Projective & Conformal Geometric Algebra (kingdon vs galgebra)

1. Two Covariant Calculi: 2D PGA vs. 2D CGA

Clifford algebra replaces ad-hoc vector cross/dot formulas with a single associative geometric product u * v = (u | v) + (u ^ v) over graded blades:

  • Plane-Based 2D PGA (`Algebra(2, 0, 1)`): Basis \(e_1^2 = e_2^2 = 1, \; e_0^2 = 0\) (e0 is the ideal line at infinity).

    • Grade-1 vectors are Lines: \(\ell = a e_1 + b e_2 + c e_0\).

    • Grade-2 bivectors are Points & Ideal Directions: \(P = e_{12} + x e_{20} + y e_{01}\) (finite point) vs. \(D = dx\,e_{20} + dy\,e_{01}\) (ideal point at infinity, weight \(e_{12} = 0\)!).

    • Division-Free Incidence & Metric: l1 ^ l2 is the meet of two lines (valid even when parallel!); P1 & P2 is the join of two points; (l | P) is the line through P perpendicular to l; and l >> X reflects any point or line across l!

  • Point-Based 2D CGA (`Algebra(3, 1)`): Lifts 2D Euclidean point \((x, y)\) onto the 4D Minkowski null cone (\(n_o^2 = n_\infty^2 = 0, \; n_o \cdot n_\infty = -1\)):

    \[P = x e_1 + y e_2 + \tfrac{1}{2}(x^2 + y^2)n_\infty + n_o, \qquad -2(P_1 \mid P_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2 = Q(P_1, P_2)\]

    In 2D CGA (kingdon with SymPy symbolic scalars), circle geometry collapses to 1-line linear/versor expressions:

    C_circ    = P1 ^ P2 ^ P3                 # OPNS grade-3 circle through 3 points!
        X3        = norm_pt(C_circ * ni * ~C_circ)  # Circumcenter = reflection of infinity ni!
        kappa1    = C1.dual()                    # IPNS grade-1 dual circle O - (1/2)r^2 ni
        power_P   = -2 * (kappa1 | P).e          # Steiner power of P wrt C1!
        lam_rad   = kappa1 - kappa2              # Radical axis of ANY two circles (even disjoint)!
        P_inv     = norm_pt(C1.normalized() >> P) # Möbius circle inversion = sandwich product!

2. Forensic Audit: Three Critical Traps in galgebra

Comparing kingdon and galgebra source code side-by-side exposed three severe pitfalls in galgebra (mv.py) for projective geometry:

  1. Crash on Degenerate PGA Duality (`galgebra/mv.py:1104`): Calling A.dual() computes self * self.ga.I_inv(), which raises AttributeError: 'Ga' object has no attribute 'i_inv' whenever \(I^2 = 0\)—breaking Poincaré duality across all degenerate Projective Geometric Algebras (Cl(2,0,1) and Cl(3,0,1))! kingdon (operators.py:390-440) dispatches cleanly to the metric-free right Hodge complement whenever \(r \ge 1\).

  2. Missing Regressive Join ``&``: galgebra implements ^ (outer) and | (inner), but has no regressive vee join ``&``, preventing standard plane-based PGA point-joining P1 & P2.

  3. Silent Operator Trap on ``>>``: In kingdon, V >> X is the versor sandwich product \(\pm V X \widetilde{V}\). In galgebra (mv.py:809), A >> B is the commutator product \(\tfrac{1}{2}(AB - BA)\)! Writing L >> P in galgebra silently computes a Lie-bracket commutator instead of a reflection!

Furthermore, kingdon (polynomial.py, codegen.py) compiles symbolic blades through RationalPolynomial + poly_cse over exact sympy.Expr leaves—needing only simp_func=sp.cancel when dividing rational functions.

3. Takeaway for GEOMETOR

Use kingdon’s blade conventions (never galgebra’s >>) to expose lightweight 2D PGA and 2D CGA multivector projections over geometor.model elements—giving GEOMETOR division-free projective chains and 1-line conformal circle algebra while preserving Model’s explicit step lineage.