What is a conic section?

A curve that shows up where a plane meets the surface of a cone, on one nappe or on both.

What is eccentricity?

The number e, the same all along the curve: the distance to the focus divided by the distance to the directrix.

How is the ellipse defined from its foci?

As the curve whose points have a distance to the first focus that, added to the distance to the second, always gives the same total. Centred at the origin with the major axis horizontal, x²/a² + y²/b² = 1 with a greater than b.

How is the hyperbola defined from its foci?

As the curve whose points have distances to the two foci that always differ by the same positive amount. Centred at the origin with the transverse axis horizontal, x²/a² - y²/b² = 1.

How is the parabola defined?

As the curve built from every point that keeps the same distance from a fixed point, the focus, and from a line not passing through it, the directrix.

Which cut gives a circle?

A plane perpendicular to the axis of the cone, with eccentricity 0.

What shape are planetary orbits?

Oval, that is, elliptical, as Kepler established after observing that orbits are not circular.

What are the rings of Saturn made of?

Small particles, as Maxwell and Kovalevskaya showed in the 1800s.

Conic Sections: What They Are and How to Tell Them Apart

In 30 seconds quick read

Conic sections are the curves you get by slicing a cone with a plane, and there are four of them: the circle, the ellipse, the parabola and the hyperbola. What changes from one to the next is the tilt of the cut and how many halves of the cone the plane goes through. The four can also be pinned down in one stroke, by a number called eccentricity: it is 0 for the circle, between 0 and 1 for the ellipse, exactly 1 for the parabola and above 1 for the hyperbola. Credit for finding these shapes generally goes to the Greek mathematician Menaechmus, who lived around 380-320 BC.

Key Points

A tilted plane that stays inside one nappe of the cone gives an ellipse or a parabola, and the steepness of the tilt decides which; a plane that takes in both nappes gives a hyperbola.
The tidiest case is the circle, and it comes from the cut that runs perpendicular to the axis of the cone.
Eccentricity comes from dividing the distance to the focus by the distance to the directrix, and that quotient never changes as you move along the curve.
In Cartesian coordinates every conic is written as a general quadratic equation: ax² + 2bxy + cy² + dx + fy + g = 0.
Ellipse and hyperbola are both defined from two foci: the first keeps the sum of the distances fixed, the second their difference in absolute value.
Kepler established that the orbit of a planet around the Sun is oval, that is, elliptical.

Deep Dive

One cone, one plane, four curves

A conic section is the curve left behind where a plane cuts through one or both nappes of a cone, which are the two halves meeting at its tip: the name describes the procedure. Hold the plane perpendicular to the axis of the cone and the intersection is a circle. Tilt it so that it stays inside a single nappe and what comes out is an ellipse or a parabola, depending on how steeply it leans. Push the tilt far enough that the plane goes through both nappes, and the curve drawn on it is a hyperbola.

The same family of curves can be recognised from the algebra too, on the ground of analytic geometry: every conic section satisfies a general quadratic equation of the form ax² + 2bxy + cy² + dx + fy + g = 0, one of those second-degree equations in which the unknowns show up squared as well.

According to MathWorld there is also a way to define all four curves at once, with a single parameter. Pick a point F, called the focus, and a line d, called the directrix: a conic section is then the set of points P for which the distance from P to F divided by the distance from P to d always comes out the same. That constant is the eccentricity, and its value separates the cases: 0 for the circle, between 0 and 1 for the ellipse, exactly 1 for the parabola, above 1 for the hyperbola.

From Menaechmus to the rings of Saturn

The discovery of these shapes is generally credited to Menaechmus, a Greek mathematician who lived around 380-320 BC: by intersecting a plane with a right circular cone and changing the steepness of the plane, he obtained different figures, carrying an almost perfect symmetry. Aristotle may have had an intuitive grasp of them before him, since he had described the orbit of a planet as circular. The correction on that point arrived much later, with Kepler, who observed that planetary orbits do not in fact have a circular shape and established that planets travel an oval, that is elliptical, orbit around the Sun: the result studied alongside Kepler’s three laws of planetary motion. In the 1800s Maxwell and Kovalevskaya added a piece to the celestial chapter, showing that the rings of Saturn, compact as they look, are made of small particles rather than one solid continuous body.

Side by side, the differences fit into two columns: the angle of the cut and the value of the eccentricity.

Curve	How the plane slices the cone	Eccentricity
Circle	Perpendicular to the axis	0
Ellipse	Tilted, on one nappe only	between 0 and 1
Parabola	Tilted, on one nappe only	1
Hyperbola	Across both nappes	greater than 1

The three open curves, one by one

An ellipse is the set of points of the plane for which the distances to two fixed points, the foci, add up to a constant total; centred at the origin and with the major axis horizontal it is written x²/a² + y²/b² = 1, with a greater than b. Among the real cases OpenStax lists the orbits of planets, moons, comets and satellites, and then shapes closer to the ground: the keels of certain boats, rudders, the wings of some aircraft.

Practical example: in a whispering gallery, a hall roofed by an elliptical dome, a sound wave made at one of the two foci bounces off the vault and gathers at the other focus. Whoever stands in that second focus hears the sound picked up by the whole dome.

