to Poisson's equation. Then again, both equations are of elliptic type. Some links for people who've never heard of minimal surfaces: https://en.wikipedia.org/wiki/Minimal_surface https://minimalsurfaces.blog/ (lots of illustrations) https://makmanx.github.io/math3435s18/talks/MSE.pdf (brief intro with hi…
I'm not saying the badminton racket follows exactly a (discrete) 2D Poisson equation. But it's certainly related enough to be more than a surface similarly.
The cords are under high tension, which means that any curvature along x (that is, dz^2/dx^2) will result in a net z-axis tension force unless balanced by an oppositely curved cord running in the y direction. Since it's in static equilibrium, there can be no unbalanced forces and so that must be the case. Therefore at each intersection, (d^2/dx^2 + d^2/dy^2)z = 0, which is Poisson's equation in 2D for z height being the function. Approximately, assuming equal tension in x and y, small z, and so on.
I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…
I stayed curious and learning. Better and better teaching methods became accessible thanks to the maturing internet. I started to understand ideas in engineering much better.
Alas, there is no way I can express this progress in a resume but learning and understanding satisfies my curiosity.
Earlier quoted context omitted.
Here's a concrete example. The first matrix in the post is f = [[1, 1, 1], [1, 1, 1], [1, 1, 1]]. In linear algebra, we would interpret this as a linear map. A true equation would be f([1, 2, 3]^T) = [6, 6, 6]^T (where I'm using ^T to mean "transpose to a column vector"). But here, the author means f(1, 2) = 1, i.e. the (1,2) coordinate of the matrix is 1.
Thank you! Yes, I agree, thank you for explaining that to me.
f(i, j) := d_i^T * M * d_j
The RHS is using classical matrix multiplication, and the function value will be the matrix' entry at column i, row j.
I could have used this 32(!) years ago when I was struggling in college. (This and 3b1b.) It amazes me just how many key topics were so inaccessible to the majority of the class at engineering school. I base this on observations from group study sessions and the hyper-aggressive test curves. I knew lots of people who never got the hang of div/grad/curl, or a Jacobians, or eignenvectors, or Z-transforms... These are k…
I feel the same. How can it be that I went to a world famous institution providing 2-to-1 student-teacher ratios, but I still think the best explanations are these modern internet explanations? I guess the best explanations just bubble up in the modern environment. > you're not supposed to learn everything in college, you're supposed to learn how to learn But to learn how to learn, you gotta learn some things to a so…