Has anyone ever come across this way of visualizing the convolution of two functions f and g: Look at the graph of the two-variable function f(x)g(y), and consider diagonal slices over the line x + y = k. The area of those slices represents (f * g)(k).

I doubt this is original, but I haven't seen it anywhere. I'm searching to see if there's a good source to cite here. If you've come across it, feel free to share links with me!

Yes, any binary operation can be visualized this way. Addition in the integers is a graph over a lattice, and in the continuum, kernels of integral operators like the fourier transform, green's functions, are seen as graphs over the plane rather than matrices. Just order the vars

It’s pretty. A great way to visualize a hike. I can imagine you doing a slice on a strange meandering path, and then “unrolling” it.

See 18:00 or so here:

I tried

Anecdotally, this is how I've always visualized it and I assumed it was the "typical" way to view it. I do have a couple of old diagrams I drew of it, e.g. at there's

Pretty sure my probability textbook showed this method for when you have two rvs X and Y, and you want to calculate a pdf of Z := X+Y by integrating along a diagonal. Super intuitive

I wonder whether this leads to some interesting generalisations of convolution for alternative couplings (e.g. optimal transport)

Je me le suis toujours représenté comme ça jpense pas que c’est nouveau