8 ms·
Spherical Harmonics
- gus_massa 2y agoThey look too big. I expected all the l=1 to be like a cone near (0,0,0). And I expected one of them to be vertical instead of horizontal.
- esperent 2y agoI'm not sure about the size but I think the shapes are correct. It's just very hard to examine them when they're rotating at such high speed. Compare them to the image on this page: https://en.m.wikipedia.org/wiki/Spherical_harmonics https://en.m.wikipedia.org/wiki/Spherical_harmonics Suggestion to OP: this would be much more useful if you add a button to stop the rotation.
- dtgriscom 2y ago... or a slider to control the rotation position or speed.
- gus_massa 2y agoNow it looks better close to the 0. Did the OP change the implementation? ALso, I noticed that there is a real/imaginary/complex menu. I was looking at the real part of the complex versions, but it's necesary to look at the complete complex version to understand them. Note that the graphic in Wikipedia is showing the real versions. In the real versions you for l=1 the functions in directions x, y and z[m=0], and all of them look identical except for the direction. But in the complex versions you have (x+iy)/sqrt(2)[m=1], (x-iy)/sqrt(2)[m=-1] and z[m=0], and when you show only the real part of them the first two are smaller.
- liontwist 2y agowhy does the page scroll when I drag a slider?
- raffihotter 2y agoHmm, looking into this.
- CamperBob2 2y agoAny updates on the ultrasound brain imager project? I remember reading about that a couple of years ago but didn't see any follow-ups. It sounded extremely nifty.
- raffihotter 2y agoim still working on ultrasound stuff, but specifically for BCI! https://news.ycombinator.com/item?id=42021450 https://news.ycombinator.com/item?id=42021450
- raffihotter 2y agofixed! sorry about that, and thanks for the feedback
- liontwist 2y agoThanks for sharing and making a fix.
- ghostpepper 2y agoAnyone know a good explanation of what spherical harmonics are?
- NotYourLawyer 2y agoSolutions to a certain differential equation that comes up in quantum mechanics and elsewhere.
- lizmutton 2y agoI think of it as a good basis for functions on a perfectly spherical surface. Going down in levels of "l", you describe more and more details in terms of angular scale. Thus, it's widely used in earth science and astrophysics, and anything that involves spherical symmetry (like a Hydrogen atom) -- in reality, nothing is a perfect sphere, but that's a very good approximation.
- defrost 2y agoComing at them from practical applications is one approach. I've used spherical harmonics to model earth centric "surfaces" and fields - magnetics and gravity, etc. You might think of them as a stacked sine and cosine waves (like a fourier transform breaking a continuous function into sin and cosine components) on a directional vector radiating outwards from the centre point of sphere. https://geomag.bgs.ac.uk/research/modelling/IGRF.html https://geomag.bgs.ac.uk/research/modelling/IGRF.html https://en.wikipedia.org/wiki/International_Geomagnetic_Reference_Field https://en.wikipedia.org/wiki/International_Geomagnetic_Refe... https://en.wikipedia.org/wiki/World_Magnetic_Model https://en.wikipedia.org/wiki/World_Magnetic_Model
- deleted 2y ago[deleted]
- ajkjk 2y agoJust for fun... They are relatively easy to understand if you already understand Fourier transforms. In a Fourier transform you can write some (suitably well-behaved) function f(x) as a sum of a bunch of sinusoids of different frequencies: f(x) = a_0 + a_1 cos(x) + a_2 cos(2x) + ... + b_1 sin(x) + b_2 sin(2x) + ... Or more generally a sum over all real values, f(x) = ∫ a(k) cos(kx) + b(k) sin(kx) dk, since signals can have fractional frequencies. And in many cases the two sides are the same. Many operations in math and in ph operate on functions in such a way that they can distribute over their behavior on different frequencies, which is why Fourier transforms are really useful. For instance we hear different frequencies in different ways so if you Fourier-transform an audio waveform you can turn some frequencies up and others down (or drop them entirely, which is more-or-less what mp3 compression is). This also works in 2d or 3d, where now you have frequencies in all