I’ve been playing a lot of Fortnite OG, and it got me wondering how exactly the “storm” circle shrinks into the non-concentric “safe” circle. The math behind it is actually extremely simple.

OG Fortnite Storm Example
Credit: echogear.com

Definition of Circles

Circles have 2 basic properties: a center point ($x_c, y_c$), and a radius $r$.

For simplicity sake, let’s assume there is a predefined circle [($x_c, y_c$), $r$]. (Not proper mathematical notation, but it will simplify writing.)

The new, smaller circle is defined as such [$(x_c’, y_c’)$, $r’$], where $r’ < r$.

Randomly Selecting New Center Point

To ensure the smaller circle is inside the larger one, we use this constraint:

$d = \text{Distance between centers}$
$d + r’ \leq r$

or

$\sqrt{(x_c’ - x_c)^2 + (y_c’ - y_c)^2} \leq r - r’$

The new center $(x_c’, y_c’)$ can be randomly chosen in this region.

Now, we can generate a random radius and its center.

  1. Generate a random angle $\theta$ between $0$ and $2\pi$
  2. Generate a random radius $d$:
    $d=\sqrt{Random(0,1)} \cdot (r - r’)$
  3. Calculate the new center as:
    $x_c’ = x_c + d \cos(\theta)$
    $y_c’ = y_c + d \sin(\theta)$

Shrinking into the Smaller Circle

To shrink the circle, we can use a technique called linear interpolation.

For 2 points $a$ (start) and $b$ (end value), it can be shrunk using the following formula:

$i = \text{interpolated value}$
$t = \text{value between [0, 1] that represents the progress of interpolation}$

$i = a + t*(b-a)$

So when $t = 0$, $i = a$, when $t = 1$, $i = b$, and when $t = 0.5$, $i = $ midpoint between a and b.

Applying this to our circle definition gives us:

$x_t = x_c + t \cdot (x_c’ - x_c)$
$y_t = y_c + t \cdot (y_c’ - y_c)$
$r_t = r + t \cdot (r’ - r)$

Specifically for Fortnite, $t$ can be represented as $\frac{t_p}{T}$, where $t_p$ is time passed in shrinking phase, and $T$ is total length of the phase.

Desmos Animation

A sample animation & implemented math can be found in this Desmos calculation.

This same shrinking math also works for the “new” storms where at endgame, the inner circle may be placed inside the current storm.

Conclusion

Epic Games may use a different formula than this, but it’s what I found first and the animation looks very similar.

Desmos Calculator