So lets create a geomtry.h so we can actually calculate the 3d math we need for our rasterizer to work.
Lets start with this struct. This is ofcourse a vector 2, 2 coordinates. Default constructor is asserting 0, 0 on them, otherwise you can declare them.
The operator functions return asign so you can return a value of either the x of y via an integer. What helped me understand this is to think of like OpenGl VertexAttrib, or assigned textures in Opengl. You assign them to an int and via that reference you can access that component in the array.
The vec3 struct works the same but with 3 components instead of 2! Excitng. What is assert? Well it checks wether or not a condition is true and if it isnt lets crash the program and display a problem so we can see what went wrong.
Now lets do math magic, this is math guys this is all math.
Except this one here this is just an overrided function that allows us to write out a vec2 as a string with our cout, its like @override in java for toString().
heres our + operator, adding two vectors together is essentially just adding together each x and y with the coresponding x, y of the other vector. So 1, 2 + 2, 1 = 3, 3. Its quite simple really. if we just want to add to like for example the X coordinate, we just make do 0 on the Y like 1, 0.
Now were done with establishing the basic math of vec2. Lets start doing the vector 3 stuff!
minus - works the exact same way as + just with -.
heres how the * functions if we multiply a vector by a scalar. Pretty simple I mean you know this. I dont really know how to explain it honestly we take it each coordinate times the scalar, and if we reverse the order we take the scalar first times the vec- you get it.
Division works just the same as the first multiplication method we defined but with /. You might be thinking, huh how come we need 2 different methods for * but only 1 for /? Well its simple dummy, You cant divide like this Scalar / Vector. Like what are you trying to accomplish here?
Now its time for some slightly harder math
Here is how we calculate our beautiful dot product. The dot proct lets us calculate how aligned two vectors are. The closer to 1 they are the more aligned and the closer to 0 the closer they are to being perpendicular. So if its 0 its completely perpendicular as in 90 degrees. However the dot product can also be negative, if this is the case it goes past those 90 degrees and faces away from the other vector. So the closer to -1 the closer the lines are to being opposite. The way to calculate a dot product is by multiplying each component of the vectors by one another and then + the x and y results of that.
So to calculate the length of a vector we use pythagoras theorem, that classic square root of (a^2 + b^2), I knew I had a reason to learn that in school!
Now this is a helper function to our normalized function which looks like this.
Normalizing means keeping a vectors direction in accordance to the origin point but making the it length of 1. We figure out the length of the vector via pythagoras theorem. We then divide the X and Y coordinate by that length. Now we get 1 length of the vector! Worth to note that if a vector is 0, 0 we return that vector but otherwise run that calculation of dividing.
Now the vec2 math stuff is done and we can move onto the vec3 stuff.
Addition with vectors is still pretty easy and its the same for minus on the vectors aswell.
Multiplication works the same as vector2 you just add another coordinate to multiply.
division works the same essentially just with an added z.
Heres our dot product, it is also the same as before essentially we just need to add each vectors z value and multiply those together and add them to our total.
Calculating length is the same principle we just gotta add the z value, this is getting quite repetetive right.
And our normalize function is exactly the same we just take a vec3 instead of a vec 2!.
Finally we have this juicer of the cross product.
We are using a right handed cross product system which means that our perpendicular vector created by comparing two vectors will be positive facing us and negative going away from us. I like to think of cross products as you are taking two lines and adding them ontop of eachother, where they both intersect you stick a needle through and thats where your new vector is. This new needle is perpendicular to both our other vectors, and depending on if its a right handed system or not were sticking the needle through the front (if left handed) or through the back so the needle is facing us (right handed).
The way to calculate the cross product in math is i like to picture it as you have to multiply all different combinations of each vectors x, y and z values and then subtract it by another. Excluding if they have the same coordinate designation, like you wouldnt multiply A.y and B.y together. It is honeslty though probably the hardest formula to remember of the top of your head, so I always google it but hey it works.
That does it for this part, really excited to actually dive deeper and draw some wirefram models but this has been very rewarding on the math aspect as that is were I am definitely lacking the most.
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