8 ms·

This is one of those when you go "Huh, this is amazing!" or "Huh, I thought this trick was really well known!" depending on your background ;) Here's a similar

This is one of those when you go "Huh, this is amazing!" or "Huh, I thought this trick was really well known!" depending on your background ;) Here's a similar impl from 1980 written in Fortran that also computes other properties like centroid: https://calgo.acm.org/550.zip https://calgo.acm.org/550.zip Algorithm 550: Solid Polyhedron Measures A. M. Messner and G. Q. Taylor ACM Trans. Math. Softw., 6(1), Mar 1980, pp.121--130 Keywords: polyhedron, graphics, numerical integration Language: Fortran 66/77; Shar Index: Z; Gams: P File size: 19.1 KB; But Messner published it first in: A. M. Messner, "A surface Integral method for computer calculation of mass properties", Paper No. 852, 29TH ANNUAL CONF. OF THE SOCIETY OF AERONAUTICAL WEIGHT ENGINEERS, Washington, D.C., May 1970. I think

jacobolus 19d ago
This is one variant of the 3-d analog of the "shoelace formula" for area: https://en.wikipedia.org/wiki/Shoelace_formula#Generalization 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.
amluto 19d ago
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.
- ted_dunning 19d ago
  The hydraulic analog is that you can weigh a volume of water (which is the same as computing its volume) by adding up the forces on the surface surrounding the water. This looks like it requires a dot product with the normal vector for each triangle, but you can expand it into the same form as the article.
georgeburdell 19d ago
As an aside, this seems related to the Holographic Principle? It states that the content of a volume is encoded in its surface https://en.wikipedia.org/wiki/Holographic_principle https://en.wikipedia.org/wiki/Holographic_principle

jacobolus 19d ago

This is one variant of the 3-d analog of the "shoelace formula" for area: https://en.wikipedia.org/wiki/Shoelace_formula#Generalization 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.

amluto 19d ago

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.

ted_dunning 19d ago
The hydraulic analog is that you can weigh a volume of water (which is the same as computing its volume) by adding up the forces on the surface surrounding the water. This looks like it requires a dot product with the normal vector for each triangle, but you can expand it into the same form as the article.

georgeburdell 19d ago

As an aside, this seems related to the Holographic Principle? It states that the content of a volume is encoded in its surface https://en.wikipedia.org/wiki/Holographic_principle https://en.wikipedia.org/wiki/Holographic_principle