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	$$\left(\begin{array}{c}x\\ y\end{array}\right)=\left[\begin{array}{cc}cos\varphi & -sin\varphi \\ sin\varphi & cos\varphi \end{array}\right]\cdot \left(\begin{array}{c}\frac{e\cdot d\cdot cos\theta }{(1-e\cdot sin(\theta ))}\\ \frac{e\cdot d\cdot sin\theta }{(1-e\cdot sin(\theta ))}\end{array}\right)$$	$\varphi$ 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 (-)