OpenGL transform matrix order is backwards

Question

If I want to rotate the object around z axis, and then translate it I must do

glm::mat4 transform;
GLfloat angle = 90f;
transform = glm::rotate(transform, angle, glm::vec3(0.0f, 0.0f, 1.0f));
transform = glm::translate(transform, glm::vec3(0.5f, -0.5f, 0.0f));

But it works backwards, it first rotates and then translates, so I need to write it as

glm::mat4 transform;
GLfloat angle = 90f;
transform = glm::translate(transform, glm::vec3(0.5f, -0.5f, 0.0f));
transform = glm::rotate(transform, angle, glm::vec3(0.0f, 0.0f, 1.0f));

How do the maths behind this work? Why I must combine matrixes in reverse to achieve the desired effect?

Answer 1

From an intuitive point of view, you are absolutely right: Transformations have to be applied the opposite way one thinks about them. The reason for this is quite easy:

In glm/OpenGL all vectors are assumed to be column vectors, thus applying a transformation (M) in matrix form to a vector t can be written as follows:

t' = M * t

Now assume that we first want to translate (T) and then rotate (R). We could now do each step separately like

t' = T * t       //Translate
t'' = R * t'     //Rotate result Translation

When we want to combine both transformations we substitute t' in the second line and get:

t'' = R * (T * t) = (R * T) * t

As you can see, the operation that is applied first is written last (or better to say closer to the vector). The same principle can be applied with as many matrices one wants.

Note, that if vectors are treated as row vectors, the whole matrix order would change.

 t' = t * M        //General case

Let's have a look at the same example as above, but this time with row vectors:

 t' = t * T
 t'' = t' * R

 t'' = (t * T) * R = t * (T * R)

Conclusion: Whenever you think about transformations and vectors, remember that the operation that is applied first has to be written closer to the vector.