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As a preface, let us begin by first learning homogenous coordinates. They are similar to our familiar Cartesian coordinates, except they allow us to also denote points at infinity. You get a homogenous coordinate system by simply adding another dimension. For a two-dimensional plane it is Z, and for three-dimensional space - W. For any point given in Cartesian coordinates (x, y, z), there are infinitely many corresponding homogenous coordinates: (x/w, y/w, z/w, w) for any w. When w = 0, however, this no longer holds. Instead, it is a point at infinity.

The render component exposes the entity is removed.

When you draw a model, typically you want the model to move somehow. It might move on the X, Y, Z axes, or it might rotate this or that way. It might even inflate in size, who knows. All of these transformations can be done by changing the positions of individual vertices.

A translation operation can be defined as a simple 3-dimensional vector. We can write down this operation in pseudo-code as so:

for each vertex v:
	v <= v + translation

Each operator has an operand which makes no effect, called the identity. For translation, it is the zero vector (0, 0, 0).

Scaling is also simple. Here you multiply (component-wise) the vertex positions by a scaling vector. The identity for scaling should be obvious.

for each vertex v:
	v <= v * scaling

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All further functions will operate on the graphics backend, often duplicate.

First are quaternions. Because they're actually for more than just rotation, there are many ways to formulate these, none are that intuitive. For our purposes, my favorite one is relating it to rotation around an axis. Choose a (normalized) axis of rotation (x, y, z), and an angle α. Then the equivalent quaternion is the 4D vector (x * sin(α/2), y * sin(α/2), z * sin(α/2), cos(α/2)). To be honest, quaternions are a whole field, so have this very nicely documented code:

for each vertex v:
	v <= Quaternion.rotate(v, q)

Second are rotation matrices. These are triplets of 3D vectors, one for each axis.

/ Rx1 Rx2 Rx3 \
| Ry1 Ry2 Ry3 |
\ Rz1 Rz2 Rz3 /

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An example would be easier to show in two dimensions, where rotation can be described with one angle. Here is the generic 2D rotation matrix defined by the angle α, and to its right is when α = 180°.

/  cos(α) sin(α) \   / -1  0 \
\ -sin(α) cos(α) /,  \  0 -1 /

The matrix changes the basis from (1, 0), (0, 1) to (-1, 0), (0, -1), which means to mirror the X and Y axes, and that is exactly what rotating by 180 degrees does.

The identity matrix is also the identity of rotation. Rotations may also be neatly combined by simply multiplying them in the order you wish. If you multiply the 180-degree rotation matrix by itself, you'll get the identity matrix, i.e. no rotation.

Like scaling, rotation also occurs relative to the origin.

The above structures are all important and have their uses, but they are toys in comparison to the mother of them all: The Transformation Matrix. Why, you may ask? I'm excited to show you. First, let us see what it is capable of. Can we write down the translation operator with a matrix? We need a matrix M, for which M * v = v + translation. As it turns out, we can if we use homogenous coordinates, specifically when v = (Vx, Vy, Vz, 1), translation = (Tx, Ty, Tz, 1).

/ 1 0 0 Tx \
| 0 1 0 Ty | * v = v + (Tx, Ty, Tz, 0) = (Vx + Tx, Vy + Ty, Vz + Tz, 1)
| 0 0 1 Tz |
\ 0 0 0 1  /

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Rotation is possible by simply extending the rotation matrix into 4x4. Note the three columns can be viewed as vectors in homogenous coordinates.

/ Rx1 Rx2 Rx3 0 \
| Ry1 Ry2 Ry3 0 | * v = (Rx1 * Vx + Rx2 * Vy + Rx3 * Vz, Ry1 * Vy + Ry2 * Vy + Ry3 + Vz, Rz1 * Vx + Rz2 * Vy + Rz3 * Vz, 1)
| Rz1 Rz2 Rz3 0 |
\  0   0   0  1 /

Scaling is also possible, like so:

/ Sx 0  0  0 \
| 0  Sy 0  0 | * v = ...
| 0  0  Sz 0 |
\ 0  0  0  1 /

Now for the grand finale. What makes the transformation matrix so powerful is that it can accumulate translations: if you multiply two matrices, their effects will add. This also implies the matrix's ability to store all three operations in one! In fact, any transformation matrix can be viewed as a combination of translation, rotation and scaling:

/ Rx1*Sx Rx2*Sy Rx3*Sz Tx \
| Ry1*Sx Ry2*Sy Ry3*Sz Ty |
| Rz1*Sx Rz2*Sy Rz3*Sz Tz |
\   0      0      0    1  /

You can see it for yourself by, say, setting some parts of the transformation to identity. And if you set all parts to identity, you will get the 4-order identity matrix.

Texture instructions in fragment programs Any instruction in a so-called index buffer, we can take advantage of our rendering loop, we blacken the window size core : override the decision to use this you ought to know.

Matrix operations are non-commutative, which is to say the order of operations are important! To the right is an example of this. Furthermore, operations have a certain quirk: the effect produced by the product of transformation matrices A, B, C, etc. appears as though the effects occur in reverse (..., C then B then A). So although the orange cube is a result of rotation before translation, when laid out with matrices, it is a translation matrix times a rotation matrix.

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Understanding transformations will greatly assist us onward.