Given ax+by+c=0, what is the set of all operations on this equation that do not alter the plotted line? Operations such as sqrt(f(x)2), f(x)+a−a, are obvious candidates for such a set. However, e.g., for the line y=−x, it seems to me to be non-trivial that x:3+y:3=0 will plot the same line but x:2+y:2=0 won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?) The following are some more thoughts on the question: It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, x/a+x/b=1 makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, a

Given ax+by+c=0, what is the set of all operations on this equation that do not alter the plotted line?
Operations such as $f(x)+a-<!--\; -\; -->a$, are obvious candidates for such a set. However, e.g., for the line y=−x, it seems to me to be non-trivial that $x^3+y^3=0$ will plot the same line but $x^2+y^2=0$ won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?)
The following are some more thoughts on the question:
It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, $\frac{x}{a}+\frac{x}{b}=1$ makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, $ax+by+c=0$ might fail to be represented $\frac{x}{a}+\frac{x}{b}=1$. So we're lead to think that these two equations plot a line by virtue of legitimate operations between them.
The equation of a plane also seems to be nicely related to the general form of a line, if $r_0=(x_0,y_0)$ and r=(x,y) are two vectors pointing to the plane and the normal is $n=(n_x,n_y)$. If $\circ <!--\; \circ \; -->$ between vectors is the dot product, $(x-<!--\; -\; -->x_0,y-<!--\; -\; -->y_0)\circ <!--\; \circ \; -->n=(x-<!--\; -\; -->x_0)\ast <!--\; \ast \; -->n_x+(y-<!--\; -\; -->y_0)\ast <!--\; \ast \; -->n_y=n_x\ast <!--\; \ast \; -->x+n_y\ast <!--\; \ast \; -->y-<!--\; -\; -->(x_0{n}_x+y_0{n}_y)=a\ast <!--\; \ast \; -->x+b\ast <!--\; \ast \; -->y+c=0$
The idea is to be able to see how the form of an equation can be altered, not the content of the variables. It seems odd to me that very complicated equations could have the same plotted curve as simple forms, but that this property wouldn't appear by virtue of the equation themselves, or the set of valid operations on this equation. This might seem weird, but say it is never immediately obvious that ax+by+c=0 plots a line, or $x^2+y^2=r^2$ plots a circle, unless we actually do the plotting, and ax+by+c=0 seems way less fundamental than y=mx+b.
Note that in the case of a circle, we have the pythagorean theorem that seems to be its clearest representation with the methods of analytic geometry, and the moment an equation can be said to share some sort of operation set with the pythagorean theorem, we know we're speaking of a circle. It seems that if we could somehow draw the operation set of a circle, we would get something like the pythagorean theorem, and that this operation set gets somehow deformed in order to give a representation onto the cartesian plane. For a translated circle with center (h,k), $x^2-<!--\; -\; -->2xh+h^2+y^2-<!--\; -\; -->2yk+k^2=r^2$ means absolutely nothing to us, but the form $(x-<!--\; -\; -->h{)}^2+(y-<!--\; -\; -->k{)}^2=r^2$ is clear as day.