MISSION · Geometry
Conic Sections
Four curves, one cone, and a single number that tells them apart
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Slice a double cone at different angles and you get the circle, the ellipse, the parabola, and the hyperbola — four curves that turn out to be one family.
The slices
Picture two cones tip to tip, and a plane cutting through:
- Horizontal cut — a circle.
- Tilted, but less steeply than the cone’s side — an ellipse.
- Exactly parallel to the side — a parabola. The cut never closes.
- Steep enough to hit both cones — a hyperbola, in two separate branches.
The Greeks studied these purely geometrically, two centuries before anyone had coordinates. Eighteen hundred years later Kepler found that planets travel on ellipses, and a piece of pure geometry became the foundation of astronomy.
Eccentricity: one number, four curves
Each conic can also be defined as the set of points whose distance to a fixed focus is a constant multiple of the distance to a fixed line. That multiple, \(e\), is the eccentricity, and it alone decides the shape:
| \(e\) | Curve |
|---|---|
| 0 | Circle |
| \(0 < e < 1\) | Ellipse |
| 1 | Parabola |
| \(e > 1\) | Hyperbola |
A circle is an ellipse with both foci in the same place. As \(e\) climbs towards 1 the ellipse stretches, and at exactly 1 it gives up on closing and becomes a parabola. Earth’s orbit has \(e \approx 0.017\) — very nearly circular. Halley’s comet is at 0.967, a long thin ellipse that spends most of its time far away.
The equations
In standard position, centred at the origin:
\(\text{circle:} \quad x^2 + y^2 = r^2\) \(\text{ellipse:} \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) \(\text{parabola:} \quad y = ax^2\) \(\text{hyperbola:} \quad \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
The ellipse and hyperbola differ by a single sign, and that sign is the whole difference between a closed loop and two branches running off to infinity.
The focus does real work
Each conic has a reflection property that industry depends on.
A parabola reflects every ray arriving parallel to its axis straight to the focus. That is why satellite dishes, radio telescopes, and solar concentrators are parabolic, and why the same shape run backwards makes a headlight throw a parallel beam.
An ellipse reflects from one focus to the other. Whispering galleries work on this: a murmur at one focus is audible at the other and nowhere in between. Lithotripsy uses it to focus shock waves on a kidney stone without cutting anyone open.
Common mix-ups
- Reading any U-shape as a parabola. A parabola is a specific curve; catenaries and hyperbolas look similar and aren’t.
- Confusing the ellipse and hyperbola equations. Plus is closed, minus is two branches.
- Putting the focus at the centre of an ellipse. There are two, and neither is the centre.
- Assuming eccentricity measures size. It measures shape; a huge circle still has \(e = 0\).
Check yourself
What curve is \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), and how wide and tall is it?
An ellipse, since the terms are added. It runs from \(-3\) to 3 across and \(-2\) to 2 up, so 6 wide and 4 tall.
LEVEL 3 · REFLECT
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