Ahh yes. I threw my math at it and I think I hit it. You're correct it does involve the Pythagorean theorem and some scaling.

You start with your normalized vector that represents your ray.

It has an x component and a y component. First we want to see how long it is when it travels one unit in the x direction. So what do we do? We want to scale the entire vector so that the x component equals 1. To figure out what to scale it by, we do the following:

scaleFactor = 1/rayDirX;

Writing that out in math it's really just

scaledX = rayDirX * (1/rayDirX) = 1

So we can just call that 1.

Then for the y component:

scaledY = rayDirY * (1/rayDirX) = rayDirY/rayDirX

So now we have our scaled components as (1, rayDirY/rayDirX)

Now, we want to know the length. Now Pythagorean comes into play. Which is

length = sqrt((x * x) + (y * y))

So plugging in our scaled components we get:

length = sqrt((1 * 1 ) + (rayDirY / rayDirX) * (rayDirY / rayDirX))

Apply some algebra and simplify and we get:

length = sqrt(1 + (rayDirY * rayDirY) / (rayDirX * rayDirX))

Same goes for the y component except we'll have (rayDirX/rayDirY, 1) which results in

length = sqrt(1 + (rayDirX * rayDirX) / (rayDirY * rayDirY))

There we have your two equations from your question. Pretty neat. Thanks for the algebra exercise.

Ahh yes. I threw my math at it and I think I hit it. You're correct it does involve the Pythagorean theorem and some scaling.

You start with your normalized vector that represents your ray.

It has an x component and a y component. First we want to see how long it is when it travels one unit in the x direction. So what do we do? We want to scale the entire vector so that the x component equals 1. To figure out what to scale it by, we do the following:

scaleFactor = 1/rayDirX;

Writing that out in math it's really just

scaledX = rayDirX * (1/rayDirX) = 1

So we can just call that 1.

Then for the y component:

scaledY = rayDirY * (1/rayDirX) = rayDirY/rayDirX

So now we have our scaled components as (1, rayDirY/rayDirX)

Now, we want to know the length. Now Pythagorean comes into play. Which is

length = sqrt((x * x) + (y * y))

So plugging in our scaled components we get:

length = sqrt((1 * 1 ) + (rayDirY / rayDirX) * (rayDirY / rayDirX))

Apply some algebra and simplify and we get:

length = sqrt(1 + (rayDirY * rayDirY) / (rayDirX * rayDirX))

Same goes for the length when the y component travels one unit, except we'll have (rayDirX/rayDirY, 1) which results in

length = sqrt(1 + (rayDirX * rayDirX) / (rayDirY * rayDirY))

There we have your two equations from your question. Pretty neat. Thanks for the algebra exercise.