three directions: f(x,y,z) = f(0) + a_(100) cos(x) + a_(110) cos(x) cos(y) + a_(111) cos(x) cos(y) cos(z) + (all the sine terms and negative frequencies and everything else) and this is useful in all kinds of ways also, e.g. taking Fourier transforms of a 2d image and then dropping the high frequency components that are hard to see is basically what JPEG compression is. As before it helps a lot that you can interact with the different frequency components separately. But. Sometimes what you want is not the frequency in linear space (e.g. the frequency in x or y) but the frequency in angular coordinates: to ask "how many times does this variable change as you go around a circle in the (xy) plane?" Which is to say, you want to know the Fourier component in term like cos(ϕ), cos(2ϕ), sin(ϕ), etc. That looks like f(x,y) = a_1 cos(ϕ) + a_2 cos(2ϕ) + b_1 sin(ϕ) + ... Which is what we would call a "circular harmonic" (with ϕ=ϕ(x,y)=arctan(y/x)), each coefficient a_i is a function of (r). Unlike the linear Fourier transform, there can be no "fractional" frequencies---since ϕ=0 and ϕ=2pi are the same point, all the circular frequencies have to be integers. When you try to do this in 3d using two angular coordinates ϕ and θ, it gets a lot funkier. Now you can't really write it as a simple series of the two frequencies separately; they kinda "step on each other", because a rotation in (xy) can be written as a sum of rotations in (yz) and (zx). But you can do it in terms of a different, slightly stranger series. One variable L=0,1,2,3... will describe the "total" frequency on any plane, and another variable m=-L,-L+1,...0,...L-1,L describes how many rotations happen in your favorite choice of (xy) plane. m is allowed to range from -L to L because we have already said that L is the total frequency on any axis, so we just have to say whether they're happening in our chosen axis or not. So the series becomes f(x,y,z) = a_(0, 0) + a_(1,-1) Y_(1,-1) + a_(1,0) Y_(1,0) + a_(1,1) Y_(1,1) + a_(2,-2) Y_(2,-2) + ... = ∑ a_(L,m) Y_(L,m) (θ,ϕ) The functions Y_(L,m) (θ,ϕ) are the "spherical harmonics". They serve the role of sin(ωx) and cos(ωx) when you Fourier-expand a function in terms of spherical coordinates ϕ and θ. There are lots of reasons that that's useful, but the case that is most well-known is that the state of an electron wave function in an atom can be indexed in terms of which spherical harmonic it's in, and only two electrons are allowed to be in each one (one spin up and one spin down, for much-more-bizarre reasons). So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.
- lizmutton 2y agoNeat!! thanks for sharing
- JeremyHerrman 2y agoFor those of you curious about WHY these shapes look like the do (e.g. "why does l=0, m=0 have a donut in the middle of two lobes?"), this video from Münster University finally gave me an intuitive understanding of how these shapes arise. https://youtu.be/Opufc3onVow https://youtu.be/Opufc3onVow
- JeremyHerrman 2y agotypo here, l=2,m=0 is the orbital with the donut
- vecter 2y agoAre these related to (or exactly) the distribution of electron orbits?
- aeve890 2y agoYes. These are solutions of the Schrodinger equation for the electron in the hydrogen atom.
- drdeca 2y agoAren’t the spherical harmonics functions with domain S^2, the sphere? I think the solutions to the (time-independent) Schrödinger equation for an electron in a hydrogen atom are given by like, a product of a function of distance from the center with one of the spherical harmonics, or something like that?
- momoschili 2y agoyou are correct. The Schrödinger equation for the hydrogen atoms in spherical coordinates demonstrates separability which allows you to separate the radial and angular coordinates. The radial term, which is most interesting due to the 1/r potential is typically a Laguerre polynomial. The angular term is 'free' from any potential is typically a spherical harmonic. The spherical harmonics in general are typically derived as part of the solution to the Laplace equation in spherical coordinates. A bit of a semantic point (though perhaps the distinction is important) though, since the Laplace equation's angular dependence is identical to that of the Schrödinger equation for the hydrogen atom.
