Homemade 3D rendering engine (V)

At this point, it does not harm for us to extend the concept of homogeneous coordinates a little bit to fit our purposes. In particular, for points, let’s define $ (cx, cy, cz, c)^T $ the same as $ (x, y, z, 1)^T $. This is nothing but just mapping the previously unused representation into a single canonical representation.

A linear transformation is defined as $ T(a\vec{x} + b\vec{y}) = aT(\vec{x}) + bT(\vec{y}) $, and it is typically represented as a matrix $ T(\vec{x}) = A\vec{x} $.

A translation is defined as $ T(\vec{x}) = \vec{x} + \vec{v} $, apparently, translation is not linear as $ T(\vec{0}) = \vec{v} \ne \vec{0} $.

As we can see, to do the projection, we need both translation and linear transformation, the fact that translation cannot be represented as matrix makes computation cumbersome.

Now the interesting thing happens, with homogeneous coordinates, we can represent both translation and general linear transformation as a matrix!

To see how, we can represent general linear transformation as follow

$ \left(\begin{matrix} \mathbf{A} & 0 \\ 0 & 1 \end{matrix}\right) \left(\begin{matrix} \vec{x} \\ 1 \end{matrix} \right) = \left(\begin{matrix} \mathbf{A}\vec{x} \\ 1 \end{matrix}\right) $

That’s doesn’t sound particularly useful, we could have done it without the extra dimension, but this is amazing

$ \left(\begin{matrix} \mathbf{I} & \vec{v} \\ 0 & 1 \end{matrix}\right) \left(\begin{matrix} \vec{x} \\ 1 \end{matrix} \right) = \left(\begin{matrix} \vec{x} + \vec{v} \\ 1 \end{matrix}\right) $

This also means we can compose translation and general linear transformation into a single matrix, these more general transformations are mathematically called affine transformations, and that’s what we will use in the projection. Note, however, that by composing the matrix, it might not be the case that we always ends with having a resulting point ends with 1, that’s why we need to introduce the homogeneous coordinates.