Are there analogues for sin, cos, sinh and cosh?

## 3 Answers

What you are looking for is a parametrization of the curve. This problem was solved by Newton, and there is a very nice algorithm that takes in a polynomial in two variables and outputs two power series x(t), y(t) that parametrize the curve. You can look up "Puiseux series".
You almost surely have to at least restrict to polynomials of 2 variables to be able to say anything useful of this nature. A single polynomial equation of n variables will typically give an n-1 dimensional surface in n-dimensional space, and you're not going to parametrize that by continuous functions of a single parameter t.
I thought (sec(t), tan(t)) parametrized x^2 - y^2 = 1, and I see now that it does too, but over a different domain of t and it's weird to me that both of these work.

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