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”?)?
ANSWER: Projective Geometric Algebra can define points as the intersection of lines. Dually, lines can be defined as the join of points. Likewise for lines with planes, planes with volumes, etc. in as many dimensions as we like.
Objects defined via joins have a linear orientation; e.g. the join of points A and B is a line through A and B, directed from A to B; the join of lines C and D is a plane containing A and B, DIRECTED HOW? DO WE NEED A ‘RIGHT-HAND RULE’? These objects are invariant in the numberof dimensions.
Objects defined via meets have a rotational orientation; e.g.the meet of planes A and B is a line lying in both (their intersection). Treating this line as an axis, its orientation is the angle between A and B (maybe better with a dot product definition?)
Conformal geometric algebra has objects representing spheres. The projective ‘line at infinity’ can be used as a radius, giving planes (‘infinite spheres’), lines (infinite circles), etc. Points are spheres with zero radius.
The fundamental operation in geometric algebra is reflection. Chaining together reflections can implement rotations (of any magnitude, about any axis); rotations about a ‘point at infinity’ are translations. Reflections in conformal geometric algebra are inversions.
The inverse of a vector, in geometric algebra, points in the same direction, but its magnitude is the reciprocal. Projective GA makes this well-defined for zero, by giving the line at infinity (I think?).