Ellipse is a very important shape in geometry. Its function defines the orbit of planet and star. Many branches of science and Math need it.

Ellipse Equation in Cartesian Coordinates

General Equation of Ellipse

The normal equation is actually the general 2nd order equation of conical section [1]. Only when certain condition is satisfied will it become an ellipse.

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

When $\Delta = B^2-4AC < 0$, this equation will represent an ellipse.

Standard Equation of Ellipse

The standard equation of an ellipse means its center is also the basis point of the Cartesian coordinates O. And the focus points are also on the x axis.

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Where a is the longer half radius, b is the shorter half radius.

Parameter Equation of Ellipse

The parameter equation of an ellipse is a good option to describe an ellipse. Which is:

\begin{equation} \left \{ \begin{array}{l} x = acos\phi \\ y = bsin\phi \end{array} \right . \end{equation}

Where a and b have same definition as that in the standard equation.

Connection between Standard and General Equation of an Ellipse

Suppose the standard equation of ellipse get rotation $\theta$ angle rotation around the original point and then a shift transformations $\Delta\vec{r}=(x_0, y_0)$, then the transformed ellipse equation will becomes:

$\frac{(xcos\theta - ysin\theta)^2}{a^2} + \frac{(xsin\theta - ycos\theta)^2}{b^2}=1$

Where the $\theta$ angle rotation follows the matrix:

\begin{equation} \left ( \begin{array}{l} cos\theta & -sin\theta\\ sin\theta & cos\theta\\ \end{array} \right ) \end{equation}

After some derivation, we can get:

Reference

[1] https://en.wikipedia.org/wiki/Conic_section