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

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.

Exact geometor.model SVG output of model.py for W.28 (Rational Polar Coordinates, Classical Curves & The Ellipson).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.28: Rational Polar Coordinates, Classical Curves & The Ellipson.

        Enunciation: 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.
        """

        from __future__ import annotations

        import sys
        from pathlib import Path

        _ROOT_SRC = Path(__file__).resolve().parents[3] / "src"
        if str(_ROOT_SRC) not in sys.path:
            sys.path.insert(0, str(_ROOT_SRC))

        from geometor.wildberger.models import (  # noqa: E402
            TopicBuildResult,
            assert_zero_floats,
            build_principle_model,
        )
        from geometor.wildberger.render import render_svg  # noqa: E402


        def construct_w_28() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.28.

            Returns:
                Populated `TopicBuildResult` for W.28 (Rational Polar Coordinates, Classical Curves & The Ellipson).
            """
            result = build_principle_model("W.28")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.28 symbolic construction, render `w-28.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_28()
            svg_out = render_svg(
                bundle.model,
                here / "w-28.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.28: Rational Polar Coordinates, Classical Curves & The Ellipson")
            print(f"  Elements in model: {len(bundle.model)}")
            print(f"  Rendered SVG:      {svg_out.name}")
            for label, val, status in bundle.verifications:
                print(f"  [OK] {label}: {val} ({status})")


        if __name__ == "__main__":
            main()