We can embed euclidean space into a higher dimensional space, called conformal space, this makes transform operations involving rotations and translations easier to work with. It allows us to represent all angle preserving transformations.
When we looked at complex functions we saw that certain functions such as the inverse: (w=1/z) mapping preserve angles and we described these as conformal functions. Here we define a space where all transformations using the transform:
x = v * x * (1/v)
are conformal.
When we discussed euclidean space we said that there is no specific origin (unless we add an arbitrary coordinate system) and any point is as good as any other for zero point. Euclidean space also does not have a way to represent points at infinity.
Conformal space adds two new dimensions, one represents zero and the other represents infinity. I haven't yet got an intuitive understanding of why we need whole dimensions to represent these points.
We can now represent translations as rotations around an axis at infinity. (Its interesting to speculate if there might be a duality here to allow us to represent translations as rotations at zero?).
It is possible to represent operations, in this space, using different types of algebra, for example matrices, the most common type of algebra which we use to represent this geometry is Geometric Algebra. This is a very good match, the particular GA we use has 4 dimensions which square to +ve and one which squares to -ve, known as G4,1,0.
So we use 5 dimensions which we will denote as follows:
- e1,e2,e3 = the 3 dimensions in euclidean space
- e0 = the dimension at zero
- e∞ = the dimension at infinity.
The way that we use this algebra to represent translations and rotations is described on this page.
