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Precalculus: An Investigation of Functions, by David Lippman and Melonie Rasmussen, licensed under CC BY-SA 4.0. Read the original free This shelf’s study material is shared under the same license (CC BY-SA 4.0).
Chapter 1 · pp. 4–4
A conic section is a curve you get by slicing a cone with a flat plane. Changing the angle of the slice changes the shape of the curve. This guide covers how the four main conics (circle, ellipse, parabola, hyperbola) arise and how to tell them apart by the tilt of the plane.
Picture two identical cones joined tip to tip, with one pointing up and one pointing down. This shape is a double cone. The shared tip is the vertex. Each cone is called a nappe. A straight line through the vertex, running down the middle of both nappes, is the axis. Any line on the surface of a cone that passes through the vertex is called a side (or generator) of the cone.
A conic section is the curve where a plane cuts through a double cone. The four main types are:
The name "conic" comes from the cone. Every one of these curves is just a different flat slice of the same solid.
The only thing that changes from one conic to the next is the angle of the cutting plane. Assume the plane does not pass through the vertex.
Circle. Hold the plane perpendicular to the axis, like slicing a party hat straight across. Every point of the cut is the same distance from the axis, so the curve is a circle.
Ellipse. Now tilt the plane a little. If it still cuts all the way around one nappe, the curve closes up into an oval. This oval is an ellipse. A circle is really a special ellipse with no tilt. The more you tilt, the longer and thinner the oval becomes.
Parabola. Keep tilting until the plane is parallel to a side of the cone. The plane now cuts only one nappe, and it never closes up. The curve opens forever in one direction and is a parabola. It has one branch.
Hyperbola. Tilt even more, so the plane is steeper than the side of the cone. It now cuts through both nappes. You get two separate curves, one in each nappe. Together they make a hyperbola, and each piece is called a branch.
You can describe the sorting rule with two angles:
Then, for a plane that misses the vertex:
| Angle condition | Conic |
|---|---|
| circle | |
| ellipse | |
| parabola | |
| hyperbola |
The parabola is the dividing case. A slightly flatter plane gives an ellipse, which is closed. A slightly steeper plane gives a hyperbola, which has two branches.
Example. Suppose the side of a cone makes an angle with the axis.
If the plane passes exactly through the vertex, you get a "degenerate" conic instead of a smooth curve. Depending on the angle, the result is a single point, a single line, or two lines that cross. These are not among the four main types, but they show what happens when the slice moves to the tip.
Shine a flashlight at a wall. The beam is a cone of light. Pointed straight at the wall, it makes a circle. Tilted a bit, the lit patch is an ellipse. Tilted until the top edge of the beam runs parallel to the wall, the edge of the patch is a parabola. Tilted further, the patch becomes one branch of a hyperbola. This shows why all four curves belong to one family.
