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3. Math considerations

Math road to nowhere

I was trying to figure out the maths behind keeping track of speed and direction of a shot fired in an arbitrary direction. The shot would be fired by an enemy into the direction of the center of the player ship. I though I might go about it like this:

keep track of the initial x and y positions of the bullet
keep track of the x and y positions of the initial target of the bullet
keep track of the angle the bullet's direction in relation to it's origin

If we think of the line between enemy and player as the long side of a triangle, and the x and y distances as the short sides, we could try to apply trigonometrics to find the angle the shot was fired at.

`x_dist` and `y_dist` are known, these are the short sides of the triangle
we do not know the length of the long side of the triangle
the angle `β` between the short sides is 90 deg. (or PI / 2 in radians)
we do not know the any of the other angles
we need to find the angle of `γ`, which is the angle of the origin where the shot was fired from

The math would look like this:

γ = asin( (sin(β) / b) * y_dist )

The only unknown here is `b`, which is the long side of the triangle aka the distance between enemy and player. But we can calculate it like this:

b = sqrt(x_dist^2 + y_dist^2 - 2 * x_dist * y_dist * cos(β))

Combining the two formulas we get:

γ = asin( (sin(β) / (sqrt(x_dist^2 + y_dist^2 - 2 * x_dist * y_dist * cos(β)))) * y_dist)

That unfortunately looks computationally expensive, let's see if we can simplify it now that we know that `β = PI/2` (or 90 degrees):

b = sqrt(x_dist^2 + y_dist^2 - 2 * x_dist * y_dist * cos(pi/2))
γ = asin(y_dist / b)

And combined:

γ = asin(y_dist / (sqrt(x_dist^2 + y_dist^2 - 2 * x_dist * y_dist * cos(pi/2))))

That is still way to computationally expensive and overkill for our purpose, let's think about something way simpler.

Simpler approach

Thinking about this a little more I could probably just do this:

record the horizontal and vertical distances between bullet origin and target
at each game update multiply both distances with a `speed` variable, which is a float between 0 and 1

This should result in the same movement, but with almost no effort. Some pseudo code:

speed = 0.05

bullet_x = enemy.x
bullet_y = enemy.y

dist_x = enemy.x - player.x
dist_y = enemy.y - player.y

while true do
  bullet.x = bullet.x + dist_x * speed
  bullet.y = bullet.y + dist_y * speed
end

That should work in theory, will test this soon.

⬅ 2. The initial status of the game

⬆ A vertical shoot'em'up game in Lua with LÖVE

🏠 callistix Gemini capsule

Created: 25/Jan/2024

Modified: 25/Jan/2024