Parameterizing Implicit Polynomial Cruves

Hi,

So we can parameterize the unit circle x^2+y^2=1 with x=cos(t), y=sin(t). Suppose we have a curve P(x)+P(y)=1 where P is a polynomial. Can we write an explicit parameterization for this curve, perhaps in terms of trigonometric polynomials (e.g., cos(t)+cos(2t)+...)? We may also want to have additional constraints on P(x), such as its degree is even, or that it only contains even powers of x. We can also ask to parameterize the curve P(x)+Q(x)=1, where P and Q are different polynomials. Thanks for any help!!!

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$