G.530 solid contained parallel — Code

XI.24

If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.

"""Programmatic construction for Book XI, Prop. 23 Simson Case III & Prop. 24 (XI.24).

        Upper figure: Simson's Case III (O outside triangle LMN, with auxiliary vertices
        P, Q, R and point S such that triangle NRS is congruent to LPM).
        Lower figure: Book XI, Proposition 24 parallelepiped CDHG contained by six
        parallel planes with face diagonals AH and DF.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct Simson's Case III and Book XI, Prop. 24 figures using geometor.model."""
            model = Model("XI.24")

            # Simson Case III (Upper figure)
            O = model.set_point(sp.Rational(355, 100), sp.Rational(-326, 100), classes=["given"], label="O")
            L = model.set_point(sp.Rational(378, 100), sp.Rational(-214, 100), classes=["given"], label="L")
            M = model.set_point(sp.Rational(246, 100), sp.Rational(-278, 100), classes=["given"], label="M")
            N = model.set_point(sp.Rational(459, 100), sp.Rational(-288, 100), classes=["given"], label="N")
            P = model.set_point(sp.Rational(352, 100), sp.Rational(-296, 100), label="P")
            Q = model.set_point(sp.Rational(356, 100), sp.Rational(-282, 100), label="Q")
            R = model.set_point(sp.Rational(368, 100), sp.Rational(-298, 100), label="R")
            S = model.set_point(sp.Rational(392, 100), sp.Rational(-438, 100), label="S")

            model.construct_circle(O, L)

            for u, v in [
                (L, M), (M, N), (N, L),
                (L, O), (M, O), (N, O),
                (L, P), (M, P), (M, Q), (N, Q), (L, R), (N, R),
                (L, S), (R, S), (N, S),
            ]:
                model.set_segment(u, v)

            for tri in [
                [L, M, N], [L, O, M], [M, O, N], [N, O, L],
                [L, P, M], [M, Q, N], [N, R, L], [N, R, S], [L, N, S], [L, R, S],
            ]:
                model.set_polygon(tri)

            # Proposition 24 Parallelepiped CDHG (Lower figure)
            A = model.set_point(sp.Rational(306, 100), sp.Rational(-692, 100), classes=["given"], label="A")
            B = model.set_point(sp.Rational(319, 100), sp.Rational(-638, 100), classes=["given"], label="B")
            C = model.set_point(sp.Rational(345, 100), sp.Rational(-716, 100), classes=["given"], label="C")
            D = model.set_point(sp.Rational(332, 100), sp.Rational(-770, 100), classes=["given"], label="D")
            E = model.set_point(sp.Rational(488, 100), sp.Rational(-770, 100), classes=["given"], label="E")
            F = model.set_point(sp.Rational(501, 100), sp.Rational(-716, 100), classes=["given"], label="F")
            G = model.set_point(sp.Rational(462, 100), sp.Rational(-692, 100), classes=["given"], label="G")
            H = model.set_point(sp.Rational(475, 100), sp.Rational(-638, 100), classes=["given"], label="H")

            # 12 edges + 2 face diagonals AH, DF
            for u, v in [
                (A, B), (B, C), (C, D), (D, A),
                (H, G), (G, E), (E, F), (F, H),
                (B, H), (A, G), (C, F), (D, E),
                (A, H), (D, F),
            ]:
                model.set_segment(u, v)

            # 6 parallelogram faces + 2 triangles ABH, DCF + solid hull
            for poly in [
                [A, B, C, D],
                [H, G, E, F],
                [A, B, H, G],
                [D, C, F, E],
                [B, C, F, H],
                [A, D, E, G],
                [A, B, H],
                [D, C, F],
            ]:
                model.set_polygon(poly)

            model.set_wedge(B, A, A, H)
            model.set_wedge(C, D, D, F)

            return model


        if __name__ == "__main__":
            construct_g1()