CindyJS: Complex Projective \(\mathbb{CP}^2\), Cayley–Klein & PSLQ

  • Repository: _refs/CindyJS (https://github.com/CindyJS/CindyJS, Apache-2.0, RichterGebert2011)

  • Scale Audited: src/js/libgeo/ (GeoOps.js, Conics.js, Tracing.js, Prover.js) + src/js/libcs/ (List.js, PSLQ.js)

  • Mathematical Foundation: Arthur Cayley, Felix Klein, and Jürgen Richter-Gebert’s complex projective geometry over \(\mathbb{CP}^2\) with Riemann-surface homotopy continuation

1. Total Duality in \(\mathbb{CP}^2\) & Circular Points \(I, J\)

CindyJS embeds every planar point \(p = [x, y, z]\) and every line \(\ell = [a, b, c]\) in the complex projective plane \(\mathbb{CP}^2\) (List.js):

  • Unified Join & Meet: Joining two points and intersecting two lines are the exact same 3D vector cross product List.cross(u, v) (GeoOps.js:156-202), handling parallel lines (z = 0) seamlessly with zero conditionals.

  • First-Class General Conics (`Conics.js`, `GeoOps.js:1032`): Unlike every ITP and sympy.geometry, CindyJS represents all conics (circles, rotated ellipses, parabolas, hyperbolas, and degenerate line pairs) uniformly as \(3 \times 3\) complex symmetric matrices \(C\):

    • Polar of point \(p\): matrix-vector product l = C * p;

    • Dual envelope conic: cofactor adjugate C_dual = List.adjoint3(C) (division-free even on rank-2 degenerate conics!);

    • Pole of line \(\ell\): p = adjoint3(C) * l;

    • 5-Point Conic (`ConicBy5`): Plücker linear combination \(C = \mu D_1 - \lambda D_2\) of two rank-2 line-pair matrices in pure multiplication/subtraction!

  • Cayley–Klein Metric via Circular Points at Infinity (`List.js:111-126`): Defining circular points at infinity List.ii = [1, i, 0] and List.jj = [1, -i, 0], every Euclidean circle is simply a conic passing through \(I\) and \(J\) (CircleMP calls ConicBy5 with ii and jj!), perpendicularity is harmonic conjugacy l1 * diag(1,1,0) * l2 == 0, and Euclidean angle is Laguerre’s cross-ratio formula \(\theta = \frac{i}{2}\ln\operatorname{CR}(A_\infty, B_\infty; I, J)\)!

2. Three Algorithmic Gems in Tracing.js, Prover.js & PSLQ.js

  1. Riemann-Surface Semicircle Homotopy (`Tracing.js:198-200`): In ordinary dynamic geometry, dragging two intersecting circles apart until they become disjoint and bringing them back causes quadratic sqrt branches to swap unpredictably (the jumping-elements bug). CindyJS keeps complex-conjugate intersections alive over \(\mathbb{C}^3\) and detours each parameter step \(t \in [0, 1]\) along an upper-half-plane complex semicircle:

    // src/js/libgeo/Tracing.js:198-200 (complex semicircle detour around branch points!)
        var alpha = t * Math.PI;
        var tc = CSNumber.complex(0.5 - 0.5 * Math.cos(alpha), 0.5 * Math.sin(alpha));

    Passing around real branch-point singularities on the Riemann surface makes element identity globally continuous!

  2. Randomized Complex Theorem Proving (`Prover.js`): Evaluates construction DAGs at 3 random complex perturbations (Schwartz–Zippel lemma) to auto-discover concurrencies and incidences at interactive frame rates.

  3. Live Integer-Relation Surd Recovery (`PSLQ.js:1-215`): Implements Ferguson–Bailey PSLQ (guess(x)), recovering rationals and quadratic surds like 3/2 + 1/2*sqrt(5) from floating-point coordinates—capped at vector dimension n = 3 (degree \(\le 2\)).

3. Takeaway for GEOMETOR

Port CindyJS’s \(3 \times 3\) symmetric conic matrix, adjoint3 dual, and ConicBy5 Plücker pencil directly into exact SymPy arithmetic in geometor.pappus (Blueprint 5), and generalize PSLQ.js’s n=3 guess(x) to arbitrary algebraic degree \(d\) via 100-digit mpmath.pslq in geometor.divine (Blueprint 9).