On the other hand, if you want to compute the area of a polygon that have vertices at lattice points, you can count the number of interior points I, the number of boundary points B. Then the area A is
one of my favorite results. It does not generalize as nicely to higher dimensions unfortunately.
If like the post you want the volume of a polyhedron you can use the three dimensional analogue of the shoelace formula (essentially equivalent).
Let Va, Vb and Vc be the vertices of a triangle ∆ of a triangulation of the surface. Then the volume V is the sum over all such triangles of the signed volumes
V_∆ = 1/6 Va ^ Vb ^ Vc.
That's the beauty of signed areas and volumes, determinants and exterior algebra.
To understand why this is so there's this beautiful short video
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
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.
Isn't the same as just taking every triangle from the mesh, calculating the volume of a prism-like polytope between it and its projection on one the planes, and then taking it with a + sign if its projection is oriented in one direction, and with a - sign if it's oriented in another? This kind of formula works based on the basic geometry.
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.
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
> For a ballpark number, if volume needs to be calculated every frame in a high-performance 60 frames per second application, without the aid of a GPU, only using the CPU capabilities of a $35 Raspberry Pi, around 30 million triangles could be measured every frame.
If knowing the volume of a mesh is important, we could pre-calculate it (even using this exact technique) and store it as an attribute on the object. Lots of things in game dev that are modeled as an integral over three+ dimensions tend to work better as a baked setup rather than real time. We kickstarted an entire AI industry trying to chase real time lighting.
My belly says the naive formula is summing the triangle pyramid volumes to the origin with sign in orientation. It looks like that's what they derived. Which is a generalization of 2d polygon area calculated by summing triangle areas for each edge, I was taught this in a math camp where we calculated map polygon areas on gis data. I remember math knowledge being hard to get pre AI era but I didn't remember it being this hard.
No idea what the author means by "which are equivalent to rendering the mesh and then sampling the render".
> 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.
Don't really need vector calculus for this. Geometric intuition is sufficient. It is simply the summation of signed volumes of triangular columns/prisms parallel to the X axis.
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.
There's a really elegant solution using Geometric Algebra, that to this day is one of the most satisfying things I've ever learnt.
Steven de Keninck outlines it in his 2019 Siggraph talk [1].
https://en.wikipedia.org/wiki/Pick's_theorem
one of my favorite results. It does not generalize as nicely to higher dimensions unfortunately.
If like the post you want the volume of a polyhedron you can use the three dimensional analogue of the shoelace formula (essentially equivalent).
Let Va, Vb and Vc be the vertices of a triangle ∆ of a triangulation of the surface. Then the volume V is the sum over all such triangles of the signed volumes
That's the beauty of signed areas and volumes, determinants and exterior algebra.To understand why this is so there's this beautiful short video
https://youtu.be/Sv7VseMsOQc
Here's a similar impl from 1980 written in Fortran that also computes other properties like centroid: 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
There is the key insight that you don't need to explicitly compute this projection.
If knowing the volume of a mesh is important, we could pre-calculate it (even using this exact technique) and store it as an attribute on the object. Lots of things in game dev that are modeled as an integral over three+ dimensions tend to work better as a baked setup rather than real time. We kickstarted an entire AI industry trying to chase real time lighting.
No idea what the author means by "which are equivalent to rendering the mesh and then sampling the render".
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.
Visualization: https://jsfiddle.net/L7r1hwca/
I don't know what they could possibly mean by the naïve algorithms with rendering and sampling (???).
[1] https://youtu.be/tX4H_ctggYo?t=4795
I must be missing something here, inside joke or something in the title?