Conical Existence of Closed Curves on Convex Polyhedra

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that “live on a cone,” in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is “visible to” a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of “source unfolding” of the entire surface of P to one non-overlapping piece, as reported in a companion paper.

💡 Research Summary

The paper investigates a fundamental geometric property of simple closed directed curves C drawn on the surface of a convex polyhedron P. The authors introduce the notion that a curve “lives on a cone”: there exists a cone Λ and a vertex‑free neighborhood N of C on one side (left or right) such that the union C∪N can be isometrically mapped onto Λ, with the cone’s apex a lying inside the image of C. A cone here is a developable surface with zero curvature everywhere except at its apex, whose total incident surface angle (the cone angle) does not exceed 2π.

To study which curves have this property, the authors define eight curve classes (four basic types and their loop variants):

1. Geodesic – L(p)=R(p)=π at every point.
2. Quasigeodesic – L(p)≤π and R(p)≤π (both sides convex).
3. Convex – L(p)≤π (left side convex).
4. Reflex – R(p)≥π (right side reflex).

Loop variants allow a single exceptional point where the angle condition may be violated. The relationships among these classes are summarized in Table 1; for instance, every geodesic is a quasigeodesic, and a convex curve is reflex on the opposite side when it contains no vertices.

The paper’s central results are two lemmas and a theorem that connect these angle conditions to the existence of a cone. Using the Gauss‑Bonnet theorem, the authors relate the total left turn τL of C and the total curvature ΩL of the region to the left of C by τL+ΩL=2π. For a convex curve, τL≥0, implying ΩL≤2π. If ΩL<2π, the authors repeatedly apply a vertex‑merging operation (originally due to Alexandrov) to the interior vertices of the left half‑surface PL. Each merge replaces two vertices by a single vertex whose curvature is the sum of the two, preserving the total curvature and never moving the merged vertex onto C. After finitely many merges only one interior vertex remains; this vertex becomes the apex of a cone ΛL on which C∪NL lives isometrically. When ΩL=2π, a limiting argument shows that the cone degenerates to a cylinder (apex at infinity), still satisfying the “live on a cone” condition.

A symmetric argument shows that a reflex loop whose opposite side is convex lives on a cone on its reflex side. Consequently, any convex curve (including quasigeodesics) that contains at most one polyhedron vertex lives on a cone on both sides. The authors also prove that the cone associated with a given C is unique (Lemma 3), so the construction does not depend on the order of vertex merges.

Beyond existence, the paper establishes a visibility property: every generator (half‑line from the apex) of the cone meets C in exactly one point. Hence C is “visible” from the apex, a crucial fact for later unfolding applications because it guarantees that when the cone is flattened onto the plane, C does not self‑overlap.

The practical payoff is twofold. First, any curve that lives on a cone can be developed onto the plane without self‑intersection, extending earlier results about non‑self‑intersecting developments of geodesics and quasigeodesics. Second, when a curve lives on cones on both sides (for example, a quasigeodesic or a pair of distinct cones), the authors’ companion paper shows how to unfold the entire polyhedron P into a single non‑overlapping planar piece. The method cuts along the two cones, flattens each side separately, and then glues the two planar developments together along C, yielding a new type of “source unfolding” that generalizes the classic two‑source unfolding.

The paper includes illustrative examples: a convex curve on an icosahedron that is a quasigeodesic, a convex curve on a cuboctahedron that fails the classification on one side, and a detailed vertex‑merging construction on a doubly‑covered flat pentagon where ΩL=2π leads to a cylindrical cone. These examples clarify how the theoretical machinery operates in concrete polyhedral settings.

In summary, the authors provide a rigorous geometric framework that determines when a closed curve on a convex polyhedron can be regarded as lying on a cone, prove that such curves are always apex‑visible, and leverage these facts to guarantee non‑self‑intersecting planar developments and to enable a novel, globally non‑overlapping source unfolding of the entire polyhedron. The work blends classic results (Gauss‑Bonnet, Alexandrov’s gluing theorem) with new combinatorial insights into curve classes, opening avenues for further exploration of more complex curves and non‑convex polyhedral surfaces.