Util Math for Games
Table of Contents
Numbers and Vector 1D
pipe means absolute value
Distance
Use subtraction of numbers to get the distance.
distance(a,b) = |b - a|
Absolute
Use the absolute value to remove sign.
abs(a) = |a|
Sign
Use the absolute value to figure out the direction of vector.
sign(a) = a / |a| 10 / abs(10) = 1.0 -10 / abs(-10) = -1.0 0 abs(0) = 0
Vector 2D
When we subtract vectors, like a - b, it's the same to ADD a flipped vector a + (-b). Let's see:
b = (-4, 1) -b = (4, -1) a = (3,2) d = a - b = (7,1)
Vectors has only one value that represents the end of it. I mean, it doesn't have START and END.
Vector Length (magnitude)
If we have only Vector 1D, just compute the length of vector by subtract them. b - a.
But, in the Vector 2D, we need the Pythagorean Theorem.
Cˆ2 = aˆ2 + bˆ2
So, the formula is:
length(c) = Math.sqrt(c.xˆ2 + c.yˆ2) a = (3,1) b = (-2,2) c = (a.x - b.x, a.y - b.y) = (5,-1) length(a - b) = Math.sqrt(5ˆ2 + (-1)ˆ2) = 5.0990195135927845
When we use two pipe we are talking about the magnitude/length of something. ||b - a||
Normalize / Direction
Use the magnitude value to get normalized value.
normalize(a) = a / ||a|| â = a / ||a||
Dot Product
The dot product is the Vector multiplication like a.b
a.b = a.x * b.x + a.y * b.y
Scalar Projection:
We can use Scalar Projection to figure out if the vector is behind a object (represented by normalized vector - â).
â.b -> Normalized vector(a) and normal vector(b) multiplication
