Editing Ellipse (section)

=== 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.