Unsound mathematics could result in bugs

The presenter says

We're only using whole numbers in our game. An invader can't be halfway between two grid squares. Truncating the decimal value we get from this formula gives us the correct distance in discrete units.

This has a number of problems.

1. A non-integer distance doesn't mean "halfway between two grid squares". It means some distances simply aren't integers. Even though (0, 0) and (1, 1) are not "halfway between two grid squares", the distance from (0, 0) to (1, 1) is √2.

2. That truncation gives a 'correct distance in discrete units' is asserted without proof. It's actually untrue: distance with truncation is no longer a valid distance function. Consider the collinear points p0=(0, 0), p1=(1, 1), p2=(3, 3).

   • We should be able to breakdown distances and have

     d(p0, p2) = d(p0, p1) + d(p1, p2)
     

     right? Not with truncation:

     ⌊d(p0, p1)⌋ = 1
     ⌊d(p1, p2)⌋ = 2
     ⌊d(p0, p2)⌋ = 4 ≠ 1 + 2 = ⌊d(p0, p1)⌋ + ⌊d(p1, p2)⌋
     

   • It's even worse: a distance function d should satisfy the triangle inequality:

     d(p0, p2) ≤ d(p0, p1) + d(p1, p2)
     

     However, ⌊d⌋ does not

     ⌊d(p0, p1)⌋ + ⌊d(p1, p2)⌋ = 3 < 4 = ⌊d(p0, p2)⌋
     

Our game will run into problems if we ever need to add "distances" or compare sums of "distances". A better alternative for working with integers might be to skip sqrt and work in terms of squared distances.