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Random Point from a 2D Donut Distribution

A donut distribution (for lack of a better term) is a random distribution of points in a 2-dimensional plane, forming a donut-like shape. The distribution is defined by two parameters: the radius r and spread s, in which the distance to the origin follows a normal (Gaussian) distribution around r, with a standard deviation s. The angular distribution is uniform in the range [0,2π).

The challenge

Given a radius r and spread s, your code should yield the Cartesian ((x,y)) coordinates of a single point chosen from this distribution.

Remarks

Running your code multiple times with the same input should result in the specified distribution.
Outputting polar coordinates is too trivial and not allowed.
You can output Cartesian coordinates in any way allowed by the default I/O rules.
- This includes complex values.

Valid approaches

Several algorithms can be used to yield the desired distribution, including but not limited to

Choose a from the uniform distribution [0,2π) and b from the normal distribution (r,s).
Let x = b*cos(a) and y = b*sin(a).
Choose a from the uniform distribution [0,4) and b from the normal distribution (r,s).
Let x+y*i = b*i^a.
Choose a,b,c all from the normal distribution (0,1).
Let d = a+b*i and x+y*i = d/abs(d) * (c*s+r).

Example distributions (N=1000)

Below: r=1, s=0.1

Below: r=3, s=1

Below: r=1, s=0

Below: r=100, s=5

Answer*

# [APL(Dyalog Unicode)][1], <sup><s></s></sup>38 bytes <sup>[SBCS][2]</sup>

Takes `r` on the left and `s` on the right and returns a complex number.

```
{(⍺+⍵×(.5*⍨¯2×⍟?0)×1○○2×?0)×¯12○2×○?0}
```
[Try it on APLgolf!](https://razetime.github.io/APLgolf/?h=AwA&c=q9Z41LtL@1Hv1sPTNfRMtR71rji03ujw9Ee98@0NNA9PN3w0vRuIgCJg7qH1hkYQLpC0N6gFAA&f=M1RIUzDQM@QyBtKGXIYgHpehgQGQNgUA&i=AwA&r=tryapl&l=apl-dyalog&m=dfn&n=f)

Uses the first approach presented in the question. Dyalog doesn't have a builtin for sampling from a normal distribution, so this uses the [Box–Muller transform][3] to convert to random numbers from \$(0,1)\$ to a normally distributed value:

`(⍺+⍵×(.5*⍨¯2×⍟?0)×1○○2×?0)` draws a normally distributed value \$b \sim N(\alpha, \omega^2)\$: 
`?0` random number \$c \in (0,1)\$ 
`1○○2×?0`: \$\sin(2\pi c)\$ 
`?0` random number \$d \in (0,1)\$ 
`.5*⍨¯2×⍟?0`: \$\sqrt{-2\ln{d}}\$ 
`⍺+⍵×` Scale from \$N(0, 1)\$ to \$N(\alpha, \omega^2)\$

`?0` generates a random number \$a \in (0,1)\$ 
`¯12○2×○?0`: \$e^{i 2\pi a} = \sin(2\pi a)i + cos(2\pi a)\$ 

The product of these two values is the result.



Plotting code and images:

 'InitCauseway' ⎕CY 'sharpplot'
 InitCauseway ⍬
 sp←⎕NEW Causeway.SharpPlot(700)
 sp.SetTrellis(2 2)
 sp.TrellisStyle←4
 
 F ← {(⍺+⍵×(.5*⍨¯2×⍟?0)×1○○2×?0)×¯12○2×○?0}
 
 :For r s :In (1 0.1)(3 1)(1 0)(100 5)
 sp.NewCell
 sp.Heading←'r = ',(⍕r),'; s = ',⍕s
 sp.SetAxesScales(1)
 sp.DrawScatterPlot↓9 11∘.○{r F s}¨⍳1000
 :EndFor
 
 sp.SaveSvg(⊂'plot.svg')

[![enter image description here][4]][4]


 [1]: https://dyalog.com
 [2]: https://github.com/abrudz/SBCS
 [3]: https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform
 [4]: https://i.sstatic.net/3Ql6B.png

If this is an answer to a challenge…

…Be sure to follow the challenge specification. However, please refrain from exploiting obvious loopholes. Answers abusing any of the standard loopholes are considered invalid. If you think a specification is unclear or underspecified, comment on the question instead.
…Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one. Explanations of your answer make it more interesting to read and are very much encouraged.
…Include a short header which indicates the language(s) of your code and its score, as defined by the challenge.

More generally…

…Please make sure to answer the question and provide sufficient detail.
…Avoid asking for help, clarification or responding to other answers (use comments instead).

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