W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas

Quadratic Focus-Directrix and Bifocal Loci Over Arbitrary Fields

A conic is the zero locus of a quadratic polynomial proportion C = <a:b:c:d:e:f>; metrically, Q(X, l) = R defines a ribbon, Q(X, F_0) = Q(X, l_dir) defines a parabola, the squared-sum/difference bifocal locus A(Q(X, F_1), Q(X, F_2), S) = 0 defines a quadrola (unifying ellipses and hyperbolas into one irreducible algebraic curve), and Q(X, l_1) +/- Q(X, l_2) = R defines a grammola.

Interactive GEOMETOR construction for W.20 (Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas). Hover any geometric label in the text or figure to highlight the corresponding element.

F_0 : Point D_1 : Point D_2 : Point F_1 : Point F_2 : Point ===== l_dir = line(D_1, D_2) V_p = midpoint(F_0, foot(F_0, l_dir)) parabola = parabola_locus(F_0, l_dir) C_q = midpoint(F_1, F_2) quadrola = quadrola_locus(F_1, F_2, S)

Mathematical Synthesis

Classically, an ellipse (\(d_1 + d_2 = k\)) and a hyperbola (\(|d_1 - d_2| = k\)) are separated by absolute values and square roots. In Universal Geometry (Wildberger2005), squaring \((\pm\sqrt{Q_1} \pm \sqrt{Q_2} \pm \sqrt{S}) = 0\) via Archimedes’ function produces a single algebraic curve—the quadrola quadrola centered at Cq with foci F1, F2—given by \(\mathcal{A}(Q(X, F_1), Q(X, F_2), S) = 0\)! Over \(\mathbb{R}\), the same equation defines an ellipse when \(S > Q(F_1, F_2)\) and a two-branched hyperbola when \(0 < S < Q(F_1, F_2)\), while over \(\mathbb{F}_p\) both regimes coexist seamlessly without ordering!

Similarly, equating point-to-focus quadrance \(Q(X, F_0)\) and point-to-directrix quadrance \(Q(X, \ell_{\text{dir}})\) yields the parabola parabola with vertex Vp and focus F0 on non-null directrix ldir, while constant quadrance to a single line yields a ribbon (two parallel lines) and constant sum/difference of quadrances to two lines yields a grammola.

Key Algebraic Identities

\[\text{Parabola:}\quad Q(X, F_0) = Q(X, \ell_{\text{dir}}) \iff (a^2 + b^2)\big((x - x_0)^2 + (y - y_0)^2\big) = (a x + b y + c)^2\]

\[\text{Quadrola:}\quad \mathcal{A}\big(Q(X, F_1),\; Q(X, F_2),\; S\big) = 0, \qquad \text{Grammola:}\quad Q(X, \ell_1) \pm Q(X, \ell_2) = R\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Parabola 6-proportion	<100:0:0:0:-400:400>	Exact focus-directrix polynomial
Quadrola 6-proportion (S = 100)	Center = [10, 2]	Unifies ellipse & hyperbola into 1 polynomial
Rational sample check on quadrola	q([390/53, -74/53]) = 0	Verified == 0

Cross-Repository Canon

Executable Demonstration	W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas — Code (Standalone model.py & w-20.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Classical Canon	Apollonius (Conics, Books I–III)
GEOMETOR Elements	G.3 extremities of a segment, G.4 segment
GEOMETOR Fundamentals	F.1 Albert’s Parabola, F.17 Ellipse, F.37 Euler’s Inradius Parabola
Universal Unification	Ellipses and hyperbolas share the single polynomial A(Q_1, Q_2, S) = 0

Constructive Dependency Graph

Depends on: W.03 Quadrance, Midpoints & Point-to-Line Separation, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Required for: W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws, W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity, W.28 Rational Polar Coordinates, Classical Curves & The Ellipson