Conic Sections

Conic Section: a section (or slice) through a cone.

Did you know that by taking different slices through a cone you can create a circle, an ellipse, a parabola or a hyperbola?

Double cone with two vertical nappes meeting at a single vertex point

Cones

Horizontal plane slicing straight through a cone to form a circular cross-section

Circle
straight through

Angled plane slicing through a cone to form an elliptical cross-section

Ellipse
slight angle

Plane slicing through a cone parallel to its side to form a parabolic cross-section

Parabola
parallel to edge of cone

Steep vertical plane slicing through both halves of a double cone to form a hyperbola

Hyperbola
steep angle

So all those curves are related.

Focus!

A curve showing a point, a focus, and a directrix line, with distance measurements d1 and d2

The curves can also be defined using a straight line (the directrix) and a point (the focus).

When we measure the distance:

from the focus to any point on the curve, and
perpendicularly from the directrix to that point

the two distances will always have the same ratio.

For an ellipse, the ratio is less than 1
For a parabola, the ratio is 1 (so the two distances are equal)
For a hyperbola, the ratio is greater than 1

That ratio is called the eccentricity. Play with it here:

images/eccentricity-graph.js

Eccentricity

We can say that any conic section is:

"all points whose distance to the focus is equal
to the eccentricity times the distance to the directrix"

Graphs of a circle, ellipse, parabola, and hyperbola with their respective eccentricity values

For:

0 < eccentricity < 1 we get an ellipse,
eccentricity = 1 a parabola, and
eccentricity > 1 a hyperbola

A circle has an eccentricity of zero, so the eccentricity shows us how "un-circular" the curve is. The bigger the eccentricity, the less curved it is.

Example: Orbits have an eccentricity less than 1

An eccentricity above 1 is isn't really an orbit as it doesn't loop back, but passes by.

Artist's impression of the elongated, rocky interstellar object Oumuamua in space
Artist's Impression of 'Oumuamua
Credit: ESO/M. Kornmesser

The interstellar asteroid 'Oumuamua has an eccentricity of about 1.2 in its path around the Sun, meaning it isn't part of our solar system:

Diagram of the hyperbolic path of Oumuamua entering and leaving the solar system around the Sun
Credit: Wikpedia authors nagualdesign and Tomruen

The orbit of Earth has an eccentricity of about 0.0167 (nearly a circle)
The orbit of Mars has an eccentricity of about 0.0934 (a little less circular)

Latus Rectum

A parabola with its vertex, focus, directrix, and the latus rectum chord passing through the focus

The latus rectum (no, it isn't a rude word!) runs parallel to the directrix and passes through the focus. Its length:

In a parabola

is four times the focal length

In a circle

is the diameter

In an ellipse

is 2b²a (where a and b are one half of the major and minor diameter)

An ellipse showing its major and minor axes, two foci, two directrices, and latus rectum

Here's the major axis and minor axis of an ellipse.

There's a focus and directrix on each side (ie a pair of them).

Equations

An ellipse centered on an x-y Cartesian grid with semi-major axis a and semi-minor axis b

When placed like this on an x-y graph, the equation for an ellipse is:

x²a² + y²b² = 1

The special case of a circle (where radius=a=b) is:

x²a² + y²a² = 1

A hyperbola centered at the origin on an x-y Cartesian coordinate plane, opening left and right, with asymptotes.

And for a hyperbola it is:

x²a² − y²b² = 1

General Equation

We can make an equation that covers all these curves.

Because they are plane curves (even though cut out of the solid) we only have to deal with Cartesian ("x" and "y") Coordinates.

But these aren't straight lines, so just "x" and "y" won't do ... we need to go to the next level, and have:

x² and y²,
and also x (without y), y (without x),
x and y together (xy)
and a constant term

There, that should do it!

Give each one a factor (A,B,C,D,E,F) and we get a general equation that covers all conic sections:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

From that equation we can create equations for the circle, ellipse, parabola and hyperbola.

9064, 9065, 9066, 9067, 637, 638, 3326, 3327, 3328, 3329