I drew "girl, born in the Balkans in 3208 BCE", which I find super remarkable since this fits the place and time for a relatively small population of people who spoke Proto-Indo-European as their native language.
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).
Then the area needs to be the trapezoids projected to x axis is ab+bc-ad-dc. What is the correct way to keep track of the signs? I.e. what is the definition of "right-to-left"?
Polygon edges are usually considered directed and forming a loop, the edges of said quad being either ab, bc, cd, da (clockwise winding) or ad, dc, cb, ba (counterclockwise winding). They give the same result up to sign, as long as you’re consistent. In the first case ab and bc are left-to-right and cd and da right-to-left, and vice versa in the second case. (Vertical edges of course contribute zero area.) The signed area of a right-to-left trapezoid is automatically negative (if above the x-axis) because the delta-x is negative (eg. a_x - d_x < 0 for the da edge), so there’s no need to worry about whether to add or subtract each individual term.
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
OP's description vaguely triggered memories of the shoelace formula from a decade ago, but geometry was never my strong suit. All I remembered was positive/negative triangles (or was it trapezoids)? make magic happen for area calculation.
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!
Except this one 1) didn't try to name it after themselves, 2) explicitly told us it wasn't new, and linked to an example of prior work, and 3) isn't generally taught in high school.
see, when I say something reminds me of something else I don't mean a one-to-one equivalence at every aspect one can think of.
anyhoo, it's always a great skill to review the state-of-the-art BEFORE investing in a work/write-up/article - one of the very first things that a post-graduate program would teach you.
Nah. Being able to derive and prove easy results like this for yourself is both way faster and more reliable than trying to wade through the literature to find the equation you want.
"The following presents a fast algorithm for volume computation of a simple, closed, triangulated 3D mesh."
- we are presented something
"I would be (pleasantly) surprised if the algorithm is novel. Further research after posting reveals the paper Efficient Feature Extraction for 2D/3D Objects in Mesh Representation by Cha Zheng and Tsuhan Chen, which appears to describe the same algorithm, although the derivation is different. It was fun while it lasted!"
- there was an initial expectation, though slim, that it may be novel. then on the discovery that it wasn't, related fun time was insinuated to be over.
It’s someone’s blog post before a calculus exam. I think it’s fun to do this stuff. On your own site you can write whatever you want.
See, when I was a kid I found something I thought was great and inventive only to find that it was not only long trodden mathematics it was famous long trodden mathematics. I named it after myself for humorous value, the distinction between others who dreamed of finding some novel structure and me being solely that I did not know the famous results and conjectures in the space.
With my minimal steering Fable wrote 20k LoC porting JAX library to Mac with Apple Silicon. It currently passes 98.4% test from the original JAX test suite. The performance is about 10x compared to the CPU on the same machine (I'm testing on M5 Ultra).
Within OS consistency is much less of thing a thing than Web design conventions. Windows by itself has had several different UI frameworks over the years, so different "native" Windows programs can look completely different from each other.
> Within OS consistency is much less of thing a thing than Web design conventions.
Sorry, are you saying that two random web apps will typically share more UI consistency than two random Windows apps? Because, although I'm not currently familiar with Windows, I would be amazed if that were true.
I've never seen an Atom formatted podcast. NYTimes and WSJ each have a whole page devoted to their RSS feeds, I've never seen an Atom feed from either of them. It caught on sorta but didn't get the traction of what it was designed to replace. (Not saying this makes it Bad, btw.)
That's a good point. Podcasts are still (almost?) exclusively RSS 2.0. IDK if this is just momentum or Apple rules but I don't think I've ever seen an Atom podcast.
But many podcast clients actually still support Atom (probably using a feed library that supports various formats?) and basically all non-podcast feed readers support Atom.
I think it caught on well enough, platforms such as Wordpress still support it out of the box (I just checked my blog, it works).
I liked Atom's clean design but it felt it was mostly pushed by Google (I may be misremembering) and in the end the syndicated web faded into obscurity anyway.
RSS 2.0 is kinda an unspecified mess, and at least 15 year ago, if you wanted to be compatible with the majority of content you needed some weird heuristics to detect which interpretation of the spec a given feed was using (lol).
And Dave Winer was strongly against ever clarifying the spec, and that’s part of what led to Atom.
I don't often use AI to cleanup my texts, but when I do, I fully own the output. I make a conscious decision whether to leave in every AI suggestion or not. The final text _is_ what I want to say.
and the reason for that is that we passively understand more than we actively use, but when reading something we often can not distinguish our active and passive knowledge of an expression. so when you read a filtered text, it will sound fine because you are familiar with the expressions used, but you don't realize that some of those expressions are not actually in your active vocabulary.
