Universal Geometry

While Affine Geometry is devoid of an notion of perpendicularity, there are three geometries with unique notions of the concept:


 * Green Geometry,
 * Blue Geometry, and
 * Red Geometry.

Remarkably, they are all tied together by certain formulas. Blue is the familiar Unit Circle (as described with sine and cosine), Red is the less familiar Unit Hyperbola (described with hyperbolic trigonometric functions), and the Green is described by another hyperbola xy=1. Three different definitions of the dot product correspond to these three different geometries:

$$ \begin{align} (x_1,y_1) \cdot_b (x_2,y_2) &= x_1x_2 + y_1y_2 \\ (x_1,y_1) \cdot_r (x_2,y_2) &= x_1x_2 - y_1y_2 \\ (x_1,y_1) \cdot_g (x_2,y_2) &= x_1y_2 + y_1x_2 \\ \end{align} $$

From a linear algebra perspective, this system can be written using $$1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$, $$i = \begin{bmatrix} 0 & 1 \\ -1 & 0 \\ \end{bmatrix} $$, $$j = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} $$, and $$k = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} $$. (This system is very similar to Hamiltonian quaternions, in that $$i^2=-1$$, but note that $$j^2=k^2=1$$ and yet $$j\ne k\ne 1$$.)  As an aside, it is convenient also to have an infinitesimal, $$\epsilon = \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}$$.

In each system, perpendicularity is defined as precisely those situations where $$v\cdot w = 0$$