Editing Ellipse (section)

=== Parameters ===

==== Principal axes ====
Throughout this article, the [[semi-major and semi-minor axes]] are denoted <math>a</math> and <math>b</math>, respectively, i.e. <math>a \ge b > 0 \ .</math>

In principle, the canonical ellipse equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1 </math> may have <math>a < b</math> (and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names <math>x</math> and <math> y</math> and the parameter names <math>a</math> and <math> b.</math>

==== Linear eccentricity ====
This is the distance from the center to a focus: <math>c = \sqrt{a^2 - b^2}</math>.

==== Eccentricity ====
[[File:Pythagorean_theorem_ellipse_eccentricity.svg|thumb|upright|Eccentricity ''e'' in terms of semi-major ''a'' and semi-minor ''b'' axes: {{nowrap|1=''e''&sup2; + (''b/a'')&sup2; = 1}}]]
The eccentricity can be expressed as:
<math display="block">e = \frac{c}{a} = \sqrt{1 - \left(\frac{b}{a}\right)^2},</math>

assuming <math>a > b.</math> An ellipse with equal axes (<math>a = b</math>) has zero eccentricity, and is a circle.

==== Semi-latus rectum ====
The length of the chord through one focus, perpendicular to the major axis, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' <math>\ell</math>. A calculation shows:<ref>{{harvtxt|Protter|Morrey|1970|pp=304,APP-28}}</ref>
<math display="block">\ell = \frac{b^2}a = a \left(1 - e^2\right).</math>

The semi-latus rectum <math>\ell</math> is equal to the [[radius of curvature]] at the vertices (see section [[#Curvature|curvature]]).