Editing Ellipse (section)

== In Cartesian coordinates ==
[[File:Ellipse-param.svg|thumb|Shape parameters:{{unbulleted list
| ''a'': semi-major axis,
| ''b'': semi-minor axis,
| ''c'': linear eccentricity,
| ''p'': semi-latus rectum (usually <math>\ell</math>).
}}]]

=== Standard equation ===
The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the ''x''-axis is the major axis, and:
{{unbulleted list | style = padding-left:1.5em
| the foci are the points <math>F_1 = (c,\, 0),\ F_2=(-c,\, 0)</math>,
| the vertices are <math>V_1 = (a,\, 0),\ V_2 = (-a,\, 0)</math>.
}}
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 other focus <math display="inline">\sqrt{(x + c)^2 + y^2}</math>. Hence the point <math>(x,\, y)</math> is on the ellipse whenever:
<math display="block">\sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} = 2a\ .</math>

Removing the [[radical expression|radicals]] by suitable squarings and using <math>b^2 = a^2-c^2</math> (see diagram) produces the standard equation of the ellipse:<ref name="mathworld">{{cite web | url=http://mathworld.wolfram.com/Ellipse.html |title=Ellipse - from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2020-09-10 |access-date=2020-09-10}}</ref>
<math display="block">\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
or, solved for ''y'':
<math display="block">y = \pm\frac{b}{a}\sqrt{a^2 - x^2} = \pm \sqrt{\left(a^2 - x^2\right)\left(1 - e^2\right)}.</math>

The width and height parameters <math>a,\; b</math> are called the [[semi-major and semi-minor axes]]. The top and bottom points <math>V_3 = (0,\, b),\; V_4 = (0,\, -b)</math> are the ''co-vertices''. The distances from a point <math>(x,\, y)</math> on the ellipse to the left and right foci are <math>a + ex</math> and <math>a - ex</math>.

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

=== 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]]).

=== Tangent ===
An arbitrary line <math>g</math> intersects an ellipse at 0, 1, or 2 points, respectively called an ''exterior line'', ''tangent'' and ''secant''. Through any point of an ellipse there is a unique tangent. The tangent at a point <math>(x_1,\, y_1)</math> of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> has the coordinate equation:
<math display="block">\frac{x_1}{a^2}x + \frac{y_1}{b^2}y = 1.</math>

A vector [[parametric equation]] of the tangent is:
<math display="block">\vec x = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \left(\begin{array}{r}
-y_1 a^2 \\
 x_1 b^2
\end{array}\right) , \quad s \in \R. </math>

'''Proof:'''
Let <math>(x_1,\, y_1)</math> be a point on an ellipse and <math display="inline">\vec{x} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \begin{pmatrix} u \\ v \end{pmatrix}</math> be the equation of any line <math>g</math> containing <math>(x_1,\, y_1)</math>. Inserting the line's equation into the ellipse equation and respecting <math display="inline">\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1</math> yields:
<math display="block">
  \frac{\left(x_1 + su\right)^2}{a^2} + \frac{\left(y_1 + sv\right)^2}{b^2} = 1\ \quad\Longrightarrow\quad
  2s\left(\frac{x_1u}{a^2} + \frac{y_1v}{b^2}\right) + s^2\left(\frac{u^2}{a^2} + \frac{v^2}{b^2}\right) = 0\ .</math>
There are then cases:
# <math>\frac{x_1}{a^2}u + \frac{y_1}{b^2}v = 0.</math> Then line <math>g</math> and the ellipse have only point <math>(x_1,\, y_1)</math> in common, and <math>g</math> is a tangent. The tangent direction has [[normal (geometry)|perpendicular vector]] <math>\begin{pmatrix} \frac{x_1}{a^2} & \frac{y_1}{b^2} \end{pmatrix}</math>, so the tangent line has equation <math display="inline">\frac{x_1}{a^2}x + \tfrac{y_1}{b^2}y = k</math> for some <math>k</math>. Because <math>(x_1,\, y_1)</math> is on the tangent and the ellipse, one obtains <math>k = 1</math>.
# <math>\frac{x_ 1}{a^2}u + \frac{y_1}{b^2}v \ne 0.</math> Then line <math>g</math> has a second point in common with the ellipse, and is a secant.

Using (1) one finds that <math>\begin{pmatrix} -y_1 a^2 & x_1 b^2 \end{pmatrix}</math> is a tangent vector at point <math>(x_1,\, y_1)</math>, which proves the vector equation.

