Separation

One of the classic motivations for our boid is separation. Steer clear from your buddy boids to minimize the chances of collisions.

Calculation

Let the position of our boids be given by a sequence of coordinates \(p_{1} = (x_{1}, y_{1}), p_{2} = (x_{2}, y_{2}), \ldots, p_{n} = (x_{n}, y_{n})\).

We need to calculate the direction we see a boid in as seen from an different boid with position \(p = (x, y)\). The direction can be calculated by taking the difference of the position of the boid you are interested in and your own boid. I.e.

\[ d_{i} = p_{i} - p = (x_{i} - x, y_{i} - y) \]

From the sequence of directions we can determine a sequence of headings by taking the arctangent.

\[ h_{i} = \operatorname{arctan}\left(\frac{y_{i} - y}{x_{i} - x}\right) \]

We would like to choose a heading that avoids all of the headings \(h_{i}\), if we want to avoid a collision.

One way of doing this is averaging the headings and going in the opposite direction. The average can be calculated by

\[ h := \frac{1}{n} \sum_{i=1}^{n} h_{i} \]

our target direction could be \(h + \pi\), normalized to lie within \(\left(-\pi, \pi\right)\).