Previously, I made a crude model of Pipe Dream, a geometry-intensive puzzle genre using Machinations,
a tool by Joris Dormans, based on Petri nets for modeling game mechanics.

Shmups are a different, older, more popular genre of video game.
They SEEM pretty geometry-intensive, but let’s see if we can create a geometry-free model despite that.

It seems hard, so let’s try and steal ideas from smart people.
This is gonna seem like a couple of non sequiturs, but stick with me.

When a nuclear physicist talks about a “cross sectional area”, they’re actually talking about a probability.
Specifically, the probability of a collision.

In most circumstances, you would say that areas and probabilities are two different kinds of things.
However, imagine a filled-in blob on a piece of paper.
We’ll call the filled-in part the “figure”, and the non-filled in part the “ground”.
If you put your finger (uniformly) randomly on the paper,
the probability your chosen point is within the figure,
is equal to the fraction of the paper’s area that is the figure.

There’s another simple model where probability-of-collision comes up.

Let’s consider a well-mixed population of zombies and humans.
When a zombie encounters a human, the human often turns into a zombie.
When a zombie encounters a zombie… nothing happens.
Similarly, when a human encounters a human, nothing happens (or at least, they both stay human).
So the zombie population increases (and the human population decreases) at a rate
that’s proportional to the zombie/human collision probability.

This is a standard model of a lot of things, including epidemics and marketing (“enthusiasts are zombies”),
and it creates a pretty S-curve.