A hyperbola collects every point whose two distances to the foci differ, in absolute value, by a positive constant; with its centre at the origin and the transverse axis horizontal its equation reads x²/a² - y²/b² = 1. The shape turns up in heavy engineering as well: hyperboloid cooling towers are designed this way because they hold against extreme winds while taking less construction material.

Practical example: when an object goes faster than the speed of sound it generates a cone-shaped shock wave. Where that cone meets the ground, the trace it leaves is part of a conic section.

A parabola holds together the points that lie equally far from a fixed line, the directrix, and from a fixed point outside it, the focus. Its profile comes back in satellite dishes and telescopes, in microphones and spotlights, in car headlights and suspension bridges. Anyone who wants the school case worked through step by step will find the equation and the graph in the Recap on the parabola and the quadratic equation.

Practical example: a parabolic mirror picks up energy, either light or radio waves, and concentrates it at a single point, the focus.

All this apparatus of foci, directrices and axes rests on an older frame, the Euclidean geometry of the Elements, where points and lines receive their first definitions.

Slide deck

Slides ready to download and make your own in PowerPoint or Google Slides, with speaker notes. Pick the Flash cut or the Full one.

Slide 1 of the presentation on Conic Sections: Conic sections

Slide 2 of the presentation on Conic Sections: What shape is a planet's orbit?

Slide 3 of the presentation on Conic Sections: From slicing a cone to satellite dishes

Slide 4 of the presentation on Conic Sections: Chapter 01: How the plane slices the cone

Slide 5 of the presentation on Conic Sections: Count how many halves the plane meets

Slide 6 of the presentation on Conic Sections: Chapter 02: Eccentricity, one number for all

Slide 7 of the presentation on Conic Sections: One number compares focus distance with directrix distance

Slide 8 of the presentation on Conic Sections: Chapter 03: Sums and differences of distances

Slide 9 of the presentation on Conic Sections: Ellipse and hyperbola start from the same two foci

Slide 10 of the presentation on Conic Sections: Where these curves end up in engineering: Parabolic mirror, Elliptical dome, Hyperboloid tower

Slide 11 of the presentation on Conic Sections: The orbit of a planet is oval, not circular.

Slide 12 of the presentation on Conic Sections: Chapter 04: From Menaechmus to Kepler

Slide 13 of the presentation on Conic Sections: From the 4th century BC to the 1800s

Slide 14 of the presentation on Conic Sections: What is the eccentricity of a parabola?

Slide 15 of the presentation on Conic Sections: Keep going on the site

Flash10 slidesThe essential thread, to present in class Full15 slidesEvery chapter and the deeper detail

Common myths

✗ Myth A hyperbola is just a figure of speech; geometry has nothing of the kind.

✓ Reality The word covers two different things. In geometry a hyperbola gathers the points whose distance from one focus, minus the distance from the other, always gives the same positive constant in absolute value, and it comes from a plane that passes through both halves of the cone.
✗ Myth A circle does not count as a conic section.

✓ Reality A circle too comes out of slicing a cone, with the cut perpendicular to the axis, and it is the case of eccentricity equal to 0. In the unified definition built on focus and directrix it holds a precise spot, the bottom end of the scale.
✗ Myth Planets go around the Sun along a circle.

✓ Reality Aristotle called a planet's orbit circular. Kepler observed that orbits do not in fact have that shape and established that the path around the Sun is oval, that is, an ellipse.

Mind map

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1 Which curve do you get when the plane cuts through both halves of the cone?

2 What is the eccentricity of a parabola?

3 On an ellipse, what stays constant?

4 Who is generally credited with discovering the conic sections?

5 Why are cooling towers shaped like a hyperboloid?

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Frequently asked questions

What are conic sections, in plain words?

They are the curves that appear when a plane slices a cone. The outcome depends on how the plane is oriented, and there are four possibilities: circle, ellipse, parabola, hyperbola.

How many conic sections are there?

Four of them are non-degenerate. A cut perpendicular to the axis gives the circle, a tilted cut inside a single nappe gives the ellipse or the parabola, and a cut that takes in both nappes gives the hyperbola.

What is the difference between an ellipse, a hyperbola and a parabola?

Ellipse and hyperbola are both built from two foci: the ellipse keeps the sum of the distances to those points constant, the hyperbola keeps the difference constant in absolute value. The parabola works from one focus and one line, the directrix, and takes the points equally far from both. In eccentricity the ellipse sits between 0 and 1, the parabola at exactly 1, the hyperbola above 1.

When do you study conic sections at school?

They normally arrive once quadratic equations and coordinates are already in hand, so they land in the later years of secondary school. The usual route starts from the circle and then takes in the ellipse, the parabola and the hyperbola, with the general quadratic equation ax² + 2bxy + cy² + dx + fy + g = 0 holding the four together.

How do you build a schema or concept map of the conic sections?

Four branches work well: how the plane slices the cone, the value of the eccentricity, the definition from focus and directrix, and the equation in Cartesian coordinates. Under each branch you hang the four curves.