Basic 3D Math: Vectors
3D Graphics is all about Vectors, Matrices, and Trigonometry – so you need at least a Basic understanding of how to accomplish basic tasks with this things. I will try to show you some Basic operations you need while programming OpenGL. As I am not a mathematician I will probably often explain things without the correct mathematical words.
Part 1: Vectors
A Vector is a mathematical entity which consists of two or more real numbers – as a programmer, imagine the vector as an array of floats. Let’s take a look at how a vector looks like:
Here the vector “v” has 4 components: 5, 6, 7 and 1. In 3D Graphics you will use 3 or 4 component vectors most of the time:
- 3 Component vectors for positions and directions
- 4 Component vectors for performing transformations using matrices (more on that later)
As stated above, vectors can be used to represent positions and directions. Below you see a Cartesian coordinate system with vectors:
The Vector v1 with the components [2,7,2] represents the direction (go 2x, 7y and 2z from the origin) and a location (i am at 2x,7y,2z). In the above picture, it’s a special case that the directional and positional vector is the same because we start at the origin. Imagine another positional vector with the components [4,0,4]: To get to [2,7,2] you need a directional vector with [-2,7-2].
Positional Vectors are mostly used for the positions of vertices, directional vectors for light calculations (just to name two of the many usages)
As with most other mathematical entities, you can perform various operations on vectors:
- Multiply with a scalar
- Add and Subtract another vector
- Multiply with another Vector
- There are two ways to multiply a vector with another vector: One that yields another vector (“Cross Product”) and one that yields a scalar (“Dot Product”)
Multiplication with a Scalar
The multiplication of a vector with a scalar is a simple operation: Just multiply each component with a real number:
Geometrically, this can be seen as a scaling of the Vector:
On the Left, the Vector A has a value of [3,4]. On the right it has been multiplied by the scalar value of 1.5 – now the vector has the value of [4.5,6]
Add and Subtract another Vector
Assume you have two Vectors:
and – these two vectors are shown below:
If you add these two vectors together, you simply do a component-wise addition of the elements:
Geometrically this can be seen as the moving of V2 so that its tail is positioned at the head of V1:
The same applies to subtraction – just in reverse order. Assume and . When subtracting V2 from V3 we create . Subtraction is simply a component-wise subtraction of each element in V2 from V3. Take a look at the visualization:
As you can see, if we position V2 in Reverse order on the head of V3, we get V1.
Multiplication with another Vector – The “Dot Product”
As stated above there are two ways to multiply a vector with another one – the first one we will cover is the “Dot Product” which can be used to get the angle between two vectors. Imagine the following situation:
- You have two vectors in space (2D or 3D) and need to know the angle between those two for Lighting calculation
The Formula to calculate the angle α is formal:
But what does that all mean?
is the actual Dot Product which is calculated by simply multiplying each component of V1 with the corresponding component of V2 – in our example:
and denote the length of the Vector which is calculated with:
This is actually Pythagoras law if you haven’t noticed So now we have everything in place to perform the actual calculation:
Doesn’t seem like an angle, didn’t it? The result is in Radians but we are used to angles in Degrees. So we have to perform the last calculation to get Degrees:
This all applies to 3D Vectors as well, I’ve just used 2D as it’s easier to get it at the first. Sorry for being gotten a little bit Mathematical this time, but it was necessary.
Multiplication with Another Vector – The “Cross Product”
The second way of multiplying Vectors is the cross Product which is only useful in 3D Space. It results in another vector which is perpendicular to the plane defined by two other Vectors. First of all – what’s a Plane? Take a look at the following picture:
The Vectors V1 and V2 define an Orientation in Space (the Vectors are Directional ones, not positional ones – their tail is the origin and their value shows the direction). The Plane itself has no size, it’s just an abstract thing.
Now let’s take a look at how to cross product looks like visually:
So why is V3 so big? Because normally the vector yielded from a Cross Product operation ist normalized – what that means is shown later
which looks actually more complex than it is in reality:
So we have a gotten a new Vector: which is really perpendicular to our plane defined by V1 and V2. In almost all situations you will “normalize” your yielded vector – normalization means that each component of the vector is divided by the length of the vector – the result is a vector of length 1:
First of all, we need to calculate the length of the vector (shown above in the section about the Dot Product) – in this case, the vector has a length of 16. Now we divide every component of the vector by 16 and get a new one:
Formally, the process of normalizing a vector looks like: