Conics and Quadrics - Applications

TheoremFocal Property of Ellipse

For any point $P$ on an ellipse with foci $F_1$ and $F_2$ , the sum of distances to the foci is constant:

|PF_1| + |PF_2| = 2a

where $a$ is the semi-major axis.

Reflective Property: The tangent line at $P$ makes equal angles with the lines $PF_1$ and $PF_2$ . Equivalently, light from one focus reflects through the other focus.

This theorem has profound practical applications. Elliptical reflectors (ellipsoids of revolution) concentrate sound or light from one focus to the other, used in whispering galleries, medical devices, and astronomical observatories.

Lithotripsy uses ellipsoidal reflectors to break kidney stones: a shock wave generator at one focus directs energy to the stone at the other focus, shattering it non-invasively. This medical application saves countless patients from surgery.

ExamplePlanetary Orbits (Kepler's First Law)

Planets orbit the Sun in elliptical paths with the Sun at one focus. For Earth:

Semi-major axis: $a \approx 149.6$ million km
Eccentricity: $e \approx 0.0167$ (nearly circular)
Perihelion (closest): $a(1-e) \approx 147$ million km
Aphelion (farthest): $a(1+e) \approx 152$ million km

This elliptical motion results from gravitational attraction following Newton's inverse-square law.

For parabolas, the analogous property involves a focus and directrix:

TheoremParabolic Reflector Property

For a parabola with focus $F$ and directrix $\ell$ , every point $P$ on the parabola satisfies $|PF| = d(P, \ell)$ .

Light rays parallel to the axis reflect through the focus, and conversely, light from the focus reflects parallel to the axis.

Parabolic dishes exploit this property:

Satellite dishes: Collect incoming signals at the focal receiver
Radio telescopes: Focus cosmic radio waves for detection
Car headlights: Bulb at focus produces parallel beam
Solar collectors: Concentrate sunlight for heating

DefinitionConic Classification via Focus

All conics can be unified via the focus-directrix definition:

\frac{|PF|}{d(P,\ell)} = e \quad \text{(eccentricity)}

$e < 1$ : ellipse (bounded)
$e = 1$ : parabola (critical case)
$e > 1$ : hyperbola (unbounded)

This parametrization by $e \in [0, \infty)$ provides a continuous transition between conic types.

Hyperbolic navigation (LORAN system) uses the difference property: a hyperbola is the locus of points with constant difference of distances to two foci. By measuring time delays of radio signals from two stations, a receiver determines its position on a hyperbola. Multiple pairs of stations give multiple hyperbolas whose intersection locates the receiver.

Remark

Modern GPS uses a similar principle in higher dimensions. Satellites at known positions broadcast signals; receivers compute distances via signal delays. The locus of points at distance $r$ from satellite $S$ is a sphere (3D generalization of a circle). Intersection of multiple spheres determines position.