Parametric curves

Before I discuss parametric equations I want to talk about how some textbooks or even teachers may be, unintentionally, misleading. When textbooks introduce functions there is always the discussion that a function of one variable cannot be a circle, because by plotting a circle a function is assuming two different values for the same argument, which is not possible.

Some textbooks present a graph similar to this:

Now, a graph of a piecewise function may have this appearance:

What I'm trying to say is that, unintentionally, a textbook may be making you think that because a trajectory in 2D can have very complicated functions describing each coordinate. Then the trajectory can be a very complex graph. Not quite. In the same way some graphs of single variable functions can only be made with a piecewise function, we'd need piecewise functions to describe a super complicated trajectory with many loops and turns. One equation that describes one trajectory passing through all the points is impractical.

This tangentially relates to numerical methods because sometimes the data is spread in such a way that one single function for all is not enough. We have to look at pieces and different equations for each piece. That or we have to disregard some data to simplify the description.