W.28 Rational Polar Coordinates, Classical Curves & The Ellipson

Parametrizing Curves by [s, Q] and Slicing the Cubic Triple-Spread Surface

Replacing polar coordinates (r, theta) with rational polar coordinates [s, Q] where s = x^2 / (x^2 + y^2) (spread to the y-axis) and Q = x^2 + y^2 converts classical polar curves into exact polynomial relations—Bernoulli’s lemniscate s = (Q + 2)/4 (or Q = a*(2*s - 1)), Descartes’ folium (2*s - 1)^2*(s^2 - s + 1)^2*Q^2 - 18*s*(1 - s)*(1 - 3*s + 3*s^2)*Q + 81*s^2*(1 - s)^2 = 0, and the deltoid/ellipson S_3(s) law—while (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3 cuts out the tetrahedral Ellipson.
lemniscate cardioid folium ellipson_slice ellipson_c_half l_ref O = [0, 0] A_ref = [0, 4] Lemniscate: Q = 16*(2*s - 1) Ellipson: S(u,u,4u(1-u))=0 & s_3=1/2 circle OO ArefAref + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.28 (Rational Polar Coordinates, Classical Curves & The Ellipson). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A_ref : Point ===== l_ref = line(O, A_ref) lemniscate = curve_lemniscate(O, l_ref) cardioid = curve_cardioid(O, l_ref) folium = curve_folium(O, l_ref) ellipson_slice = curve_ellipson_section()

Mathematical Synthesis

Traditional polar coordinates \((r, \theta)\) mix a radical \(r = \sqrt{x^2 + y^2}\) with a transcendental angle \(\theta\), defined only modulo \(2\pi\). Norman J. Wildberger’s rational polar coordinates \([s, Q]\) (Wildberger2005, Chapter 27 & Appendices A–B) pair the polar spread \(s \equiv x^2 / (x^2 + y^2)\) measured from the vertical \(y\)-axis lref (through O and Aref) with the radial quadrance \(Q \equiv x^2 + y^2\), aligning directly with navigation azimuths and 3D rational spherical coordinates.

Classical curves simplify dramatically in \([s, Q]\): Jakob Bernoulli’s figure-eight lemniscate (lemniscate, \((x^2 + y^2)^2 = a(x^2 - y^2)\)) becomes the linear equation \(Q = a(2 s - 1)\) (\(s = (Q + 2)/4\) for \(a = 2\)), while René Descartes’ folium (folium, \(x^3 + y^3 + 3 x y = 0\)) eliminates quadrant signs via \((1 - 3s + 3s^2)^2 - 4 s^3(1 - s)^3 = (2s - 1)^2(s^2 - s + 1)^2\), and the cardioid (cardioid) admits a pure rational parametrization. Crowning the codex is the Ellipson (ellipsonslice): the inflated-tetrahedron cubic surface inside \([0, 1]^3\) defined by the Triple Spread Formula, whose horizontal slices \(s_3 = c\) are ellipses inscribed in \([0, 1]^2\) (an exact circle at \(c = 1/2\)!) and whose oblique slice \(s_1 + s_2 + s_3 = 2\) satisfies \(S_3(s) = (36 Q - 5)^2 / (6 Q)^3\)!

Key Algebraic Identities

\[s \equiv \frac{x^2}{x^2 + y^2}, \quad Q \equiv x^2 + y^2 \implies \text{Lemniscate: } Q = a(2 s - 1), \quad \text{Folium: } (2s - 1)^2(s^2 - s + 1)^2 Q^2 - 18 s(1 - s)(1 - 3s + 3s^2) Q + 81 s^2(1 - s)^2 = 0\]
\[\text{Ellipson }\mathcal{E} \subset [0, 1]^3:\quad (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4 s_1 s_2 s_3 \quad \Big(s_3 = \tfrac{1}{2} \implies \big(s_1 - \tfrac{1}{2}\big)^2 + \big(s_2 - \tfrac{1}{2}\big)^2 = \tfrac{1}{4}\Big)\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Lemniscate [s, Q] law Q == 16*(2*s - 1)

Q = 4608/337, 16*(2*s - 1) = 4608/337

Linear in [s, Q] (s to y-axis)

Folium of Descartes [s, Q] quadratic (t=2)

s=1/5, Q=20/9 => residual = 0

Matches Wildberger App. A p. 286

Ellipson horizontal slice s_3 = 1/2

S(4/5, 9/10, 1/2) == 0

Exact inscribed circle in [0, 1]^2!

Cross-Repository Canon

Executable Demonstration

W.28 Rational Polar Coordinates, Classical Curves & The Ellipson — Code (Standalone model.py & w-28.svg)

Thematic Part

Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves

Classical Curves

Jakob Bernoulli (1694) & René Descartes (1638)

GEOMETOR Fundamentals

F.22 Functions, F.51 Apollonian Circles and the Problem of Apollonius

Triple Spread Geometry

Geometric locus of Law V (W.10) in spread space (s_1, s_2, s_3)

Universal Synthesis

Completes the 28-Principle GEOMETOR Wildberger Codex

Constructive Dependency Graph

digraph dependency { rankdir=TB; bgcolor="transparent"; node [fontname="Helvetica,Arial,sans-serif", fontsize=10, shape=box, style="rounded,filled", margin="0.15,0.08"]; edge [color="#888888", arrowsize=0.7]; "W.03" [label="W.03\nQuadrance, Midpoints & Point-to...", fillcolor="#d8dee9", color="#4c566a", URL="../w-03/index.html"]; "W.04" [label="W.04\nSpread, Cross, Twist & Line Sep...", fillcolor="#d8dee9", color="#4c566a", URL="../w-04/index.html"]; "W.10" [label="W.10\nLaw V: Triple Spread, Triple Cr...", fillcolor="#d8dee9", color="#4c566a", URL="../w-10/index.html"]; "W.17" [label="W.17\nCircle Geometry: Subtended Spre...", fillcolor="#d8dee9", color="#4c566a", URL="../w-17/index.html"]; "W.20" [label="W.20\nMetrical Conics: Circles, Ribbo...", fillcolor="#d8dee9", color="#4c566a", URL="../w-20/index.html"]; "W.28" [label="W.28\nRational Polar Coordinates, Cla...", fillcolor="#ffff99", color="#d4a017", penwidth=2.0, URL="../w-28/index.html"]; "W.03" -> "W.28"; "W.04" -> "W.28"; "W.10" -> "W.28"; "W.17" -> "W.28"; "W.20" -> "W.28"; }

Depends on: W.03 Quadrance, Midpoints & Point-to-Line Separation, W.04 Spread, Cross, Twist & Line Separation Ratios, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points, W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas