The canonical form and physical meaning of hyperbolas
Will Vaughan, Oct. 30, 2011
---
A hyperbola, defined in general by the equation Ax^2 + Bxy + Cy^2 + Dx +
Ey + F = 0 with discriminant B^2 - 4AC > 0, can be converted by
rotation, translation, and scaling to a canonical form x^2/a^2 - y^2/b^2
= 1. (Rotation kills B, translation kills D and E, and scaling sets F to
-1.) This hyperbola is geometrically similar to the original hyperbola;
more generally, all conic sections with the same eccentricity are
similar (hence all parabolas, for which e=1, and all circles, for which
e=0, are similar).
The geometric meaning of the quantities a and b in the canonical form is
clear: for large x and y, x^2/a^2 ~ y^2/b^2 >> 1, so (x, y) must lie
very near a line passing through (0,0) with slope |b/a|. You can confirm
that the equation x^2/a^2 - y^2/b^2 = 1 implies that the difference of
the distances from any point on the hyperbola to the hyperbola's two
foci (at +c, -c along the x-axis where c^2 = a^2 + b^2) is constant and
equal to 2a.
However, the physical meaning of this equation is unclear. What are some
physical or geometric situations where the difference of the distance
from some point to two foci is constant? The only good example I know of
is radionavigation (e.g., LORAN) where the measured difference between
the time-synchronized signals from radio stations at known positions is
used to plot hyperbolic lines of position.
Equilateral hyperbolas (not hyperbolae, since the word is Greek) are
hyperbolas where a=b, or, equivalently, where the asymptotes of the
hyperbola bisect its axes (the axes always bisect the asymptotes, but
not necessarily vice-versa); notice that these hyperbolas are similar with
e=sqrt(2). Equilateral hyperbolas can be converted by rotation to a more
physically revealing form (why only equilateral hyperbolas? when A!=C in
the general equation above, A and C could not be killed simultaneously
to leave B). You can confirm that a rotation of 45 degrees changes the
equation of an equilateral hyperbola x^2 - y^2 = a^2 to xy = a^2/2; this
equation describes inversely proportional variables x and y, which
frequently occur in nature (for example, by the ideal gas law, pressure
and volume in an ideal gas are inversely proportional at constant
temperature).
In short, the canonical form of an equation may not be its most
revealing form!