The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode control, which is a control scheme for nonlinear system. Briefly explaining, in sliding mode control we have a $\sigma(x)$ which is a scalar function of the vector $x(t)$, and $x$ represents the system states. (I think you do not need to be totally familiar with these concepts and a short glimpse might be enough to answer the question.) Then someone asked her why we call the $\sigma(x)$ a surface? Why don't we call it a sliding curve?
So this question led me to the basic question of, what is the general definition of a curve and a surface and what is the difference between them? Wikipedia says:
A plane algebraic curve is the locus of the points of coordinates $x,y$ such that $f(x,y)=0$, where $f$ is a polynomial in two variables defined over some field $F$. Algebraic geometry normally looks not only on points with coordinates in $F$ but on all the points with coordinates in an algebraically closed field $K$.
and here it says:
A complex projective algebraic curve resides in n-dimensional complex projective space $CP^n$. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over $C$ likewise has topological dimension two; in other words, it is a surface.
I am not an expert in math. From what I have learned previously, a curve refers to a one-dimensional object and surface is something two-dimensional (Not precise I know, intuitively speaking...) But these definitions left me confused.
I general n-dimensional space, or in topology, what is called a curve and what is a surface? And referring to the original question, what is wrong with calling the $\sigma(x)$ a sliding curve?