Slicing a cone

What's going on

The cone here has sides at 60° to the horizontal. Call the plane's tilt θ. While θ is less than 60° the plane crosses every line on one cone and the cut closes up: an ellipse, and a circle when θ = 0. At exactly 60° the plane is parallel to one line of the cone, so it never crosses that line and the curve never closes: a parabola. Past 60° the plane is steeper than the cone's side and must cut the upper cone too, giving the two branches of a hyperbola.

One number tells the four apart: the eccentricity e. Every point on the curve is e times as far from a fixed point, the focus, as from a fixed line, the directrix. A circle has e = 0 (the directrix has gone off to infinity), an ellipse has 0 < e < 1, a parabola has e = 1 exactly, and a hyperbola has e > 1. For this cone, e = sin θ ÷ cos 30°, which is why e reaches 1 at θ = 60°.

The right-hand picture draws the curve from that focus-and-directrix rule alone. Change e continuously and the circle, ellipse, parabola and hyperbola slide into one another. For a hyperbola the picture zooms out, keeping the focus-side vertex pinned near the left edge, so that both branches fit; the readout and the corner label say by how much.

Predict first, then slide. Start at 57° and creep up toward 60°. Where does the far end of the ellipse go? Now start at 63° and creep down toward 60°. Where does the second branch go? Both answers are "off to infinity", from opposite sides: the far vertex leaves to the left, and the second branch arrives from the right. At 60° itself neither is anywhere, which is the parabola. The readout tracks the far vertex's distance from the focus, and you can watch it blow up on either side of 60°.

Planets travel on ellipses with the Sun at a focus. A ball thrown in the air follows a parabola. A comet passing once and leaving forever follows a hyperbola. Whispering galleries, satellite dishes and the shadow of a lampshade on a wall are all conics too.

Sheets: Slicing a cone and Kepler's laws and measuring the universe.