Plane Sections of Symmetric Tetrahedral Cones: Inverse Fibers, Poncelet 3-Porisms, Phase Transitions, and Triangle Loci

Abstract

We study triangles cut from a one-parameter family of symmetric trihedral cones, equivalently from tetrahedra whose equilateral base is allowed to recede arbitrarily far. Exact elimination reduces the fixed-height inverse problem to a quartic and exposes fold, escape, and parabolic boundaries in triangle-shape space. The circumconic–inconic pair is shown to be projectively equivalent to concentric circles with an exact order-three Poncelet return map. The work gives a symmetry-marked list of twelve finite transition heights, certified special cases and limits, barycentric projection formulae, and a strand analysis whose projected locus lies on a quintic, including an exact curvature jump. This deposit contains the 38-page manuscript and a complete reproducibility archive with source, figures, Wolfram tools, certificates, and manifest.