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

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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".
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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.
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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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