This is one variant of the 3-d analog of the "shoelace formula" for area:

https://en.wikipedia.org/wiki/Shoelace_formula#Generalizatio...

The 2-d version is from the 18th century. I'd expect the 3-d version was probably known in the 19th century, but I haven't searched for a reference.

This technique should be straightforwardly adaptable to compute arbitrary moments, not just the centroid.

If you have a scalar-valued function that you can conveniently express as the divergence of any closed-form function, you can integrate it like this. And you can generalize beyond scalar-valued functions and beyond Euclidean space using the generalized Stokes’ theorem.

You can even do this in real life: if you want to integrate the electric current density through a surface (that is, measure the total current crossing the surface), you can integrate its anti-curl (is that a word?) around the boundary of that surface, which is what a current transformer or a clamp-on current meter does.

I bet there’s a hydraulic or pneumatic analog as well, but a nontrivial example isn’t immediately coming to mind.

Yes, this is essentially what the author derived (by means of calculus rather than geometric argument but the result is unsurprisingly the same). The 2D analog is easy to grok: to compute the area of a polygon, find the sum of the signed areas of each of the trapezoids formed by an edge and its projection on the x-axis. Turns out the negative areas of the right-to-left trapezoids cancel precisely out any excess area of the left-to-right trapezoids (or in the case of edges below the x-axis, add precisely the "missing" area).

Yep. Using the same kind of calculus ideas, I can also think about a vector field that has a Dirac mass of divergence at some point and zero divergence everywhere else. Then you get an expression that you can sum over faces to determine if a polyhedron contains some point. Again, for the right vector field there is a simple geometric interpretation, namely the solid angle that a face makes with respect to the point.

> calculating the volume of a prism-like polytope between it and its projection

There is the key insight that you don't need to explicitly compute this projection.

Better still, you calculate the signed volume of the the tetrahedron formed by the triangle and the origin (4th vertex).

I wonder if this could be reversed to give an intuitive “proof” of the divergence theorem.

Yes, the algorithm and its derivations are elementary and rather obvious for anyone decent at undergraduate level mathematics. But still, I am glad to see more people enjoying math!

Yes I remember doing something like that in 90s for a survey/map engineering cad application. After delaunay triangulation, calculating approximate voulume is easy. But this probably is a more general solution

reminds me of that 1994 paper that reinvented the trapezoidal rule

From a computational standpoint, Pick's theorem seems more useful to find the number of interior points via

    I = 2 (A - B + 1)

Where area would be calculated using the sum of signed areas of triangles.

    A = I + B/2.

Voxelising, most likely. The naive way's to rasterise the mesh into a 3D grid and count the cells inside, which really is rendering and then sampling the result. It costs resolution cubed instead of triangle count, and the answer's only ever as good as the grid.

> My belly says the naive formula is summing the triangle pyramid volumes to the origin with sign in orientation.

Yeah, that would also work but it's a slightly slower formula, sum(det(v1,v2,v3))/6. This one is summing sort of prism+pyramid shapes made by projecting each triangle to the yz plane.

I'd say what author means is the standard solution - which is equivalent to his on triangles but is on pixels ... except there's nothing naive about it and by using GPU parallelism and depth hardware it is lower cost on dense meshes.

>> Similar formulas exist for moments, to compute the inertia matrix for a rigid body.

Fun fact. The inertia for any rigid body can be represented by 4 point masses forming a tetrahedron. If you diagonalize the inertia matrix, the coordinates of the 4 point masses can be (x, y, -z) (-x,-y,-z) (x, -y, z) (-x, y, z) where x,y,z are easy to calculate (I wrote this all down ages ago). You can also represent any point on the rigid body by its barycentric coordinates relative to those points. I believe an impulse can be applied, by finding the barycentric coordinates of the point its applied and using those coordinates to distribute the impulse to the 4 masses.

This is all really cool with one huge exception. The 4 points become coplanar for large flat objects, which means the z-height is really small for a piece of sheet metal for example.

Closed is clearly important. Why does it have to be simple? It looks like it should handle disjoint components, interior holes, etc. just fine?

(2018)

Yeah, agreed. But words are cheap. It's one thing to say we don't need vector calculus, and another to develop that claim through the actual vector calculus, step by step.

It's an intensifier. As a native English speaker you should probably be aware that we eventually sand-blast the semantics off of words until they all becomes synonyms for "good", "bad", "very", or "um". (This is similar to what French does to phonemes, but unrelated.)

The author is just expressing amusement at the surprising simplicity of the resulting algorithm

    Adverb
        hilariously (comparative more hilariously, superlative most hilariously)

        1. In a hilarious manner; so as to amuse greatly.

The author was greatly amused how quick the resulting algorithm works.

I assume it's because the title contains "hilariously" and the author felt it necessary to state the content wasn't meant as humorous.

Yup!

I used to watch their vtuber videos on YouTube!

Sadly, there is no way to find out, f.ex. by a quick Google search.

For this particular case of 3 dimensions, I found

Newson, H. B. “On the Volume of a Polyhedron.” Annals of Mathematics, vol. 1, no. 1/4, 1899, pp. 108–10. JSTOR, https://doi.org/10.2307/1967277