The physical analogue would be taking a piece of paper and drawing a curve on it.
If I'm able to identify the curve to an equation, I would be able to perform algebra on that equation in such a way that the result would be another curve that would lie on the plane, or I could perform an algebraic operation on the equation such that it could no longer represent any curve. I'm thinking of for example the way we define vector spaces to make vectors behave in the way we expect them to. What are the assumptions behind for instance having $ax + by + c = 0$ be able to be plotted as a continuous line on the Cartesian plane, or any other curve?
A bit of an explanation of the question:
The idea stems from a comparison between an Euclidean plane (i.e. the one where Euclid does his proofs) and a Cartesian one. So as to produce some sort of correspondence between these two planes, surely we have to assume a certain structure on the Cartesian plane, so that results in one can apply to the other, but so that in the Cartesian plane we can draw geometrical figures in terms of equations.
This wouldn't be a problem if geometrical figures on the Cartesian plane could be dealt with so that there is a 1 to 1 correspondence in the algebra with the length of a segment, or e.g. $x^2$ and the area of a square, as for instance in the proofs of Book 2 of Euclid, where we could really speak of a sort of geometrical algebra. Somewhere in our transition between the Euclidean plane and the Cartesian one, we lose a lot of information to be able to use equations to model curves, and get unintuitive results like the equation of a line $ax + by + c = 0$, which I can't bring myself to believe unless I plot the points which satisfy the equation, or otherwise change its form.
I'm trying to get a better understanding of the Cartesian plane, and the equation of common geometrical forms like $ax + by + c = 0$ or $x^3 + y^3 = 0$ for a line, or $x^2 + y^2 = r^2$ for a circle, which seem to be very far away from the notion of an algebra where $x$ would correspond to a line, $x^2$ to a square and $x^3$ to a cube.
In a limited sense, e.g., $ax + by + c = 0$ does behave like a line would insofar as if I increase $a$, I get a rotation, if I increase $b$, I get a rotation, and if I increase $c$, I get a translation, but this seems completely coincidental to me, and it feels like any such properties of equations modeling curves are a posteriori, as in impossible to discover unless you actually perform the modification and then plot anew.
$x^2 + y^2 = r^2$ is almost intuitive if we imagine a right triangle with hypotenuse of length $r$ in it, but then if we make a translation, the equation ceases to make any sense to me, especially if we distribute: $(x-h)^2 + (y-k)^2 = r^2 \ \ \leftrightarrow \ \ x^2 - 2xh + h^2 + y^2 - 2yk + k^2 = r^2$.
I'm supposing that by making explicit the structure we assume about the Cartesian plane (or other planes where we could plot a curve) I could get a deeper insight into the form these equations take.
References would be very welcome.