I would like to get more familiar with parametrization techniques for closed curves (on a 2D plane).
To be more specific:
Given a curve on a 2D plane I want to approximate it through a series (polynomial, Fourier etc) and parametrize the series. The approximation is important because it could be that the function of the curve is unknown but individual samples can be retrieved permitting a descent approximation. One example that I found, where the authors use the Fourier series to parametrize a picture of Einstein can be found here: https://www.i-programmer.info/projects/119-graphics-and-games/5735-how-to-draw-einsteins-face-parametrically.html
I am wondering if there are other approaches for that problem.
Can you recommend any good books/souce that can work as a good starting point?
If you write $x$ and $y$ as a linear function of a single parameter you get a line. For example: $x=2t-3, y=4t+1$ is line. If you eliminate the parameter $t$ you get the cartesian equation of line as $\frac{x+3}{2}=\frac{y-1}{4}.$
Similarly, if we write $x=r \cos t, y=r \sin t$. If $t$ is a parameter and $r$ is fixed (say $r=3$), then we have a circle of radius 3. If you eliminate $t$ by squaring and adding the two equations, you get the cartesian equation of circle as $x^2+y^2=9$. If you fix $t=t_0$ and let $r$ be the parameter then it is a straight line passing through origin as you divide the two equation and get the line $y=x \tan t_0$
$x=at^2, y=2at$, where $a$ is fixed and $t$ is parameter represents a parabola. If you eliminate $t$ you get $y^2=4ax$.
$x=t, y=1/t$ represent rectangular hyperbola $xy=1$.
$x=a \cos t, y= b\sin t$ represents ellipse $x^2/a^2+y^2/b^2=1$
$x=a sec t, y=b\tan t$ represents hyperbola $x^2/a^2-y^2/b^2=1$.
$x=ce^t, y=ce^{-t}$, represents rectangular hyperbola $xy=c^2$
So on and so forth.