While researching about geometrical significance of addition of two curves, I started off with family of lines,
I came across this question: Concepts involved in "Family of Lines". The accepted answer explains it well, that we create a new coordinate system with the two lines as coordinate axes, and these "family members" pass through origin in new system.
Multiplying the equation of two curves yields the graph of both the curves, since $C_1\times C_2=0$ is satisfied when either one of $C_1$ or $C_2$ are zero. Hence we get graph of both.
In case of lines, it is simple, the coordinate axes.
What does it mean, geometrically, to multiple or add higher degree curves?
Wouldn't it be a coordinate system with non-linear axes? I'm not able to imagine it.
P.S. I'm a high schooler.
You have it right, that if two curves are given by the equations $C_1(x,y)=0$ and $C_2(x,y)=0$, then their union (join) can be described by the single equation $C_1(x,y)C_2(x,y)=0$.
I think you realize that if the curves are graphs of functions, $y=f_1(x)$ and $y=f_2(x)$, then taking the product of the two functions, $y=f_1(x)f_2(x)$, will give you a picture totally different from either of the original two. In this setting, multiplying two things is not as useful or meaningful as the way you presented multiplication.
For plain, straightforward addition, I don’t think that there’s any reasonable picture that comes out, either from $C_1(x,y)+C_2(x,y)=0$, or $y=f_1(x)+f_2(x)$, though performing the latter operation is something we do all the time.
If you’d like a trick that gives you the intersection (common part) of two curves, you can use $C_1(x,y)^2+C_2(x,y)^2=0$, as I’m sure you see. This is good only for the real picture though: if you should venture out into complex curves, as we do in the field of algebraic geometry, the trick will be no good.
I hope this has been of some help.