Reading the shapes of star orbits from a single angle. An Evidence Press briefing. Many stars live in wide pairs, so far apart that one orbit takes thousands of years. Astronomers would like to know how stretched, or eccentric, these orbits are, because that records how the pairs formed. But nobody can watch a whole orbit. What the Gaia satellite can measure is a snapshot: the line joining the two stars, and the direction in which they are moving relative to each other. The angle between those two directions carries information about the orbit's shape. This release works out that information exactly, for pairs caught at a random moment and seen from a random direction. It gives a short closed formula for how likely each angle is, for an orbit of any eccentricity. It then proves that the full spread of angles across many pairs pins down the spread of eccentricities completely, answering a question raised by Hwang, Ting and Zakamska in 2022. The catch is that fine detail is almost invisible, so in practice only broad features can be recovered. The paper also gives simple averages of the angle that estimate useful summaries directly, such as the average squared eccentricity, and it identifies which common ways of selecting stars keep these results exact. The proofs are written out, and computer simulations of Kepler orbits check the formulas. No real star catalogue is analysed, and real data would need a model of measurement errors. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on 31 October 2026, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.