- momoschili 2y agonot quite as they are missing the radial dependence
- ajross 2y agoIt's actually more confusing IMHO, because these graphs overload the radial dimension to show probability as "distance from the origin". You have to multiply that by the radial function to get an actual probability distribution, which kinda/sorta looks like these pictures but not really. Really the harmonics are best understood as something like "wave height on the surface of a sphere". They tell you how the electrons (or whatever) are going to distribute themselves radially, not where they're going in 3D space. Also FWIW: the much harder thing to grok here (at least it was for me), and that no one tries to tackle, is why the "l" number corresponds directly to angular momentum. In particular "l==0" doesn't look like there's any rotation going on at all.
- Scene_Cast2 2y agoIf anyone is curious about applications - these can be used to approximate low-frequency components of a point's surroundings. They were used in Halo 3 to do real-time HDRI lighting and shadowing (see "Lighting and Material of Halo 3" from Siggraph 2008). After the success of this method, there was a fairly long stretch of researchers looking for a better orthonormal basis (such as 2D Haar wavelents, as spherical harmonics is basically a Fourier Transform on a spherical basis). I think the pinnacle of this direction was Anisotropic Spherical Gaussians from 2013. These days though, you'd at least use a neural net to learn a basis (or use a neural net to learn something else entirely). And of course, Gaussian Splats are the technique du jour for realtime relighting.
- deleted 2y ago[deleted]
- xeonmc 2y agoI wonder if Cartesian-basis multipole expansion could get the best of both worlds of GS and SH, as the former basis captures anisotropy but not detail while the latter captures detail but not anisotropy, whereas Cartesian multipole expansion naturally captures both right from the low orders, and is much easier to align to game worlds. (to be precise, both can be captured by either if you include enough orders, what I mean is mainly how the information distribution scales with respect to each attribute) Also, the age-old physics question: what is the minimum order of spherical harmonics required to approximate a cow?
- RossBencina 2y agoAnother application is Ambisonic sound spatialisation formats, where a finite set of signals corresponding to spherical harmonics are used to encode spatial sound fields. https://en.wikipedia.org/wiki/Ambisonics https://en.wikipedia.org/wiki/Ambisonics
- ziotom78 2y agoThey are used also to characterize the statistical properties of fields over the sphere. A notable example is the pattern of hot/cold spots in the Cosmic Microwave Background Radiation (CMBR, [1]). They are distributed stochastically, and the best way to fit cosmological models against the measurements is to decompose the temperature/polarization fields into spherical harmonics and compute the power spectrum associated with each ℓ (which plays the role of a “spatial frequency” over the sky sphere). [1] https://en.wikipedia.org/wiki/Cosmic_microwave_background https://en.wikipedia.org/wiki/Cosmic_microwave_background
- jms55 2y agoObligatory useful SH paper for 3d rendering: http://www.ppsloan.org/publications/StupidSH36.pdf http://www.ppsloan.org/publications/StupidSH36.pdf Also lots of other cool research around SH in rendering, e.g. the recent ZH3 paper.
- raffihotter 2y agoThanks for sharing this! Linked it on the website.
- ykonstant 2y agoStupid sexy harmonics.
- tomxor 2y agoShameless plug, in 140 bytes https://www.dwitter.net/h/wikimedia https://www.dwitter.net/h/wikimedia
- setopt 2y agoSee also the “cubic harmonics”, which is an equivalent basis to spherical harmonics but they are real instead of complex, and also more natural to use in cubic crystals due to their symmetries. I have also seen “triangular harmonics”, “zonal harmonics”, etc. in use in other materials.
- openrisk 2y agoNext project: spin harmonics https://en.m.wikipedia.org/wiki/Spinor_spherical_harmonics https://en.m.wikipedia.org/wiki/Spinor_spherical_harmonics
- pletnes 2y agoIf you need to work on numerical computation with spherical harmonics, I’ve used this library with some success. https://github.com/SHTOOLS/SHTOOLS https://github.com/SHTOOLS/SHTOOLS
- dagss 2y agoTL;DR about spherical harmonics: It is what you use instead of Fourier transforms if what you transform is on the surface of a sphere. My experience is from cosmology (CMB) where they are heavily used just like Fourier transforms, I think they are also used in meteorology.