---
title: "Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials"
authors:
  - "Cristopher Moore"
  - "Malcolm Smith"
  - "Stephan Mertens"
year: 2026
venue: "arXiv:physics.hist-ph"
arxiv: "2610.05358"
doi: null
source: "https://arxiv.org/abs/2610.05358"
topics:
  - "curiosity"
  - "physics"
  - "classical-mechanics"
  - "history-of-science"
  - "orbits"
seed_rank: 1845
seed_batch: "curiosity-2026-10-07"
reviewed: "2026-10-07"
pool: "physics"
relevance_score: 9
lineage: celestial-mechanics-history
cites:
  - title: "Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials"
    url: "https://arxiv.org/abs/2610.05358"
    year: 2026
    arxiv: "2610.05358"
    doi: null
---

# Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials

## One-sentence takeaway

There is a family of orbits that obey Kepler's second law (equal areas, i.e. angular momentum conserved about the central mass) and also Ptolemy's equant (constant angular speed about another point); for special angular momenta they are written with Legendre polynomials and look like the egg-shaped "ovoid" orbits Kepler tried before the ellipse.

## Why it's lovely

Why you might love this: it is a counterfactual from the history of science made exact. Kepler fought Mars for years and passed through egg- and face-shaped orbits before landing on the ellipse; Moore, Smith and Mertens (Moore and Mertens wrote The Nature of Computation) ask what orbits would have satisfied both his area law and the ancient equant at once. The answer is a family of curves expressible with Legendre polynomials and Legendre functions of the second kind, analytic everywhere except at perihelion, where they are only twice differentiable. A small, perfect physics toy: two incompatible-looking constraints, one closed form, and a picture of the road not taken.

## Key ideas

- Impose Kepler's second law (conserved angular momentum about the central mass) together with an equant point around which angular speed is constant.
- The resulting orbit family can be written, for specific angular momenta, in terms of Legendre polynomials and functions of the second kind.
- Orbits are analytic except at perihelion, where they are twice (but not more) differentiable.
- Shapes resemble Kepler's own "ovoid" (egg-shaped) and "metopoid" (face-shaped) trial orbits.
- Framed as a historical-mathematical exploration of Kepler's battle with the orbit of Mars.

## Caveats

These are not Newtonian inverse-square orbits; the equant is imposed as a kinematic constraint, so this is a mathematical-historical curiosity, not a new dynamics result. Short paper; the abstract does not claim Kepler actually considered these exact curves, only that they resemble his ovoids.

## Links

- arXiv abs: https://arxiv.org/abs/2610.05358
- PDF: https://arxiv.org/pdf/2610.05358