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:

the two distances will always be the same ratio.

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:

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:

 

ellipse directrix, focus and latus rectum

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

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

Equations

ellipse on xy graph

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):

x2a2 + y2a2 = 1

hyperbola on xy graph

And for a hyperbola it is:

x2a2y2b2 = 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 are not straight lines, so just "x" and "y" will not do ... we need to go to the next level, and have:

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.