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Consider a 3D spherical Gaussian distribution function that depends on radius only,

$$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$

where $R_\mu$ is the radial offset of the distribution and $\sigma$ is the usual variance parameter. $N$ is the normalization constant. To help visualize this function, see the plot below which shows $f(r)$ plotted in one quadrant of the x-y plane, using $R_\mu = 10$ and $\sigma^2 =0.5$.

A plot of f(r) in the xy plane

What is the projection of the distribution function $f(r)$ on to the x-y plane, $f(x,y)$?

Note that it is obviously not as simple as what I have plotted above which is simply $f(x,y,0)$, the slice through the x-y plane. I'm asking for the projection of this distribution function in to the x-y plane, $f(x,y)$.

A simpler example

To try and solve this problem I have tried to evaluate the simpler case of a uniform distribution on the surface of a sphere with radius $R_\mu$, i.e. the limit of $f(r)$ above as $\sigma$ goes to zero. I believe this sample distribution would be defined as

$$ f_s(r) = \begin{cases} \frac{1}{N}, & \text{if }r = R_\mu \\ 0, & \text{if }r \neq R_\mu \end{cases} $$

The integral over the whole surface must equal 1. $$ \int_\Omega f_s(r) d\mathbf{a} = \int_0^{\pi} \int_0^{2\pi} f_s(r) r^2 Sin(\phi) d\theta d\phi = 1 $$

To find the projection of this much simpler function on to the xy plane, I express it in terms of an infinitesimal area element. In spherical coordinates the area element is $ d\mathbf{a} = r^2 Sin(\phi)$. But $r^2 = R_\mu$, and $Sin(\phi)$ can be re-written in terms of the x-y coordinates since $Sin (\phi ) = \frac{z}{R_\mu}$. e.g. from the equation for a sphere

$$x^2 + y^2 + z^2 = R_\mu ^2$$

$$\therefore \frac{z^2}{R_\mu^2} = 1- \frac{x^2+y^2}{R_\mu^2}$$

$$\therefore Sin(\phi) = \sqrt{1- \frac{x^2+y^2}{R_\mu^2}}$$

Hence, I find the projection of this distribution in to the x-y plane is:

$$ f_s(x,y) = \begin{cases} \frac{1}{N} R_\mu^2 \sqrt{1- \frac{x^2+y^2}{R_\mu^2}}, & \text{if }x^2 + y^2 < R_\mu^2 \\ 0, & \text{if }x^2 + y^2 > R_\mu^2 \end{cases} $$

Is this correct? If so, how do I do the more complex case above for a spherical Gaussian in 3D projected on to the xy plane? Do I need to express $f(r)$ in Cartesian coords and integrate the elemental volume over $z$? If I use this logic can I write it in 3D as

$$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$

$$ dV = r^2 Sin(\phi) dr d\theta d\phi$$

$$ f(x,y) = \int (x^2 + y^2 +z^2) \frac{z}{\sqrt{x^2 + y^2 +z^2}} e^{-(\frac{r-R_\mu}{\sigma})^2} dz $$

I don't think I've got this last step correct because it doesn't look right when I evaluate it numerically and plot it. I suspect it needs to visually converge on the previous result as $\sigma$ -> 0.

Any help you can give is appreciated. This is for a research project where I plan to use these distribution functions in another expression. I expect my question is a very specific case of someone's more general question.

EDIT: Extra graph inspired by Joriki's answer below

See Joriki's answer below. Its easy to see why it makes sense by generating 1000 random unit vectors and plotting their (x-y) components on a scatter plot. You get something like this. Its clear that my original result has the wrong behaviour.

1000 random unit vectors projected on to x-y plane

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2 Answers 2

up vote 1 down vote accepted

No, that's not correct. You can see that it can't be correct because it's zero on the boundary and maximal in the centre, whereas it should be minimal at the centre and maximal at the boundary where the slope of the surface becomes infinite and thus concentrates a lot of mass.

There are two errors in fact. One is that you seem to have mixed different conventions for the spherical coordinates – if $z$ is proportional to the sine of the polar angle, the volume element is proportional to the cosine, and vice versa; you've got the sine for both.

The other problem is that you've simply dropped $\mathrm d\phi\mathrm d\theta$ whereas you need to transform it to $\mathrm dx\mathrm dy$. Going with the $z\propto\cos\phi$ convention, we have

$$ \begin{align} x&=r\sin\phi\cos\theta\;,\\ y&=r\sin\phi\sin\theta \end{align} $$

and thus

$$ \frac{\partial(x,y)}{\partial(\phi,\theta)}=(r\cos\phi\cos\theta\,r\sin\phi\cos\theta+r\sin\phi\sin\theta\,r\cos\phi\sin\theta)=r^2\sin\phi\cos\phi\;, $$

so the $\sin\phi$ in $\sin\phi\,\mathrm d\phi\,\mathrm d\theta$ is divided out and you get a factor $1/\cos\phi$ instead, which diverges as you approach the boundary, as it should.

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Thanks Joriki, I found your answer very helpful, it appears I had forgotten way more calculus than I thought. Its easy to see that this result makes physical sense. I just generated 1000 random unit vectors and plotted their x-y components as data points in an x-y scatter plot. It is very easy to see that the appropriate distribution function needs to diverge near the boundary. It becomes immediately clear (as @joriki pointed out) that my first answer can't be correct. I think I understand how to derive 1/cos(phi) and will post my solution attempt to the 3D case soon. –  MatthewCNG Oct 29 '12 at 6:09
    
@Matthew: You're welcome! –  joriki Oct 29 '12 at 6:19

Answer for the 3D case based on @Joriki's response. The 3D distribution function of interest is

$$f(r)=\frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$

The integral over all space in spherical coordinates must be 1.

$$1= \iiint_V f(r) r^2 Sin(\phi) d\phi d\theta dr$$

The Jacobian for the transformation back to cartesian is $\frac{\partial (r,\theta,\phi)}{\partial (x,y,z)} = \frac{1}{r^2 Sin(\phi)}$, hence

$$\iiint_V f(r) r^2 Sin(\phi) d\phi d\theta dr = \iiint_V f(r) r^2 Sin(\phi) \frac{\partial (r,\theta,\phi)}{\partial (x,y,z)} dz dy dx$$ $$= \iiint_V \frac{1}{N} e^{-(\frac{\sqrt{x^2+y^2+z^2}-R_\mu}{\sigma})^2} dz dy dx$$

Therefore we can pull out the integral over dz as this gives us $f(x,y)$.

$$\therefore f(x,y) = \frac{1}{N}\int_{-\infty}^{\infty} e^{-(\frac{\sqrt{x^2+y^2+z^2}-R_\mu}{\sigma})^2} dz $$

We can see this is correct by plotting this function for several converging values of $\sigma$. We expect the function to converge on to $\frac{1}{Cos(\phi)}$. For the plot below I used $\sigma$ values of 2,1,0.75,0.5. $R_\mu = 10$

Sigma goes to zero showing f(x,y) -> 1/Cos(phi)

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Seems like a roundabout way of getting from $f(r)$ to $f(\sqrt{x^2+y^2+z^2})$... :) –  Rahul Oct 31 '12 at 8:05
    
Yes, you are right that it could really have been done more directly. In hindsight the 3D case is very trivial. The reason I found it difficult was because I was trying to do the the surface distribution fs(r) (isotropic unit vectors, or surface of a sphere) first, and that derivation was actually more tricky. –  MatthewCNG Oct 31 '12 at 23:40

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