Editing Hyperbola (section)

===Equation===
If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the ''x''-axis is the major axis, then the hyperbola is called ''east-west-opening'' and
:the ''foci'' are the points <math>F_1=(c,0),\ F_2=(-c,0)</math>,{{sfn|Protter|Morrey|1970|p=310}}
:the ''vertices'' are <math>V_1=(a, 0),\ V_2=(-a,0)</math>.{{sfn|Protter|Morrey|1970|p=310}}
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is <math display="inline">\sqrt{(x-c)^2 + y^2}</math> and to the second focus <math display="inline">\sqrt{(x+c)^2 + y^2}</math>. Hence the point <math>(x,y)</math> is on the hyperbola if the following condition is fulfilled
<math display="block">\sqrt{(x-c)^2 + y^2} - \sqrt{(x+c)^2 + y^2} = \pm 2a \ .</math>
Remove the square roots by suitable squarings and use the relation <math>b^2 = c^2-a^2</math> to obtain the equation of the hyperbola:

<math display="block">\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ .</math>

This equation is called the [[canonical form]] of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is [[congruence (geometry)|congruent]] to the original (see [[#Quadratic equation|below]]).

The axes of [[symmetry (geometry)|symmetry]] or ''principal axes'' are the ''transverse axis'' (containing the segment of length 2''a'' with endpoints at the vertices) and the ''conjugate axis'' (containing the segment of length 2''b'' perpendicular to the transverse axis and with midpoint at the hyperbola's center).{{sfn|Protter|Morrey|1970|p=310}} As opposed to an ellipse, a hyperbola has only two vertices: <math>(a,0),\; (-a,0)</math>. The two points <math>(0,b),\; (0,-b)</math> on the conjugate axes are ''not'' on the hyperbola.

It follows from the equation that the hyperbola is ''symmetric'' with respect to both of the coordinate axes and hence symmetric with respect to the origin.

====Eccentricity====
For a hyperbola in the above canonical form, the [[eccentricity (mathematics)|eccentricity]] is given by

<math display="block">e=\sqrt{1+\frac{b^2}{a^2}}.</math>

Two hyperbolas are [[similarity (geometry)|geometrically similar]] to each other – meaning that they have the same shape, so that one can be transformed into the other by [[translation (geometry)|rigid left and right movements]], [[rotation (mathematics)|rotation]], [[reflection (mathematics)|taking a mirror image]], and scaling (magnification) – if and only if they have the same eccentricity.