If <math>(x_1, y_1)</math> and <math>(u, v)</math> are two points of the ellipse such that <math display="inline">\frac{x_1u}{a^2} + \tfrac{y_1v}{b^2} = 0</math>, then the points lie on two ''conjugate diameters'' (see [[#Conjugate diameters|below]]). (If <math>a = b</math>, the ellipse is a circle and "conjugate" means "orthogonal".)

=== Shifted ellipse ===
If the standard ellipse is shifted to have center <math>\left(x_\circ,\, y_\circ\right)</math>, its equation is
<math display="block">\frac{\left(x - x_\circ\right)^2}{a^2} + \frac{\left(y - y_\circ\right)^2}{b^2} = 1 \ .</math>

The axes are still parallel to the ''x''- and ''y''-axes.

=== General ellipse ===
{{Main|Matrix representation of conic sections}}
[[File:General ellipse.png|thumb|right|upright=1.25|A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle]]

In [[analytic geometry]], the ellipse is defined as a [[Quadratic form|quadric]]: the set of points <math>(x,\, y)</math> of the [[Cartesian plane]] that, in non-degenerate cases, satisfy the [[Implicit and explicit functions|implicit]] equation<ref>{{cite book|url=https://books.google.com/books?id=yMdHnyerji8C | title=Precalculus with Limits|last1=Larson|first1=Ron| last2=Hostetler|first2=Robert P. | last3=Falvo|first3=David C.| publisher=Cengage Learning|year=2006 | isbn=978-0-618-66089-6|page=767 | chapter=Chapter 10 | chapter-url=https://books.google.com/books?id=yMdHnyerji8C&pg=PA767}}
</ref><ref>{{cite book| url=https://books.google.com/books?id=9HRLAn326zEC | title=Precalculus| last1=Young|first1=Cynthia Y.|author-link=Cynthia Y. Young| publisher=John Wiley and Sons| year=2010| isbn=978-0-471-75684-2|page=831| chapter=Chapter 9| chapter-url=https://books.google.com/books?id=9HRLAn326zEC&pg=PA831}}
</ref>
<math display="block">Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0</math>
provided <math>B^2 - 4AC < 0.</math>

To distinguish the [[degenerate conic|degenerate cases]] from the non-degenerate case, let ''∆'' be the [[determinant]]
<math display="block">\Delta = \begin{vmatrix}
               A & \frac{1}{2}B & \frac{1}{2}D \\
    \frac{1}{2}B &            C & \frac{1}{2}E \\
    \frac{1}{2}D & \frac{1}{2}E &            F
  \end{vmatrix} = ACF + \tfrac14 BDE - \tfrac14(AE^2 + CD^2 + FB^2).
</math>

Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.<ref name="Lawrence">Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.</ref>{{rp|p=63}}

The general equation's coefficients can be obtained from known semi-major axis <math>a</math>, semi-minor axis <math>b</math>, center coordinates <math>\left(x_\circ,\, y_\circ\right)</math>, and rotation angle <math>\theta</math> (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
<math display="block">\begin{align}
  A &=   a^2 \sin^2\theta + b^2 \cos^2\theta &
  B &=  2\left(b^2 - a^2\right) \sin\theta \cos\theta \\[1ex]
  C &=   a^2 \cos^2\theta + b^2 \sin^2\theta &
  D &= -2A x_\circ   -  B y_\circ \\[1ex]
  E &= - B x_\circ   - 2C y_\circ &
  F &=   A x_\circ^2 +  B x_\circ y_\circ + C y_\circ^2 - a^2 b^2.
\end{align}</math>

These expressions can be derived from the canonical equation
<math display="block">\frac{X^2}{a^2} + \frac{Y^2}{b^2} = 1</math>
by a Euclidean transformation of the coordinates <math>(X,\, Y)</math>:
<math display="block">\begin{align}
  X &=  \left(x - x_\circ\right) \cos\theta + \left(y - y_\circ\right) \sin\theta, \\
  Y &= -\left(x - x_\circ\right) \sin\theta + \left(y - y_\circ\right) \cos\theta.
\end{align}</math>

Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:<ref name="mathworld"/>

<math display="block">\begin{align}
  a, b    &= \frac{-\sqrt{2 \big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) F\big)\big((A + C) \pm \sqrt{(A - C)^2 + B^2}\big)}}{B^2 - 4 A C}, \\
  x_\circ  &= \frac{2CD - BE}{B^2 - 4AC}, \\[5mu]
  y_\circ  &= \frac{2AE - BD}{B^2 - 4AC}, \\[5mu]
    \theta &= \tfrac12 \operatorname{atan2}(-B,\, C-A),
\end{align}</math>

where {{math|[[atan2]]}} is the 2-argument arctangent function.