Projective geometric algebra
2D PGA overview
A compact synthesis of the 2D PGA path keeps the main operations close together, so invariants and changing representatives can be compared directly.
Start with the foundation pages if grades, bivectors, wedge, or the geometric product need review: Grades and wedge and Geometric product and motion. The comparison uses those ideas with the plane-based PGA convention from the 2D articles.
No new notation is introduced here. The familiar operations are collected in one place: homogeneous scale, meet/join, reflection, motor action, and frame-relative coordinates.
The Common Shape
The same data keeps reappearing:
P = 1e12 - x * 1e02 + y * 1e01
line = a * 1e1 + b * 1e2 + c * 1e0
result = operator >>> PSometimes the operator is a mirror line. Sometimes it is an even motor. Meet and join use their own products. Keeping each value in multivector form keeps the algebraic representative explicit.
What To Revisit
For scale, go back topoints, lines, and homogeneous scale. For operation details, revisit meet and join orreflections. For off-origin motion, comparetwo reflections withmotors and exponentials. For polygon clipping, revisitshape intersections.
Frame coordinates connect the algebra back to graphics vocabulary:world space, local space, model transforms, view transforms, and inverse frame actions.
What Changes In 3D
The 1-up homogeneous model of Algebra(2,0,1) is an intermediate model before physical 3D PGA. Physical 3D PGA changes the object grades: planes become grade-1, lines become grade-2, and points become grade-3. Grade tracking and representative normalization still matter, with one more spatial basis direction.