Editing Ellipse (section)

==Eccentricity and the directrix property==
[[File:Ellipse-ll-e.svg|300px|thumb|Ellipse: directrix property]]
Each of the two lines parallel to the minor axis, and at a distance of <math display="inline">d = \frac{a^2}{c} = \frac{a}{e}</math> from it, is called a ''directrix'' of the ellipse (see diagram).
: For an arbitrary point <math>P</math> of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: <math display="block">\frac{\left|PF_1\right|}{\left|Pl_1\right|} = \frac{\left|PF_2\right|}{\left|Pl_2\right|} = e = \frac{c}{a}\ .</math>

The proof for the pair <math>F_1, l_1</math> follows from the fact that <math display="inline">\left|PF_1\right|^2 = (x - c)^2 + y^2,\ \left|Pl_1\right|^2 = \left(x - \tfrac{a^2}{c}\right)^2</math> and <math>y^2 = b^2 - \tfrac{b^2}{a^2}x^2</math> satisfy the equation
<math display="block">\left|PF_1\right|^2 - \frac{c^2}{a^2}\left|Pl_1\right|^2 = 0\, .</math>

The second case is proven analogously.

The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):
: For any point <math>F</math> (focus), any line <math>l</math> (directrix) not through <math>F</math>, and any real number <math>e</math> with <math>0 < e < 1,</math> the ellipse is the locus of points for which the quotient of the distances to the point and to the line is <math>e,</math> that is: <math display="block">E = \left\{P\ \left|\ \frac{|PF|}{|Pl|} = e\right.\right\}.</math>

The extension to <math>e = 0</math>, which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the [[line at infinity]] in the [[projective plane]].

(The choice <math>e = 1</math> yields a parabola, and if <math>e > 1</math>, a hyperbola.)
[[File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex and common semi-latus rectum]]

;Proof

Let <math>F = (f,\, 0),\ e > 0</math>, and assume <math>(0,\, 0)</math> is a point on the curve. The directrix <math>l</math> has equation <math>x = -\tfrac{f}{e}</math>. With <math>P = (x,\, y)</math>, the relation <math>|PF|^2 = e^2|Pl|^2</math> produces the equations
:<math>(x - f)^2 + y^2 = e^2\left(x + \frac{f}{e}\right)^2 = (ex + f)^2</math> and <math>x^2\left(e^2 - 1\right) + 2xf(1 + e) - y^2 = 0.</math>

The substitution <math>p = f(1 + e)</math> yields
<math display="block">x^2\left(e^2 - 1\right) + 2px - y^2 = 0.</math>

This is the equation of an ''ellipse'' (<math>e < 1</math>), or a ''parabola'' (<math>e = 1</math>), or a ''hyperbola'' (<math>e > 1</math>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).

If <math>e < 1</math>, introduce new parameters <math>a,\, b</math> so that <math>1 - e^2 = \tfrac{b^2}{a^2}, \text{ and }\ p = \tfrac{b^2}{a}</math>, and then the equation above becomes
<math display="block">\frac{(x - a)^2}{a^2} + \frac{y^2}{b^2} = 1\, ,</math>

which is the equation of an ellipse with center <math>(a,\, 0)</math>, the ''x''-axis as major axis, and
the major/minor semi axis <math>a,\, b</math>.

[[File:Leitlinien-konstr-e.svg|thumb|Construction of a directrix]]
;Construction of a directrix

Because of <math>c\cdot\tfrac{a^2}{c}=a^2</math> point <math>L_1</math> of directrix <math>l_1</math> (see diagram) and focus <math>F_1</math> are inverse with respect to the [[circle inversion]] at circle <math>x^2+y^2=a^2</math> (in diagram green). Hence <math>L_1</math> can be constructed as shown in the diagram. Directrix <math>l_1</math> is the perpendicular to the main axis at point <math>L_1</math>.

;General ellipse

If the focus is <math>F = \left(f_1,\, f_2\right)</math> and the directrix <math>ux + vy + w = 0</math>, one obtains the equation
<math display="block">\left(x - f_1\right)^2 + \left(y - f_2\right)^2 = e^2 \frac{\left(ux + vy + w\right)^2}{u^2 + v^2}\ .</math>

(The right side of the equation uses the [[Hesse normal form]] of a line to calculate the distance <math>|Pl|</math>.)