An ellipse is a curve that is the locus of all points in the plane the sum of whose distances $r_1$ and $r_2$ from two fixed points $F_1$ and $F_2$ (called the foci) separated by a distance of $2c$ is a given positive constant $2a$, i.e., $r_1 + r_2 = 2a$.

The eccentricity of an ellipse, i.e., its elongation can be expressed as $e=\frac{c}{a}$ assuming $a > b$.

When $F_1 = F_2$, we get a degenerate ellipse, i.e., a circle.

$2a$ and $2b$ are called the major and minor axis, respectively.

The point $C$, midpoint of the segment $F_1F_2$ is called the center/centre of the ellipse.

The equation of an ellipse with center/centre $\left(x_0,\,y_0\right)$ with major axis $2a$ and major axis $2b$ in Cartesian coordinates is

The parametric equations of an ellipse with center/centre $\left(x_0,\,y_0\right)$ can be given by

In polar coordinates, with the origin at the center/centre of the ellipse and with the angular coordinate $\theta$ measured from the major axis, the ellipse’s equation is

If instead we use the origin at one focus, with the angular coordinate $\theta= 0$ still measured from the major axis, the ellipse’s equation is

where the sign in the denominator is negative if the reference direction $\theta=0$ points towards the center/centre, and positive if that direction points away from the center/centre.