The canonical form and physical meaning of hyperbolas
Will Vaughan, Oct. 30, 2011
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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!