W.04 Spread, Cross, Twist & Line Separation Ratios — Code

For two non-null lines l_1 = <a_1 : b_1 : c_1> and l_2 = <a_2 : b_2 : c_2>, the spread s(l_1, l_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 + b_1^2)*(a_2^2 + b_2^2)) depends only on the unoriented lines, vanishes iff l_1 || l_2, equals 1 iff l_1 perp l_2, and equals Q(B, C) / Q(A, B) for any altitude BC dropped from B on l_1 to C on l_2.

Exact geometor.model SVG output of model.py for W.04 (Spread, Cross, Twist & Line Separation Ratios).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.04: Spread, Cross, Twist & Line Separation Ratios.

        Enunciation: For two non-null lines l_1 = <a_1 : b_1 : c_1> and l_2 = <a_2 : b_2 : c_2>, the spread s(l_1, l_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 + b_1^2)*(a_2^2 + b_2^2)) depends only on the unoriented lines, vanishes iff l_1 || l_2, equals 1 iff l_1 perp l_2, and equals Q(B, C) / Q(A, B) for any altitude BC dropped from B on l_1 to C on l_2.
        """

        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_04() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.04.

            Returns:
                Populated `TopicBuildResult` for W.04 (Spread, Cross, Twist & Line Separation Ratios).
            """
            result = build_principle_model("W.04")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.04 symbolic construction, render `w-04.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_04()
            svg_out = render_svg(
                bundle.model,
                here / "w-04.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.04: Spread, Cross, Twist & Line Separation Ratios")
            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()