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!
Presentation Suggestions:
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. <https://www.math.hmc.edu/funfacts>.
References:
H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited.
(for the material on projective geometry).
Fun Fact suggested by:
Aaron Archer