When you are programming in OpenGL or any other graphics api you will hit a brick wall when you are not that good in math. Here I will explain with example code how you can achieve movement/scaling and many other cool stuff with your 3d object.

Let's take a real life case... You've made a awesome (three dimensional) cube in OpenGL and you want to move it to any direction.

```
glUseProgram(cubeProgram)
glBindVertexArray(cubeVAO)
glEnableVertexAttribArray ( 0 );
glDrawArrays ( GL_TRIANGLES, 0,cubeVerticesSize)
```

In game engines like Unity3d this would be easy. You would just call transform.Translate() and be done with it, but OpenGL does not include a math library.

A good math library is glm but to get my point across I will code all the (important) mathematical methods for you out.

First we must understand that a 3d object in OpenGL contains a lot of information, there are many variables that depend on each other. A smart way to manage all these variables is by using matrices.

A matrix is a collection of variables written in columns and rows. A matrix can be 1x1, 2x4 or any arbitrary number.

```
[1|2|3]
[4|5|6]
[7|8|9] //A 3x3 matrix
```

You can do really cool stuff with them... but how can they help me with moving my cube? To actually understand this we first need to know several things.

- How do you make a matrix from a position?
- How do you translate a matrix?
- How do you pass it to OpenGL?

Let's make a class containing all our important matrix data and methods (written in c++)

```
template<typename T>
//Very simple vector containing 4 variables
struct Vector4{
T x, y, z, w;
Vector4(T x, T y, T z, T w) : x(x), y(y), z(z), w(w){}
Vector4(){}
Vector4<T>& operator=(Vector4<T> other){
this->x = other.x;
this->y = other.y;
this->z = other.z;
this->w = other.w;
return *this;
}
}
template<typename T>
struct Matrix4x4{
/*!
* You see there are columns and rows like this
*/
Vector4<T> row1,row2,row3,row4;
/*!
* Initializes the matrix with a identity matrix. (all zeroes except the ones diagonal)
*/
Matrix4x4(){
row1 = Vector4<T>(1,0,0,0);
row2 = Vector4<T>(0,1,0,0);
row3 = Vector4<T>(0,0,1,0);
row4 = Vector4<T>(0,0,0,1);
}
static Matrix4x4<T> identityMatrix(){
return Matrix4x4<T>(
Vector4<T>(1,0,0,0),
Vector4<T>(0,1,0,0),
Vector4<T>(0,0,1,0),
Vector4<T>(0,0,0,1));
}
Matrix4x4(const Matrix4x4<T>& other){
this->row1 = other.row1;
this->row2 = other.row2;
this->row3 = other.row3;
this->row4 = other.row4;
}
Matrix4x4(Vector4<T> r1, Vector4<T> r2, Vector4<T> r3, Vector4<T> r4){
this->row1 = r1;
this->row2 = r2;
this->row3 = r3;
this->row4 = r4;
}
/*!
* Get all the data in an Vector
* @return rawData The vector with all the row data
*/
std::vector<T> getRawData() const{
return{
row1.x,row1.y,row1.z,row1.w,
row2.x,row2.y,row2.z,row2.w,
row3.x,row3.y,row3.z,row3.w,
row4.x,row4.y,row4.z,row4.w
};
}
}
```

First we notice a very peculiar thing in the default constructor of a 4 by 4 matrix. When called it doesn't start all on zero but like:

```
[1|0|0|0]
[0|1|0|0]
[0|0|1|0]
[0|0|0|1] //A identity 4 by 4 matrix
```

All matrices should start with ones on the diagonal. (just because >.<)

Alright so let's declare at our epic cube a 4 by 4 matrix.

```
glUseProgram(cubeProgram)
Matrix4x4<float> position;
glBindVertexArray(cubeVAO)
glUniformMatrix4fv(shaderRef, 1, GL_TRUE, cubeData);
glEnableVertexAttribArray ( 0 );
glDrawArrays ( GL_TRIANGLES, 0,cubeVerticesSize)
```

Now we actually have all our variables we can finally start to do some math! Let's do translation. If you have programmed in Unity3d you might remember a Transform.Translate function. Let's implement it in our own matrix class

```
/*!
* Translates the matrix to
* @param vector, The vector you wish to translate to
*/
static Matrix4x4<T> translate(Matrix4x4<T> mat, T x, T y, T z){
Matrix4x4<T> result(mat);
result.row1.w += x;
result.row2.w += y;
result.row3.w += z;
return result;
}
```

This is all the math needed to move the cube around(Not rotation or scaling mind you) It works at all the angles. Let's implement this in our real life scenario

```
glUseProgram(cubeProgram)
Matrix4x4<float> position;
position = Matrix4x4<float>::translate(position, 1,0,0);
glBindVertexArray(cubeVAO)
glUniformMatrix4fv(shaderRef, 1, GL_TRUE, &position.getRawData()[0]);
glEnableVertexAttribArray ( 0 );
glDrawArrays ( GL_TRIANGLES, 0,cubeVerticesSize)
```

Our shader needs to use our marvellous matrix

```
#version 410 core
uniform mat4 mv_matrix;
layout(location = 0) in vec4 position;
void main(void){
gl_Position = v_matrix * position;
}
```

And it should work.... but it seems we already have a bug in our program. When you move along the z axis your object seems to disappear right into thin air. This is because we don't have a projection matrix. To solve this bug we need to know two things:

- How does a projection matrix look like?
- How can we combine it with our position matrix?

