Editing Ellipse (section)

== Definition as locus of points ==
[[File:Ellipse-def-e.svg|thumb|Ellipse: definition by sum of distances to foci]]
[[File:Ellipse-def-dc.svg|thumb|Ellipse: definition by focus and circular directrix]]
An ellipse can be defined geometrically as a set or [[locus of points]] in the Euclidean plane:
{{block indent | em = 1.5 | text = Given two fixed points <math>F_1, F_2</math> called the foci and a distance  <math>2a</math> which is greater than the distance between the foci, the ellipse is the set of points <math>P</math> such that the sum of the distances <math>|PF_1|,\ |PF_2|</math> is equal to <math>2a</math>: <math display="block">E = \left\{P\in \R^2 \,\mid\, \left|PF_2\right| + \left|PF_1\right| = 2a \right\} .</math>}}

The midpoint <math>C</math> of the line segment joining the foci is called the ''center'' of the ellipse. The line through the foci is called the ''major axis'', and the line perpendicular to it through the center is the ''minor axis''. {{anchor|Vertex}}The major axis intersects the ellipse at two ''[[vertex (curve)|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>e = \tfrac{c}{a}</math> is defined as the ''eccentricity''.

The case <math>F_1 = F_2</math> yields a circle and is included as a special type of ellipse.

The equation <math>\left|PF_2\right| + \left|PF_1\right| = 2a</math> can be viewed in a different way (see figure):
{{block indent | em = 1.5 | text = If <math>c_2</math> is the circle with center <math>F_2</math> and radius <math>2a</math>, then the distance of a point <math>P</math> to the circle <math>c_2</math> equals the distance to the focus <math>F_1</math>: <math display="block">\left|PF_1\right| = \left|Pc_2\right|.</math>}}

<math>c_2</math> is called the ''circular directrix'' (related to focus {{nowrap|<math>F_2</math>)}} of the ellipse.<ref>{{citation | first1=Tom M.|last1=Apostol | first2=Mamikon A.|last2=Mnatsakanian | 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> This property should not be confused with the definition of an ellipse using a directrix line below.

Using [[Dandelin spheres]], one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.