Explanation

This was a fun little experiment I wanted to do where I wanted to let a couple of children run around randomly in their lunch break maybe and plot their movements on a polar coordinates.

So the number of bois are set by n_bois and the number of seconds(or time points) they have their lunch break is set by recess_time. First I set the initial points of the bois as follows. These were simply set to be cartesian coordinates so that I can simply convert them to polar and see how it goes. $$\begin{gathered} X_{i} = Unif(-100,100)\\ Y_{i} = Unif(-100,100) \end{gathered}$$ And then for each $t^{th}$ time point I calculated the following values

$$\begin{gathered} angle_{t} = Unif(0,180)\\ speed_{t} = Unif(10,1000)\\ t = 1,2,...,recess_time\\ angles_{i} = Unif(-angle_{t},angle_{t})\\ speed_{i} = Unif(-speed_{t},speed_{t})\\ i = 1,2,...,n\_bois \end{gathered}$$

library(tidyverse)

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set.seed(42)
theme_set(theme_void())

n_bois <- 100
a <- tibble(x=runif(n=n_bois,-100,100),
            y=runif(n=n_bois,-100,100),
            time=1,
            tag=seq(1,n_bois))

recess_time <- 360

radian <- function(degree){
  (degree / 360) * 2 * pi  
}

for(i in 1:recess_time){
  angle_val <- runif(1,0,180)
  speed_val <-runif(1,10,1000)
  angles <- runif(n_bois,-angle_val,angle_val)
  speed <- runif(n_bois,-speed_val,speed_val)
  start_index <- length(a$x)-n_bois+1
  end_index <- length(a$x)
  b <- tibble(x = a$x[start_index:end_index] + speed * cos(radian(angles)),
              y = a$y[start_index:end_index] + speed * sin(radian(angles)),
              time = i,
              tag=seq(1,n_bois))
  a <- rbind(a,b)
}

ggplot(a,aes(x,y))+
  geom_point(aes(color=time,group=tag),
                 size=0.5,alpha=0.8,show.legend = FALSE) +
  geom_line(aes(color=time,group=tag),show.legend = FALSE) +
  coord_polar() + 
  scale_color_gradient(low="#83a4d4",high="#b6fbff")