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Ladder

Conic sections — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

    One cone, four curves

    The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.

    rung 1 · geometry
  2. Every ray from one focus arrives at the other. An ellipse with its two foci and 13 rays leaving the first. Each is reflected at the curve by the ordinary law of reflection and each passes through the second focus.

    Every ray comes back to the other focus

    An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.

    rung 2 · geometry
  3. Every ray aimed at one focus is turned towards the other. A hyperbola with its two foci and 13 rays aimed at the far one. Each strikes the near branch from outside and is turned towards the near focus — which is the property a Cassegrain telescope's secondary mirror uses.

    Aimed at one focus, turned towards the other

    An ellipse sends every ray leaving one focus through the other. A hyperbola does something that sounds like the same sentence and is not — it takes a ray aimed at the far focus and turns it towards the near one, which is what the second mirror of a telescope is for.

    rung 3 · geometry
  4. Two families of conics, crossing at right angles. 4 ellipses and 3 hyperbolas with the same pair of foci. Every ellipse meets every hyperbola at a right angle, checked at all 12 crossings.

    Two families that cross at right angles

    Fix two points and draw every ellipse with those foci, then every hyperbola with the same two. Each curve of the first family meets each of the second at a right angle, so between them they are a coordinate system — and the reason is one sentence about angle bisectors.

    rung 4 · geometry
  5. One sign decides which curve it is. 3 conics drawn from the general quadratic, each labelled with its discriminant B² − 4AC and the curve that sign names, checked against how many times the curve meets a large circle.

    One sign decides which curve

    The general quadratic in two variables has six coefficients and draws a conic. Which of the four it draws is settled by a single combination of three of them, and the other three cannot change the answer however they are chosen.

    rung 5 · geometry

All ladders