2030.sagews

Parametric equations are equations in which the values of both $x$ and $y$ are determined by functions of some other variable, say $t$:

$x = f(t)$

$y = g(t)$

Here are some sample function definitions for $f$ and $g$:

f(t) = cos(3*t)
g(t) = 3*sin(t)

This line creates a list of ordered pairs where $x = f(t)$, and $y = g(t)$:

domain = [0..2*pi, step = pi/180]

P = [(f(t), g(t)) for t in domain]

And this line plots points constructed from those ordered pairs.

points(P).show(aspect_ratio = 1)

Feel free to explore this.  Modify parts of it and see what happens.  

Experiment with different function definitions for $f$ and $g$.  Experiment with different domains for $t$.

Don't worry about breaking the code.  You can always access the original file.

@interact
def _(a = 4, b = 3, r = 1, s = 1):
    domain = [0..2*pi, step = pi/180]
    P = [r*(cos(a*t), s*sin(b*t)) for t in domain]
    show(points(P).show(aspect_ratio = 1))