Sorry If this question is too cumbersome, I am still a bit lost at the basics of differential geometry: Given the curve $(\cos t,\sin 3t)$, how to obtain the parametrization of this curve? I've been thinking about using the relation $\tilde\gamma(t)=\gamma(\phi(t))$ and comparing it with the arc length parametrization. I'm a bit confused.
2026-04-06 09:55:22.1775469322
If I "randomly" come up with curve, how do I obtain it's parametrization?
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As you suspected, you are a bit confused. You probably need to do some reading. But here are a few hints to get you started.
You ask how you can find a parametrization for a given curve. If the curve is given in implicit form, i.e. $F(x,y)=0$, it may not be possible to parametrize it. Not all curves can be parametrized.
You ask about the parametrization of a curve. But any curve that has a parametrization actually has several. You can just compose a given parametrization with another function (preferably monotone) to get a new parametrization.
The curve you gave as an example is already in parametric form. The function $t \mapsto (\cos t, \sin 3t)$ already is a parametrization.
Maybe what you’re trying to find is an arclength parameterization. Given any random parametric curve, you can construct an arclength parametrization, but it’s going to involve integrals that can hardly ever be expressed via simple formulas.