Chart of Ellipse Equations

In this chart,

• $a$ is always the semi-major axis length.
• $b$ is always the semi-minor axis length.
• $r$ is the length from a focal point to some point on the ellipse.
• $r$ may also be used as a function of some other variable, $t$ or $\theta$.
• Both $t$ and $\theta$ are angles and will vary from $0$ to $2\pi$.
• The center point is $(h,k)$.
• $x$ and $y$ are the Cartesian axis coordinates.
• $c$ is a half focal length, where the focal length is the distance between foci.
• $d$ is the distance from the directrix to some point on the ellipse.
• $d$ and $r$ when used as lengths have to be used together to reference the same point.
• $e$ is the eccentricity.
• The directrix is $d$ units in a direction that puts it outside the ellipse and perpendicular to the major axis.

A generally useful equation to relate many of these distances: $$e=\frac{c}{a}=\frac{r}{d}=\sqrt{1-\frac{b^{2}}{a^{2}}}$$

Shape Type	Equation	Notes
Tall Cartesian	$$\frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}=1$$	$center:\,(h,k)$
Tall Polar	$$r(\theta)=\frac{e\cdot d}{(1\pm e\cdot sin(\theta))}$$	top focal point (+) bottom focal point (-)
Tall Parametric	$$x(\theta)=\frac{e\cdot d\cdot cos\theta}{(1\pm e\cdot sin(\theta))}$$ $$y(\theta)=\frac{e\cdot d\cdot sin\theta}{(1\pm e\cdot sin(\theta))}$$	top focal point (+) bottom focal point (-)
Rotated Parametric	$$\left(\begin{array}{c} x\\ y \end{array}\right)=\left[\begin{array}{cc} cos\phi\ & -sin\phi\\ sin\phi\ & cos\phi \end{array}\right]\cdot\left(\begin{array}{c} \frac{e\cdot d\cdot cos\theta}{(1-e\cdot sin(\theta))}\\ \frac{e\cdot d\cdot sin\theta}{(1-e\cdot sin(\theta))} \end{array}\right)$$	$\phi$ is the angle of rotation and the equation implies matrix multiplication.
Rotated Parametric	$$\left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} h\\ k \end{array}\right)+a\cdot\cos(\theta)\left(\begin{array}{c} u_{x}\\ u_{y} \end{array}\right)+b\cdot\sin(\theta)\left(\begin{array}{c} v_{x}\\ v_{y} \end{array}\right)$$	$\mathbf{u}$ is the “unit” direction vector of the principle axis and $\mathbf{v}$ is orthogonal to $\mathbf{u}$.
Wide Cartesian	$$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$$	$center:\,(h,k)$
Wide Polar	$$r(\theta)=\frac{e\cdot d}{(1\pm e\cdot cos(\theta))}$$	top focal point (+) bottom focal point (-)