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Spherical Harmonics

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Re: Spherical Harmonics

#51

Post by gus_massa »

esperent wrote: post #14
gus_massa wrote: post #2

They 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.

I'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 Suggestion to OP: this would be much more useful if you add a button to stop the rotation.

Now 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.

Re: Spherical Harmonics

#53

Post by rzzzt »

Scene_Cast2 wrote: post #23

If 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 har…

I have "Spherical Harmonic Lighting: The Gritty Details" in my PDF stash: https://www.semanticscholar.org/paper/Spherical-Harmonic-Lig...

Re: Spherical Harmonics

#54

Post by pletnes »

dustingetz wrote: post #42
Scene_Cast2 wrote: post #23

If 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 har…

in QM the spherical harmonics are more of a basis space for electronic state and not the actual electronic state, right? So does that mean there are other ways to think about electron configurations that satisfy Shrodinger etc?

Spherical harmonics are exact solutions to electronic states for hydrogen-«style» atoms.

Re: Spherical Harmonics

#56

Post by lloda »

Scene_Cast2 wrote: post #23

If 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 har…

The vector version of these is used in antenna theory to represent and transform the radiation of finite sources, like when measuring antenna patterns.

Re: Spherical Harmonics

#57

Post by ajross »

momoschili wrote: post #30
vecter wrote: post #17

Are these related to (or exactly) the distribution of electron orbits?

not quite as they are missing the radial dependence

It'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.

Re: Spherical Harmonics

#58

Post by ajross »

ghostpepper wrote: post #7

Anyone know a good explanation of what spherical harmonics are?

It's the list of "waves" that can propagate around the surface of a sphere without interfering with each other. They are self-reinforcing modes. So you can represent any function of values on the surface as a combination of these, the same way you do with e.g. FFT coefficients in a JPEG file.

And it turns out that this "self-reinforcing" property is critically important for quantum mechanics, as each of the functions defines a different "state" from the perspective of a "particle". So we can do a lot of good physics work[1] by pretending[2] that all electrons exist in one of these states.

[1] Like, y'know, explaining chemistry.

[2] The details are always harder, because the electrons interact with other ways than just flying around the sphere, so the math isn't tractable in an absolute sense. But as long as you pretend that this is mostly right you can treat the remainder as just "fixups" in a giant framework called perturbation theory.

Re: Spherical Harmonics

#59

Post by dwallin »

Scene_Cast2 wrote: post #23

If 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 har…

I do want to point out, gaussian splats don't really offer anything in particular for realtime relighting, if anything it adds additional challenges. Under the hood, most implementations leverage spherical harmonics for baked-in lighting.

Did you by chance mean anisotropic spherical gaussians? ASG is a new-ish technique often used to model specular lighting but is unrelated to gaussian splatting.

Re: Spherical Harmonics

#60

Post by evanb »

pletnes wrote: post #54
dustingetz wrote: post #42

Earlier quoted context omitted.

in QM the spherical harmonics are more of a basis space for electronic state and not the actual electronic state, right? So does that mean there are other ways to think about electron configurations that satisfy Shrodinger etc?

Spherical harmonics are exact solutions to electronic states for hydrogen-«style» atoms.

... for the angular dependence; there is also radial dependence.
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