Symbolic Precision, Quadrance & Spread

Why Exact Algebra Replaces Floating-Point Approximation

Most interactive geometry environments represent points as pairs of 64-bit IEEE-754 floating-point numbers. Within two or three nested circle-circle intersections, rounding error accumulates: three lines that ought to concur at a single Steiner or Kirkman point miss one another by \(10^{-15}\), and a ratio of segments can only be tested as abs(r - 1.618033988749895) < 1e-9.

In GEOMETOR, decimal approximations belong to applied engineering—not to pure geometric inquiry.

Exact Algebraic Coordinates in SymPy

Inside geometor.model, every point, line coefficient, circle equation, and intersection is maintained as a closed-form algebraic expression in SymPy:

\[C = \left(\frac{1}{2},\, \frac{\sqrt{3}}{2}\right), \qquad O = \left(\frac{1}{2},\, \frac{\sqrt{3}}{6}\right), \qquad G = \left(\frac{1 + \sqrt{5}}{2},\, 0\right)\]

When geometor.divine or geometor.pappus tests collinearity, concurrency, harmonic conjugacy, or Extreme and Mean Ratio, sp.simplify(expr) == 0 proves exact mathematical identity across nested quadratic extensions such as \(\mathbb{Q}(\sqrt{3}, \sqrt{5})\).

Rational Trigonometry Invariants

Following Norman J. Wildberger’s Divine Proportions: Rational Trigonometry to Universal Geometry (explored in full in geometor/wildberger and built directly into geometor.model.invariants), GEOMETOR complements Euclidean distance and transcendental angle with purely algebraic quadratic invariants:

  1. Quadrance \(Q(A_1, A_2)\) — squared separation without square roots:

    \[Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2\]

  2. Spread \(s(\ell_1, \ell_2)\) — rational separation of two lines \(a_1 x + b_1 y + c_1 = 0\) and \(a_2 x + b_2 y + c_2 = 0\) without inverse trigonometric functions:

    \[s(\ell_1, \ell_2) = \frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)} \in [0, 1]\]

    Parallel lines have spread \(s = 0\); perpendicular lines have spread \(s = 1\); the equilateral triangle of the Vesica Piscis has exact rational spread \(s = \frac{3}{4}\); and the regular pentagon is governed by the golden spread \(s = \frac{5 - \sqrt{5}}{8}\), satisfying the five-fold spread polynomial \(S_5(s) = s(16s^2 - 20s + 5)^2 = 0\).

  3. Projective Cross-Ratio \((A, B; C, D)\) — the fundamental projective invariant preserved under central projection (geometor/pappus):

    \[(A, B; C, D) = \frac{(A - C)(B - D)}{(A - D)(B - C)}\]

    When \((A, B; C, D) = -1\), the four collinear points form a harmonic range, constructible by unmarked straightedge alone from a complete quadrangle.