I want to know what the parametric equation for an ellipse is if the one of the foci is centered at the origin. I know the semi-major and minor axes.
I know the parametric equation of an ellipse where the center of the ellipse is at the origin is $x = a\cos(t)$ and $y = b\sin(t)$ but I want to know the parametric equation where one of the foci is at the origin.
Best Answer
In polar coordinates, the equation for an ellipse with one focus at the origin, and whose center lies in the direction $\phi$ from the origin, is $$r = \frac{a(1-e^2)}{1 - e\, \cos(\theta - \phi)}$$ where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $e = \sqrt{1 - \frac{b^2}{a^2}}$. Using typical data available about planetary orbits, you can set $a$ equal to the planet's mean distance from the Sun and $e$ equal to the eccentricity of the orbit.
To parameterize the $x$ and $y$ coordinates, just convert from polar to Cartesian coordinates: $x = r \cos \theta$, $y = r \sin \theta$.
An advantage of this formulation is you can plot the perihelion of the orbit in any direction you want, and you can easily plot the orbits of multiple planets (each of which has its perihelion in a different direction from the Sun).