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.
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.
Comments
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.
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.