SymPy Geometry (1.14.0): Symbolic CAS Architecture & Live Bug Audit

  • Location Audited: .venv/lib/python3.13/site-packages/sympy/geometry/ (11 modules, 12,071 LOC) + sympy/combinatorics/polyhedron.py (1,020 LOC), BSD-3-Clause

  • Role in GEOMETOR: Provides the base Point2D, Line2D, Segment2D, Circle, and Polygon entity classes wrapped—and strategically bypassed—by geometor.model

1. Architectural Inventory & Solver Bottlenecks

sympy.geometry models affine Cartesian entities over general sympy.Basic expressions (point.py, line.py, ellipse.py, parabola.py, polygon.py, plane.py, curve.py, entity.py, util.py). While intuitive for 1-step textbook exercises, our line-by-line audit explains why naive ``sympy.geometry`` grinds to a halt after 3–4 compass steps:

  • Generic ``solve()`` Inside ``Ellipse.intersection`` (`ellipse.py:676-691`): Intersecting two Circle objects substitutes quadratic Cartesian equations into SymPy’s generic Buchberger/polynomial solve(...) instead of subtracting circles to form the linear radical axis—exploding exponentially on nested quadratic radicals \(\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{10 - 2\sqrt{5}})\).

  • Pervasive Eager ``simplify()`` (`37` Call Sites): Methods across line.py, ellipse.py, and polygon.py invoke general heuristic simplify() on symbolic denominators rather than canonical arithmetic in an algebraic number field (QQ.algebraic_field).

  • Axis-Aligned Conic Prison & Missing ``Hyperbola``: Ellipse only supports horizontal/vertical axes; calling Ellipse.rotate(theta) (ellipse.py:1244-1255) raises NotImplementedError("Only rotations of multiples of pi/2 are currently supported for Ellipse."), and ``Hyperbola`` does not exist at all!

  • Total Absence of Inversive, Coaxial & Projective APIs: Circle has no power, radical_axis, invert, polar, or pole; parallel Line.intersection returns []; cross_ratio is absent; and combinatorics.Polyhedron stores vertex permutation groups only with zero 3D geometric coordinates.

2. Live Bug #1: Double Algebraic Bug in Triangle.excenters!

Auditing sympy/geometry/polygon.py:2714-2726 in SymPy 1.14.0 uncovered a live double bug in Triangle.excenters:

# sympy/geometry/polygon.py:2714-2726 (LIVE BUG IN SYMPY 1.14.0!)
        excoords = {
            "a": (
                simplify(-a*x[0] + b*x[1] + c*x[2]/(-a + b + c)),  # <-- 1. MISSING PARENTHESES!
                simplify(-a*y[0] + b*y[1] + c*y[2]/(-a + b + c)),  # divides ONLY c*x[2] by (-a+b+c)!
            ), ...
        }
        # AND 2. WRONG VERTEX-SIDE PAIRING: s[0] = Segment(v[0], v[1]) has length c (opposite v[2]),
        # NOT opposite v[0]! Even SymPy's own test_polygon.py hardcoded the corrupted output!

For the standard 3-4-5 right triangle Triangle(Point(0, 0), Point(3, 0), Point(0, 4)), the true excenter opposite (0, 0) is (-6, -6), whereas SymPy 1.14.0’s Triangle.excenters[t.sides[0]] returns Point2D(12, 2)—which does not even lie on the angle bisectors!

3. Live Bug #2: Float Collinearity False-Positive in Point.affine_rank!

In sympy/geometry/point.py:331-335, Point.affine_rank (called by Point.is_collinear!) tests whether exact symbolic matrix entries vanish using a 2-digit floating-point threshold:

# sympy/geometry/point.py:331-335 (LIVE PRECISION BUG!)
        # Exact non-collinear points Point(0,0), Point(1,0), Point(1, Rational(1, 10**13))
        # evaluate abs((10**-13).n(2)) < 1e-12 == True -> FALSELY RETURNS COLLINEAR!

4. Takeaway for GEOMETOR

  1. Validates geometor.model’s architectural mandate to never call ``Circle.intersection`` or ``Point.is_collinear``, using closed-form radical-axis intersections (intersections.py), exact signed_area2 == 0, and ANP canonicalization instead.

  2. Gives GEOMETOR two crisp, high-impact upstream SymPy pull requests (Blueprint 10) fixing Triangle.excenters and Point.affine_rank.