Conic Sections

The idea

The circle, ellipse, parabola, and hyperbola are usually met as four separate equations. They are one family, and the family has a single origin: each is the curve where a flat plane cuts a cone. That is the whole content of the name conic section, and it is a statement about three-dimensional geometry that a flat diagram can only gesture at.

Which curve you get depends on one number, the tilt of the plane compared with the slope of the cone's side. A plane perpendicular to the axis cuts a circle. Tilt it a little and the cut stretches into an ellipse, still a closed loop. Tilt it until it is exactly parallel to a line on the cone's surface and the loop breaks open: one end runs off to infinity, and the cut is a parabola. Tilt it further and the plane meets both halves of the double cone, giving two separate branches, a hyperbola.

The view keeps the cone, the plane, and the intersection curve as three objects you can orbit around. The scenario sets the tilt, and the default slicing animation plays the plane through its morph so the transitions between the four curves are visible rather than asserted.

What to look for

• In the default 'ellipse' scenario, orbit to a side view and confirm the plane is tilted but still shallower than the cone's side. The intersection curve is closed.
• Switch to 'parabola' and look along the cone's surface from the side: the plane runs exactly parallel to one generating line. That is why the curve never closes. It is the boundary case, and it is a single tilt, not a range.
• In 'hyperbola', the plane pierces the upper nappe too. Orbit above and below to see both branches, and notice they are one cut of one plane, not two curves that happen to be paired.
• In 'circle', the plane is perpendicular to the axis and the cut is round from every angle. Tilt to 'ellipse' and the roundness is preserved only from the plane's own normal direction; from the axis it looks squashed, which is exactly what an ellipse is.

Getting it right

• An ellipse is not a squashed circle drawn on a tilted plane; it is a true ellipse in its own plane. The cone's cross-section is a circle, and the tilted cut of a circular cone is an exact ellipse, which is the theorem Apollonius proved.
• The parabola is not "an ellipse with one end very far away." It is the single tilt at which the plane is parallel to the cone's side; a hair either way gives an ellipse or a hyperbola.
• The hyperbola needs the double cone. A single cone gives only one branch; the second branch is the same plane cutting the other nappe, and the two branches are one curve.
• A plane through the apex gives degenerate conics — a point, a line, or a pair of crossed lines — and those are not scenarios here. They are the reason the cutting plane is always kept off the apex.

Turn the knobs

• scenario sets the tilt: 'circle', 'ellipse', 'parabola', 'hyperbola'. Step through them in order and watch the closed loop open, then split.
• animate plays the slicing morph by default; set it false to bake the final cut statically, which is the better setting for reading off one specific curve.
• scale resizes the whole assembly. The cone's half-angle and the plane's tilt are the geometry and stay fixed; only the tilt decides the curve.