Well we can make a perspective (we are using three dimensions after all) matrix The code

```
template<typename T>
Matrix4x4<T> perspective(T fovy, T aspect, T near, T far){
T q = 1.0f / tan((0.5f * fovy) * (3.14 / 180));
T A = q / aspect;
T B = (near + far) / (near - far);
T C = (2.0f * near * far) / (near - far);
return Matrix4x4<T>(
Vector4<T>(A,0,0,0),
Vector4<T>(0,q,0,0),
Vector4<T>(0,0,B,-1),
Vector4<T>(0,0,C,0));
}
```

It looks scary, but this method will actually calculate a matrix of how far you wish to look into the distance(and how close) and your field of view.

Now we have a projection matrix and a position matrix.. But how do we combine them? Well fun thing is that we can actually multiply two matrices with each other.

```
/*!
* Multiplies a matrix with an other matrix
* @param other, the matrix you wish to multiply with
*/
static Matrix4x4<T> multiply(const Matrix4x4<T>& first,const Matrix4x4<T>& other){
//generate temporary matrix
Matrix4x4<T> result;
//Row 1
result.row1.x = first.row1.x * other.row1.x + first.row1.y * other.row2.x + first.row1.z * other.row3.x + first.row1.w * other.row4.x;
result.row1.y = first.row1.x * other.row1.y + first.row1.y * other.row2.y + first.row1.z * other.row3.y + first.row1.w * other.row4.y;
result.row1.z = first.row1.x * other.row1.z + first.row1.y * other.row2.z + first.row1.z * other.row3.z + first.row1.w * other.row4.z;
result.row1.w = first.row1.x * other.row1.w + first.row1.y * other.row2.w + first.row1.z * other.row3.w + first.row1.w * other.row4.w;
//Row2
result.row2.x = first.row2.x * other.row1.x + first.row2.y * other.row2.x + first.row2.z * other.row3.x + first.row2.w * other.row4.x;
result.row2.y = first.row2.x * other.row1.y + first.row2.y * other.row2.y + first.row2.z * other.row3.y + first.row2.w * other.row4.y;
result.row2.z = first.row2.x * other.row1.z + first.row2.y * other.row2.z + first.row2.z * other.row3.z + first.row2.w * other.row4.z;
result.row2.w = first.row2.x * other.row1.w + first.row2.y * other.row2.w + first.row2.z * other.row3.w + first.row2.w * other.row4.w;
//Row3
result.row3.x = first.row3.x * other.row1.x + first.row3.y * other.row2.x + first.row3.z * other.row3.x + first.row3.w * other.row4.x;
result.row3.y = first.row3.x * other.row1.y + first.row3.y * other.row2.y + first.row3.z * other.row3.y + first.row3.w * other.row4.y;
result.row3.z = first.row3.x * other.row1.z + first.row3.y * other.row2.z + first.row3.z * other.row3.z + first.row3.w * other.row4.z;
result.row3.w = first.row3.x * other.row1.w + first.row3.y * other.row2.w + first.row3.z * other.row3.w + first.row3.w * other.row4.w;
//Row4
result.row4.x = first.row4.x * other.row1.x + first.row4.y * other.row2.x + first.row4.z * other.row3.x + first.row4.w * other.row4.x;
result.row4.y = first.row4.x * other.row1.y + first.row4.y * other.row2.y + first.row4.z * other.row3.y + first.row4.w * other.row4.y;
result.row4.z = first.row4.x * other.row1.z + first.row4.y * other.row2.z + first.row4.z * other.row3.z + first.row4.w * other.row4.z;
result.row4.w = first.row4.x * other.row1.w + first.row4.y * other.row2.w + first.row4.z * other.row3.w + first.row4.w * other.row4.w;
return result;
}
```

Ooef.. that's a lot of code that actually looks more scarier then it actually looks. It can be done in a for loop but I (probably mistakenly) thought this would be clearer for people that never ever worked with matrices.

Look at the code and notice a repeating pattern. Multiply the column with the row add it and continue.(this is the same for any size matrix)

*Note that multiplication with matrices is not like normal multiplication. A X B != B x A *

Now we know how to project and add this to our position matrix our real life code will probably look like:

```
glUseProgram(cubeProgram)
Matrix4x4<float> position;
position = Matrix4x4<float>::translate(position, 1,0,0);
position = Matrix4x4<float>::multiply(Matrix<float>::perspective<float>(50, 1 , 0.1f, 100000.0f), position);
glBindVertexArray(cubeVAO)
glUniformMatrix4fv(shaderRef, 1, GL_TRUE, &position.getRawData()[0]);
glEnableVertexAttribArray ( 0 );
glDrawArrays ( GL_TRIANGLES, 0,cubeVerticesSize)
```

Now our bug is squashed and our cube looks pretty epic in the distance. If you would like to scale your cube the formula is this:

```
/*!
* Scales the matrix with given vector
* @param s The vector you wish to scale with
*/
static Matrix4x4<T> scale(const Matrix4x4<T>& mat, T x, T y, T z){
Matrix4x4<T> tmp(mat);
tmp.row1.x *= x;
tmp.row2.y *= y;
tmp.row3.z *= z;
return tmp;
}
```

You only need to adjust the diagonal variables.

For rotation you need to take a closer look at Quaternions.