algebra

Projective Geometric Algebra

A superior new/rediscovered/reworked formulation of Vector Geometry, in which all objects ( points,lines,planes,translations,rotations...) are represented in a uniform way, e.g. as multi-vectors in ℝ(3,0,1).

It comes with a set of operators

• a∧b outer/exterior wedge product: plane “meet“ plane⇒line , line∧line⇒point (intersection)
• ...
• a⟇b span of different objects

Which Projective Geometric Algebra can represent/distinguish the following entities:

directedness is identical to non-commutedneds of sub algebras

All abstract objects have anchored variants (with possible origin ≠ 0)

point ∨ point = line ∞-A-B-∞ a∞b (allow directed lines?) point ⟇ point = path segment A-B
point → point = ray/direction A-B-∞ (line with one source) point - point = vector A->B = b-a (directed segment)
point ↦ point = arrow A->B at A (anchored vector) point ⟇ direction = ray (e.g. {x:0->∞} )
line ∨ line = plane object | object = collection / assembly of objects ≠ a+b a*b segment ⟇ segment = face element arrow ⟇ segment= face, anchored at start of first arrow arrow ⟇ arrow = face, anchored at start of first arrow segment ⟇ face = volume ray ⟇ ray = sector (e.g. first quadrant in ℝ²)

Note how ⟇ is similar to classical dx ∧ dy which is different to e12 ∨ e23

operations:

direct segment s := nearest s to origin -> furthest s to origin