To each conic section
(ellipse, parabola, hyperbola) there
is a number called the eccentricity that uniquely
characterizes the shape of the curve.
A circle has eccentricity 0, an ellipse between 0 and 1,
a parabola 1, and hyperbolae have eccentricity greater than 1.
Although you might think that y=2x2
and y=x2 have different "shapes" because
the former is skinnier, they really have the same "shape"
(and thus same eccentricity) because the first curve is
just the second curve viewed twice as far away
(i.e., x and y are both increased by a factor of 2).
One way to define a conic section is to specify a line in
the plane, called the directrix,
and a point in the plane off of the line,
called the focus.
The conic section is then the set of all points whose
distance to the focus is a constant times the distance
to the directrix. This constant is the eccentricity.
It is easy to see that as the eccentricity of an ellipse
grows, the ellipse becomes skinnier. The formula for
the ellipse also shows that every ellipse can be
obtained by taking a circle in a plane, lifting it up
and out, tilting it, and projecting it back into the plane.
Surprise: the eccentricity is equal to the sine of the
angle of this tilt!
If students are puzzled why the circle has eccentricity
zero, you might explain that its directrix is the line
"at infinity" in the projective plane.
The Math Behind the Fact:
Conic sections take their name from the fact that one can
also obtain them by slicing a cone by a plane at various
angles. Yet another way to obtain a conic section is by
starting with a circle and performing a geometric
transformation called reciprocation. The focus, directrix
and eccentricity fall out as obvious parameters of this
reciprocation operation. This approach to conic sections
comes from the field of projective geometry.
How to Cite this Page:
Su, Francis E., et al. "Eccentricity of Conics."
Math Fun Facts.