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?

 

cones
Cones

 
conic section circle
Circle

straight through
conic section ellipse
Ellipse

slight angle
conic section parabola
Parabola

parallel to edge
of cone
conic section hyperbola
Hyperbola

steep angle

So all those curves are related!

Focus!

focus and directrix

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

When we measure the distance:

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

the two distances will always be 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

Eccentricity

That ratio above is called the "eccentricity", so 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"

Eccentricity

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.

Latus Rectum

latus rectum

The latus rectum (no, it is not 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 2b2/a (where a and b are one half of the major and minor diameter).

 

ellipse directrix, focus and latus rectum

Here is the major axis and minor axis of an ellipse.

There is not just one focus and directrix, but a pair of them (one each side).

General Equation

In fact, 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 are not straight lines, so just "x" and "y" will not 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!

And each one needs a factor (A,B,C etc) ...

So the general equation that covers all conic sections is:

Ax^2 etc

And from that equation we can create equations for the circle, ellipse, parabola and hyperbola ... but that is beyond the scope of this page.