# Homogeneous Co-ordinates

For now we have been looking at co-ordinate systems in two and three dimensions but these co-ordinate systems have an annoying problem when it comes to describing certain kinds of transformations. This comes down to the fact that the systems of equations represent transformations which are fundamentally multiplicative in nature.

So how can we represent something like a translation, for example? Well, we can add an additional dimension, $w$ to deal with this problem. In this dimension, we follow a few important conventions. For any vector that represents a co-ordinate, we specify the $w$ as 1. Lets see what impact this has, when we set the $w$ co-ordinate for all the other dimensions in the column

$\left[\begin{array}{cccc}1.5& -1& -1& 1\\ 0& 1.5& 0& 1\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

Notice that unlike all the transformations we have looked at so far, this transformation translated us away from the origin?

To a sharp mind, this might appear perplexing, because transforms done by multiplication can only really scale space on either dimension. How can we do a translation when that is fundamentally additive in nature?

It all comes down to the fact that matrix multiplications are a multidimensional generalization of the vector dot product, which involves both multiplication and addition.

Remember how we set the final co-ordinate $w$ in our 4-dimensional vectors and the last entry in the 4-dimensional matrix to 1, with all the $wx,wy,wz$ co-ordinates in the last row set to zero? Lets see how this pans out in a matrix multiplication

$\left[\begin{array}{ccccccc}1& & 0& & 0& & 1\\ 0& & 1& & 0& & 1\\ 0& & 0& & 1& & 0\\ 0& & 0& & 0& & 1\end{array}\right]\cdot \left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}0×0+0×0+0×0+1×1\\ 0×0+0×0+0×0+1×1\\ 0×0+0×0+0×0+0×1\\ 0×0+0×0+0×0+1×1\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 0\\ 1\end{array}\right]$