Projective Geometric Algebra

Projective geometry is a particularly simple form of geometry: it consists only of points and lines; the only equipment needed (other than pen & paper) is a straight-edge; the only skills needed are to identify where two lines cross, and to draw a line between two points.

Although projective geometry is limited, this actually makes it more general. For example, it works in exactly the same way for a flat piece of paper, for the spherical geometry of astronomy and cartography, and for the hyperbolic geometry of relativity; since it doesn’t involve notions like “angle”, “distance”, “parallel” etc. which behave very differently across those regimes.

Constructing a Number Line

Draw two points and a line between them:

Projective geometry can be expressed using cross products (e.g. see https://youtu.be/puMYfJTFdgQ?t=964 )

Cross products are icky. Can we use geometric products instead?

Going further: projective geometry involves lines and points, which are dual/symmetric. Geometric algebra doesn’t involve points or (unbounded) lines; it starts with vectors. What’s the nicest way to capture these things? Is it by going to a higher dimension, like the homogeneous coordinates of projective geometry (e.g. subspaces crossing a “viewing plane”?)?

(Conformal geometric algebra?)