I can't find a clear, comprehensive explanation, on this site or elsewhere, for why parametrically defined curves frequently have the condition that the the derivatives of their points $x = f(t)$ and $y = g(t)$ cannot simultaneously be zero on the interval $[a, b]$.
Most of the explanations use language that assumes that the reader already understands the concept they're explaining, or the explanations make the meaningless claim that the curve must be "nice".
I would appreciate it if people could please take the time to explain, comprehensively (not rigorously), what is meant by this condition. If you're going to use words that are likely to be unfamiliar to someone who doesn't understand this concept, like "regular", then please take the time to define what it means.