Sometimes it is convenient to be able to parametrically define closed loops with more features than a circle or an ellipse. Here are some additional examples.

Ellipse

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

or:

\[\begin{aligned} x &= a\cos\theta\\ y &= b\sin\theta \end{aligned}\] 
x = a*np.cos(th)
y = b*np.sin(th)

Oval

\[\begin{aligned} x &= a\cos\theta\\ y &= b\sin\theta \sqrt{1+kx} \end{aligned}\] 
x = a*np.cos(th)
t = 1 + k*x
y = b*np.sin(th)*np.sqrt(t)

Superellipse

\[\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=1\]

or:

\[\begin{aligned} x &= a|\cos t|^{\frac{2}{n}}\text{sgn}(\cos t)\\ y &= b|\sin t|^{\frac{2}{n}}\text{sgn}(\sin t) \end{aligned}\] 
x = np.abs(np.cos(th))**(2/n)*a*np.sign(np.cos(th))
y = np.abs(np.sin(th))**(2/n)*b*np.sign(np.sin(th))

Summary

Comparison of Various Polar Shapes