Focal Properties of Conic Sections

Conic Sections of Apollonius

Apollonius of Perga (3rd century BC) systematically studied "conic sections"—intersections of a cone with a plane. Depending on the angle of inclination of the plane, we obtain an ellipse, a parabola, or a hyperbola.

Two millennia later, Kepler discovered: planets move along ellipses, comets—along parabolas or hyperbolas. Thus, ancient geometry turned out to be a law of nature.

Reflective Properties

Ellipse: a ray originating from one focus, after reflection from the ellipse, passes through the other focus. Whispering rooms (elliptical ceilings), lithotripters for crushing kidney stones—applications of this property.

Parabola: a ray parallel to the axis, after reflection, passes through the focus. And conversely: a source located at the focus gives a parallel beam. Parabolic mirrors of telescopes, spotlights, antennas.

Hyperbola: a ray directed towards one focus, after reflection appears to emanate from the other. Used in LORAN navigation systems.

Directrix Definition

Eccentricity $e = c/a$. A second-order curve is the locus of points for which the ratio of the distances to the focus and to the directrix equals $e$.

$e < 1$ → ellipse, $e = 1$ → parabola, $e > 1$ → hyperbola.

Parametric Equations

Ellipse: $x = a \cos t$, $y = b \sin t$. Hyperbola: $x = a \operatorname{ch} t$, $y = b \operatorname{sh} t$ (hyperbolic functions). Parabola: $x = t^2$, $y = 2pt$.

The parametric form is convenient for calculating lengths and areas, as well as for describing trajectories.