So I was doing some problems looking for curves of an intersection between two surfaces. I was wondering how do I know when i "got" it? What is the algorithm of finding these curves?
For example,
Take the surfaces $z=4x^2+y^2$ and $y=x^2$. So the curve would be $z=4x^2+x^4$ if we just plug in one of the equations But this is not where the story ends. Now to fully represent the curve lets parametrize it. $x=t, y=t^2, z=4t^2+t^4$.
So here i had to plug in of the equations into the other one. Is this always how you solve such problems? Sometimes we want to parametrize first and then do some plugging but in the end, we have to somehow plug one equation into the other one. Am i correct? I am sorry if the question is vague but i am looking for a more or less universal way of finding a solution to such a problem.
Curves and surfaces can be described in different ways, among others with implicit, explicit and parametric equations.
If the two surfaces are given by explicit or implicit equations, like in your example, just forming a system with the two equations
$$\begin{cases}z=4x^2+y^2,\\y=x^2\end{cases}$$ is a representation of the curve and you can stop here. This representation allows you to check if a given point $(x,y,z)$ belongs to the curve.
If you are requested to generate points on the curve, then a parametric representation is more appropriate. You can obtain one by seeing one of the variables as independent, and solving the system of equations for the other two variables. For instance
$$(x,x^2,4x^2+x^4)$$ describes the given curve.
In the case of general equations, there is no systematic method and sometimes no analytical solution at all.