Editing Hyperbola (section)

===As locus of points===
[[File:Hyperbel-def-e.svg|thumb|Hyperbola: definition by the distances of points to two fixed points (foci)]]
[[File:Hyperbel-def-dc.svg|thumb|Hyperbola: definition with circular directrix]]
A hyperbola can be defined geometrically as a [[set (mathematics)|set]] of points ([[locus of points]]) in the Euclidean plane:

{{block indent |em=1.5 |text=
A '''hyperbola''' is a set of points, such that for any point <math>P</math> of the set, the absolute difference of the distances <math>|PF_1|,\, |PF_2|</math> to two fixed points <math>F_1, F_2</math> (the ''foci'') is constant, usually denoted by {{nowrap|<math>2a,\, a>0</math>:}}{{sfn|Protter|Morrey|1970|pp=308–310}}

<math display="block">H = \left\{P : \left|\left|PF_2\right| - \left|PF_1\right|\right| = 2a \right\} .</math>
}}
The midpoint <math>M</math> of the line segment joining the foci is called the ''center'' of the hyperbola.{{sfn|Protter|Morrey|1970|p=310}} The line through the foci is called the ''major axis''. It contains the ''vertices'' <math>V_1, V_2</math>, which have distance <math>a</math> to the center. The distance <math>c</math> of the foci to the center is called the ''focal distance'' or ''linear eccentricity''. The quotient <math>\tfrac c a</math> is the ''eccentricity'' <math>e</math>.

The equation <math>\left|\left|PF_2\right| - \left|PF_1\right|\right| = 2a</math> can be viewed in a different way (see diagram):<br/>
If <math>c_2</math> is the circle with midpoint <math>F_2</math> and radius <math>2a</math>, then the distance of a point <math>P</math> of the right branch to the circle <math>c_2</math> equals the distance to the focus <math>F_1</math>:
<math display="block">|PF_1|=|Pc_2|.</math>
<math>c_2</math> is called the ''circular directrix'' (related to focus <math>F_2</math>) of the hyperbola.<ref>{{citation |last1=Apostol |first1=Tom M. |last2=Mnatsakanian |first2=Mamikon A. |title=New Horizons in Geometry |year=2012 |publisher=The Mathematical Association of America |series=The Dolciani Mathematical Expositions #47 |isbn=978-0-88385-354-2 |page=251}}</ref><ref>The German term for this circle is ''Leitkreis'' which can be translated as "Director circle", but that term has a different meaning in the English literature (see [[Director circle]]).</ref> In order to get the left branch of the hyperbola, one has to use the circular directrix related to <math>F_1</math>. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.