It is fairly straightforward to plot an explicit equation such as $y=x^3+3x^2+2x+5$ in linear time, because you can just iterate through all $x$ in your graphing space and use the equation to calculate $y$ for every $x$.
However, if we need to plot an arbitrary implicit equation, such as $y^2+y+x^3+3x^2+5=0$, past questions seem to suggest that it can only be plotted in quadratic time, such as this one.
My question: Is there any sub-quadratic algorithm to plot an arbitrary implicit equation? (Or do I need to do algebraic manipulation, which will make the use of such a sub-quadratic plotting algorithm very limited?)
This question is completely theoretical, and I'm just wondering if an algorithm exists.
Unless you have specific knowledge about the behavior of the function, the answer is negative.
[Even for the 1D case, it is an oversimplification to say that linear time is enough. By sampling at discrete points, you can always miss important features such as asymptotes.]
For a similar reason, in 2D you cannot drop a part of the domain as you can't predict what happens there. And an implicit equation can very well "fill" the whole domain.
Think of
$$2\sin100x\cdot\sin100y=1$$