Solution for How to randomly scatter points inside a circle with ggplot, without clustering around the center?

is Given Below:

I want to use `ggplot`

to draw a circle, then scatter points inside it. I have code (adopted from this answer) that gets me pretty close to what I want. However, I want the points to scatter inside the circle *randomly*, but right now I get an undesired cluster around the center.

I saw a similar SO question and answer, but it’s in c# and I don’t understand how to adapt it to `R`

code.

## My code so far

The following code defines a custom visualization function `vis_points_inside_circle()`

, and then calls it 4 times to give 4 examples of visualizing using my current method.

```
library(ggplot2)
library(ggforce)
## set up function
vis_points_inside_circle <- function(n) {
# part 1 -- set up empty circle
df_empty_circle <-
data.frame(x = 0,
y = 0,
r = 1)
p_empty_circle <-
ggplot(df_empty_circle) +
geom_circle(aes(x0 = x, y0 = y, r = r)) +
coord_fixed() +
theme_void()
# part 2 -- set up points scatter
r <- runif(n)
th <- runif(n)
df_circular_points_scatter <-
data.frame(x = r*cos(2*pi*th), ## from @Ben's answer: https://stackoverflow.com/a/68606605/6105259
y = r*sin(2*pi*th))
# part 3 -- combine circle and points
p_empty_circle +
geom_point(data = df_circular_points_scatter,
aes(x = x, y = y))
}
## visualize
library(gridExtra)
set.seed(2021)
p1000_1 <- vis_points_inside_circle(n = 1000)
p1000_2 <- vis_points_inside_circle(n = 1000)
p2000_1 <- vis_points_inside_circle(n = 2000)
p2000_2 <- vis_points_inside_circle(n = 2000)
gridExtra::grid.arrange(p1000_1, p1000_2, p2000_1, p2000_2, nrow = 2)
```

^{Created on 2021-08-02 by the reprex package (v2.0.0)}

It should be easy to notice the cluster around the center of each circle. How can we randomly arrange the points within the circle, to avoid the cluster at the center?

**My attempt with sample()**

I guess the solution involves the modification of `df_circular_points_scatter`

data. However, I don’t know how. I’ve tried to wrap each of `df_circular_points_scatter`

‘s columns with `sample()`

, but then got points that exceed the circumference, and overall a “cross” arrangement.

That is, if we wrap with `sample()`

like this:

```
r <- runif(n)
th <- runif(n)
df_circular_points_scatter <-
data.frame(x = sample(r*cos(2*pi*th)),
y = sample(r*sin(2*pi*th)))
```

Then the result is:

Any idea how to avoid clustering? I don’t necessarily want a perfectly even scatter. Just random within the circle.

## Desired output

Adopted from this answer, the desired output should look something like:

You are almost there. The sampling needs to be done as follows:

```
r <- runif(n)
th <- runif(n, 0, 2 * pi)
df_circular_points_scatter <-
data.frame(x = sqrt(r) * cos(th),
y = sqrt(r) * sin(th)
)
```

(see this question on crossvalidated)

Result:

You have the points eventually distributed in r and theta, but you want them to be evenly distributed in *area*.

As the area element in the circle is $r dr dtheta$ (not $dr dtheta$ as your code implies), so you should transform

`r <- sqrt(r)`