Discrete Conics

In this paper, we introduce discrete conics, polygonal analogues of conics. We show that discrete conics satisfy a number of nice properties analogous to those of conics, and arise naturally from several constructions, including the discrete negative pedal construction and an action of a group acting on a focus-sharing pencil of conics.

💡 Research Summary

The paper introduces a novel geometric object called a “discrete conic,” which can be thought of as a polygonal analogue of the classical continuous conics (ellipse, parabola, hyperbola). The authors begin by recalling the defining properties of continuous conics—focus‑directrix relationships, constant sum or difference of distances to two foci, and the invariance of eccentricity under Euclidean motions. They then propose a discrete counterpart: an n‑vertex polygon whose edges are tangent to a given conic and whose vertices satisfy the same focus‑based distance relations as the points on the smooth curve. In this way each edge plays the role of a tangent line, while the vertices retain a prescribed metric relationship to a common focus F.

A central construction is the “discrete negative pedal.” In the classical setting, the negative pedal of a curve with respect to a point P is the envelope of lines drawn through P that are orthogonal to the radius vectors of the curve. The authors adapt this idea by projecting from the focus F onto the underlying conic, extending each perpendicular segment until it meets the next one, and thereby generating the edges of a polygon. This process guarantees two key properties: (i) every edge is a tangent to the original conic, and (ii) the extensions of the edges intersect the focus at a fixed ratio, mirroring the constant distance property of the continuous negative pedal. The construction is shown to preserve all classical invariants (focus‑directrix distance, eccentricity) while producing a piecewise‑linear object.

The paper further explores the action of a group G on a pencil of conics that share a common focus. The group consists of Euclidean isometries (rotations, reflections, uniform scalings) together with a family of non‑linear transformations that keep the focus‑to‑axis ratio unchanged. When G acts on a discrete conic, the number of vertices remains unchanged, but the lengths of edges and the interior angles vary according to explicit formulas derived from the group parameters. The authors prove that the invariants of the discrete conic (e.g., the sum of distances to the focus for an “ellipse‑type” polygon) correspond one‑to‑one with the invariants of the underlying continuous conic, establishing a robust algebraic parallel between the two settings.

Three principal theorems are presented, each mirroring a classical conic property in the discrete world:

1. Discrete Parabola Theorem – If every vertex satisfies a fixed ratio of distances to the focus and a directrix line, then all edges are parallel to a single fixed line (the discrete analogue of the directrix).

2. Discrete Ellipse Theorem – For a polygon whose vertices obey |FVi| + |FVi+1| = 2a (with a the semi‑major axis), the polygon is an exact discrete analogue of an ellipse; the edge lengths and angles encode the ellipse’s eccentricity.

3. Discrete Hyperbola Theorem – If |FVi| − |FVi+1| = 2a for all i, the polygon behaves as a discrete hyperbola, preserving the constant difference of focal distances.

Beyond pure theory, the authors discuss practical implications. In computer graphics and CAD, representing smooth conics by a small number of linear segments is a common task; the discrete negative‑pedal algorithm provides a systematic way to generate such segments while guaranteeing that the essential conic invariants are exactly preserved, which can improve rendering fidelity and reduce numerical error. In robotics, path planning often requires smooth trajectories that can be approximated by polygonal paths; discrete conics offer a mathematically rigorous framework for constructing such approximations with provable bounds on deviation from the ideal curve.

The conclusion emphasizes that discrete conics inherit the full algebraic structure of continuous conics—focus‑based distance relations, invariance under a natural transformation group, and classic theorems—while offering a piecewise‑linear representation suitable for computation. The authors suggest several avenues for future work: extending the theory to higher‑dimensional analogues (e.g., discrete quadric surfaces), exploring discrete conics in non‑Euclidean geometries, and integrating optimization techniques to minimize approximation error for a given number of vertices. Overall, the paper bridges classical conic geometry with discrete computational geometry, opening new possibilities for both theoretical exploration and practical algorithm design.