I spent a couple hours today working out a Newton method for finding the closest rotation to a given matrix A, i.e. min ‖R − A‖² over R ∈ SO(3). Then I found out that Kugelstadt et al. already figured it out: https://animation.rwth-aachen.de/publication/0561/
Oh well. Glad to verify that I got the same result, but I like my derivation better; it's much shorter :)
@johncarlosbaez Same thing that happens every time you ask Newton's method to find the minimum of a constant function: the gradient and Hessian are both zero and you can't go any further.
"I don't know a well-established name for this, and a quick google search failed to reveal anything of releavance, so I will call this method /shape matching/. If you know some resources on this, I would love to know them, since I had to derive all the equations myself :)"
An interesting counterargument to the recent popularizations of geometric algebra as the ideal language for doing geometry in physics, computer graphics, and so on.
I haven't actually worked with GA enough to have an opinion either way, but their basic argument seems compelling:
"1. The wedge product and the rest of Exterior Algebra is 100% amazing, S-tier stuff, definitely something everybody who uses mathematics should know about [...]
2. The geometric product, though, is kinda weird and bad.
3. A lot of other parts of GA are working around the fact that the geometric product is weird and bad.
4. The “better” version of Geometric Algebra [...] which we are... slowly unearthing... will be mostly the same as GA but it will discard the geometric product as a basic operation, to everyone’s benefit. [...]"
"The “better” version of Geometric Algebra [...] which we are... slowly unearthing... will be mostly the same as GA but it will discard the geometric product as a basic operation, to everyone’s benefit. [...]"
Is this "better" version the one that uses the wedge product instead of the geometric (= Clifford algebra product)? If so, it was introduced by Grassmann in 1844 and most decent mathematicians know about it.
A few months ago I came across a paper about how rational numbers can be represented as LEFT-infinite digit sequences without a decimal point.
For example, in base ten,
−1 = ...9999
because adding 1 to it gives ...0000. Similarly,
1/3 = ...6667
because multiplying it by 3 gives ...0001. It's a fun exercise to verify that 1/3 − 1 = −2/3 indeed holds in this system.
I can't find this paper any more. Does anyone know what it might be?
What is the area of a 3D polygon p₁p₂...pₙp₁? If the polygon is non-planar, this is not well-defined: if you split up the polygon into triangles pᵢpⱼpₖ and add up their areas ½‖(pⱼ−pᵢ)×(pₖ×pᵢ)‖, the result depends on the choice of triangulation. However, if you forget to take norms and just add up the vectors ½(pⱼ−pᵢ)×(pₖ×pᵢ) instead, you always get the same result: the "vector area" of the polygon! So if you're dealing with non-planar polygons, or non-planar curves in general, it makes sense to think of area as a vector rather than a scalar.
Here's a rather niche derivation in computer graphics theory that leads to a surprising and generally interesting fact about spheres. Please tell me if this has been noticed before.
In path tracing of Lambertian (i.e. diffuse) surfaces, we need to perform cosine-weighted sampling of a hemisphere, i.e. choose a point in {(x,y,z) : x² + y² + z² = 1, z ≥ 0} with probability density proportional to z. Depending on how you simplify the equations arising from inversion sampling, you get one of two possible methods:
A.1. Sample a point (x,y) uniformly from the unit disk.
A.2. Project it up to the hemisphere by choosing z = √(1 − x² − y²).
B.1. Sample a point p = (x,y,z) uniformly from the unit sphere.
B.2. Take the unit vector halfway between it and the north pole n = (0,0,1), i.e. (p + n)/‖p + n‖.
It's very strange to me that a routine calculation can be simplified in two different ways to yield two very different geometric interpretations -- in particular, two interpretations that by themselves cannot easily be related to reach other.
In fact, putting them together yields the (non-obvious!) fact that the following bijection between a sphere and a disk is area-preserving, up to a constant factor:
C.1. Given a point on a sphere, take the point halfway between it and the north pole.
C.2. Project it down to the xy-plane to get a point on the unit disk.
Since there is only one such equal-area map with rotational symmetry, we must have just reinvented the Lambert azimuthal equal-area map projection!
Cool rainbow effect in the reflection in a metro train window.
What could be the reason for this? I assume it's an LCD screen so the light coming out of it is polarized, but I'm not sure what's going on at the glass interface to cause the colour shift.
Which actually is a good set of settings (turns out most do not need to run off to about:config just the regular settings) but I learnt about not doing the windows specific things or the jit things and I still use quad9 for the DNS.
Mostly I pick things up as I go but the new rust webrender engine was just a hidden gem.
If I want spherical geometry but I only have Euclidean geometry, I can get spherical geometry by restricting myself to the unit sphere.
If I want Euclidean geometry but I only have hyperbolic geometry, can I still get Euclidean geometry by restricting myself to some lower-dimensional submanifold?
I think I've found the answer: Coxeter writes¹ that "the intrinsic geometry of the horosphere² is Euclidean", and that this fact was known to Lobachevsky from the beginning.