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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.)

Thank you!

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  • $\begingroup$ Where a family of parametric curves "roll" along another curve (that is, where the curves with very close parameters meet), they generate an envelope. See this answer or this answer for examples. $\endgroup$
    – robjohn
    Sep 4, 2021 at 22:36
  • $\begingroup$ @robjohn this looks interesting! When are envelopes used (in higher math or in applications)? Thank you for the response :) $\endgroup$ Sep 4, 2021 at 22:42
  • $\begingroup$ The question for which I wrote the first of those answers mentions this question, which shows a real life application. I had an art project in elementary school which used envelopes to create curves with thread. $\endgroup$
    – robjohn
    Sep 4, 2021 at 22:52

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Parameterized families of functions appear in many places. Often, one of the interesting features they have is their envelope.

Envelopes

When a family of parametric curves "roll" along another curve (that is, where the curves with very close parameters meet), they generate an envelope. See this answer or this answer or this answer or this answer or this answer for examples. For a two-dimensional envelope in $\mathbb{R}^3$, see this answer.


The Family in the Question

The family of curves parametrized by $$ (a+\cos(t),a\tan(t)+\sin(t))\tag1 $$ that are given in the question is

enter image description here

This family does not seem to have an envelope, except perhaps where they self-intersect at $(0,0)$. To compute the envelope, we need to solve $$ \left.\frac{\partial}{\partial a}(a+\cos(t),a\tan(t)+\sin(t))\,\middle\|\,\frac{\partial}{\partial t}(a+\cos(t),a\tan(t)+\sin(t))\right.\tag2 $$ that is, $$ \left.(1,\tan(t))\,\middle\|\,\left(-\sin(t),a\sec^2(t)+\cos(t)\right)\right.\tag3 $$ whose solution is $$ -\cos(t)=a\tag4 $$ which, when plugged back into $(1)$, gives a one point envelope at $(0,0)$ when $-1\le a\le1$.

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