Given the curve $y^2 = x^3 - 10x + 10$, which looks like this:
Is there any way to prove whether or not there exists an equation of an algebraic curve such that its graph would correspond to a rotation of the above given curve, e.g., the rotation about the origin (please forgive my Paint skills):
Mostly, I'm looking for a way to only simply study whether there is any way to make sense of the above question, so as to see whether there exists an equation which corresponds to the rotated curve. Is there any specific concepts I could begin studying to approach the question? Thank you!
Let the $2D$ curve's equation be $g(x,y) = 0$. Note that rotation about the origin in $2D$ is the same as rotation about the $z$-axis in $3D$.
Rotation about $z$-axis: Let $R_{z,\theta}$ denote the rotation matrix for a rotation of angle $\theta$ about the z-axis and let $(x',y',z')^T := R_{z,\theta} (x_0,y_0, 0)^T$.
$ \begin{align*} R_{z,\theta} = \begin{pmatrix} \cos \theta & - \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} &\Rightarrow \begin{cases} x' &= (\cos \theta) x_0 - (\sin \theta) y_0 \\ y' &= (\sin \theta) x_0 + (\cos \theta) y_0 \\ z' &= 0 \end{cases} \\ &\Rightarrow \begin{cases} x_0 &= (\cos \theta) x' + (\sin \theta) y' \\ y_0 &= - (\sin \theta) x' + (\cos \theta) y' \\ 0 &= z' \end{cases} \end{align*} $
So on rotating the curve $g(x,y) = 0$ around the $z$-axis by angle $\theta$, the equation of the curve we get is: $$g((\cos \theta) x' + (\sin \theta) y', - (\sin \theta) x' + (\cos \theta) y') = 0$$ We can't eliminate $\theta$ from this equation to get a single equation for the curve formed by rotating $g(x,y) = 0$ by $2\pi$ in this case.