Here's the same path (in black), and an assembly of circles and straight lines that duplicate the curved path (in red)*. Moral: All curves can be reconstructed from circles and lines - so you only need to know how to deal with two types of figures - circles and lines. Of course, other types of curves - ones that can be described nicely with equations - are often used to analyze problems. We will soon see how a particularly useful set of such curves, the conic sections, have important physical applications.
*OK, I'll admit that I drew the circles and lines first, and then cut out the ant's path later, but the fact remains that it can be done the other way, and so my point should still be valid.