A double cone and a flat plane: change the cutting angle and out come all four curves. Cut perpendicular to the axis for a circle, tilt moderately for an ellipse, tilt exactly parallel to a generating line for a parabola, tilt steeply through both nappes for a two-branched hyperbola. A single number, the eccentricity e, threads the family into a continuum, and that same e classifies orbits under inverse-square gravity: Earth at e ≈ 0.017 is nearly circular, Halley’s comet at 0.967 is drawn out thin, and ʻOumuamua at e ≈ 1.2 passed by once and left for good.
A double cone and a flat plane: change the cutting angle and out come all four curves. Cut perpendicular to the axis for a circle, tilt moderately for an ellipse, tilt exactly parallel to a generating line for a parabola, tilt steeply through both nappes for a two-branched hyperbola. A single number, the eccentricity e, threads the family into a continuum, and that same e classifies orbits under inverse-square gravity: Earth at e ≈ 0.017 is nearly circular, Halley’s comet at 0.967 is drawn out thin, and ʻOumuamua at e ≈ 1.2 passed by once and left for good.
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