Why is viewing parametric equations in two dimensions better than three dimensions?

Question

As I understand, equations qualify as parametric iff multiple dependent variables depend on one or more common independent variable(s). The prototypical case involves two dependent variables and one independent variable (often time).

According to the Khan Academy video, "you might think, when you visualize something like [a prototypical parametric equation], 'ah, it's got...a single input, and it's got a two-dimensional output; let's graph it [in 3D]...but what turns out to be even better is to look just in the output space [2D]."

Of course, we can preserve some information about the input space on a 2D graph by marking a finite number of values of the independent variable, but why is this 2D graph actually better for visualizing the parametric equations than the 3D graph?

Answer 1

The 3D graph has more information than the 2D graph, but sometimes less is better: it's easier to see some of the important features without the distraction of the third dimension.

Answer 2

One main point of a parametric equation is to define a curve in the plane (or more generally in the target space of the map). For example, a circle is the curve parameterized by $(\cos t, \sin t)$. So drawing the parameterized curve is frequently more descriptive in that it shows the thing we started out trying to describe. If we're using the map $t \mapsto (\cos t, \sin t)$ to understand the circle, it makes more sense to draw it as a circle than as a helix.