Moore, Smith & Mertens (2026) construct an orbit family that satisfies Kepler's second law about one point and an exact Ptolemaic equant about a different point simultaneously

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For an ordinary Keplerian ellipse, Ptolemy's equant point is only an approximate location of uniform angular speed — exact to within O(ε²) in orbital eccentricity, never fully uniform — while Kepler's second law (equal areas in equal time) holds exactly, but only about the true focus, not the empty focus where the historical equant sits. Moore, Smith & Mertens (arXiv:2610.05358, 2026) ask whether a single orbit can satisfy both constraints exactly, around two different points, and report that it can: "we present a family of orbits that satisfy his second law, conserving angular momentum around a central mass, and also fulfill Ptolemy's belief in an equant—a point around which the angular speed is constant."

For specific integer values of the angular momentum (J = n(n+1)), the orbits are expressible in closed form using Legendre polynomials and Legendre functions of the second kind. The resulting curves are egg-shaped — resembling, in the paper's words, the "ovoid" (egg-shaped) or "metopoid" (face-shaped) orbits Kepler himself considered on the way to the ellipse; "metopoid" is Kepler's term as rendered in Donahue's translation, which the paper adopts for its own orbits, not the paper's coinage — and analytic everywhere except at perihelion, where two branches are glued together into a join that is only twice-differentiable rather than fully analytic.

A true ellipse cannot do this: satisfying both constraints exactly requires leaving the ellipse family entirely. Ptolemy's ancient, empirically-motivated device and Kepler's exact dynamical law turn out to be jointly satisfiable after all — just not by the curve either astronomer actually used. See also the Mars-accuracy figure this paper cites for why the equant was abandoned (claim-ptolemy-equant-mars-error-8-arcminutes-tycho-precision-forced-kepler-to-abandon-it) and Kepler's own documented rejection of a near-elliptical alternative on the way to the ellipse (claim-kepler-1609-astronomia-nova-rejected-puffy-cheeked-via-buccosa-oval-orbit).