1. Slice the cone
Picture two ice-cream cones joined at their tips, one pointing up and one pointing down, and running on forever. That is a double cone. Cut it with a flat plane and the edge of the cut is a curve. Tilt the plane and watch which curve you get.
Step through the four cases
How this picture stays honest. Nothing here is drawn by hand or morphed from one shape to another. The cone is the true surface x² + y² = (z tan α)² with α = 30°, and the plane is a true flat plane. For every direction around the cone the page solves for the one point where that line of the cone hits the plane, and the red curve joins those points. A separate test program checks that every point lies on both the cone and the plane.
2. From slice to equation
A conic section is any curve you can get by cutting a double cone with a plane. Only one thing decides which curve you get: how steep the plane is compared with the side of the cone. Where the plane sits only changes the size of the curve, unless it passes right through the tip.
The eccentricity e measures that comparison with one number. If the cone's half-angle (from its axis to its side) is α and the plane is tilted β from level, then
e = sin β ÷ cos α
Some books measure the plane's angle θ from the axis instead; then θ = 90° − β and the same rule reads e = cos θ ÷ cos α.
| Slice | How the plane cuts | e | Standard equation |
|---|---|---|---|
| Circle | Level, square to the axis. Every point is the same height. | e = 0 | (x − h)² + (y − k)² = r² |
| Ellipse | Tilted, but less steep than the side. Cuts one half all the way around. | 0 < e < 1 | x²/a² + y²/b² = 1, e = c/a |
| Parabola | Exactly as steep as the side, so it runs alongside one line of the cone and never closes. | e = 1 | x² = 4py or y = ax² + bx + c |
| Hyperbola | Steeper than the side. Cuts both halves, one branch on each. | e > 1 | x²/a² − y²/b² = 1, e = c/a |
Look back at the figure in step 1. Under the picture it gives the slice's own equation, written in the plane's own coordinates: v runs along the plane's level lines and u runs straight up the tilt. It always comes out as A u² + v² + D u + F = 0. That is the test you use to recognize a conic from its equation: with C = 1, a level plane gives A = C (circle), a gentle tilt gives AC > 0 (ellipse), the side's own steepness gives A = 0 (parabola), and a steeper tilt gives AC < 0 (hyperbola). Completing the square turns it into the standard form in the table.
Through the tip. If the plane passes through the tip itself, the curve collapses. A gentle tilt touches the cone only at the tip (a single point). A tilt equal to the side lies along the cone in one line. A steeper tilt cuts out two crossing lines. These are the degenerate conics.
Apollonius of Perga. The Greek geometer Apollonius of Perga wrote the Conics in eight books, about 200 B.C. Seven of the books survive, some in Greek and some only in Arabic translation. He showed that all three curves come from cutting one double cone at different tilts, and he gave them the names we still use. Ellipse comes from a Greek word for falling short, parabola from a word for placing alongside, and hyperbola from a word for going beyond. He worked entirely with figures and proportions. The equations came much later, yet they describe exactly the same curves.
The same four curves turn up far from any cone: in the path of a thrown ball, the shape of a satellite dish, and the orbits of the planets. Careful reasoning about one simple figure reaches all the way into the order of the created world.
3. Quick check
1. The side of a cone makes a 60° angle with level ground (so its half-angle is 30°). A plane tilted 45° from level cuts the cone and misses the tip. Which conic do you get, and what is its eccentricity?
2. In the top-down view, the plane's lines are exactly as far apart as the cone's rings. What curve is traced, and why does it never close up?
3. A plane tilted 75° from level passes exactly through the tip of the same cone. What do you see? What would you see if the plane were moved a little away from the tip?