I'm taking multivariable calculus, and the textbook briefly mentioned the concept of families of parametric curves, which it roughly defined as a set of equations that have the same form/similar attributes but have constants which can be varied to produce many individual curves, e.g. $x = a+\cos t, y = a \tan t + \sin t$.
Beyond this brief definition/example, the book didn't make use of the concept at all. I asked my GSI when this might come up, and he said that you can use the properties of curves within the family to approximate what he termed "degenerate curves" (which he said were curves which weren't quite within the family/explorable). He also said a more thorough explanation would require much higher level math.
I'm interested if there are any other uses of the concept of families of parametric curves, and what references might exist to start reading about the idea of degenerate curve approximation - I know these may be above my level, but I'd like to at least take a look. (I started looking on Wikipedia to find books in their references for the pages for the keywords I'm aware of, but I honestly think I don't know well enough what I'm looking for, which is why I ask this question.)