What is the difference between surface and algebraic curve in general?

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?

• It is hard to answer your confusion when you don't provide justification for your thinking. Why do you think we should call $\sigma$ a curve? Perhaps you are focusing on the difference between the maps $\sigma$ and $\sigma\circ x$. The former is a map from $R^n$ to $R^m$, and the preimage of zero is a surface (under suitable regularity conditions). The map $\sigma\circ x$ however is a map from $R$ to $R^m$, and this is indeed a curve (under suitable regularity conditions). – symplectomorphic Jul 18 '16 at 14:04
• @symplectomorphic I really wish I was smart enough to understand what you are saying. You asked why do I think we should call $\sigma$ a curve. Like I said, this is a question asked from somebody else and I have no idea about the answer. I was confused about the general concepts of curve and surface and I hoped somebody could shed a light in an understandable language. – polfosol Jul 19 '16 at 4:10
• the word "curve" has different definitions depending on the field of study. the most general idea is a geometric object that is, in some sense, one-dimensional, or dependent on only one parameter. the definitions you just cited are of algebraic curves, not curves in general: hence the reason for the adjective "algebraic" in front of the noun. in calculus and differential topology and geometry, a curve (or parametrized curve) is sometimes a nice map from R (the set of real numbers) to a manifold M; other times it is just the image of that map. – symplectomorphic Jul 19 '16 at 6:34
• the main difference between the notion of curve and the notion of surface is that the former depends only on one parameter, while the latter depends on two. for example, the map from $R$ to $R^3$ that sends $t$ to $(\cos t, \sin t, t)$ is a (parametrized) curve, namely an infinite helix, while the map defined by $(s\cos t, s\sin t, 0)$ for $s$ in $(0,1)$ and $t$ in $(0,2\pi)$ is a (parametrized) surface, namely the unit disk in the $xy$ plane with the center and the point $(1,0)$ deleted. – symplectomorphic Jul 19 '16 at 6:39
• finally, the only reason a complex curve can be thought of as a surface, as your quote above says, is that the complex plane is itself two-dimensional over the real numbers. but the notion of curve in algebraic geometry is not the same as the notion of curve in differential geometry. what you really should be asking is "how has the intuitive notion of a curve been made mathematically precise?" the answer is: in many different ways, and which way you choose depends on your other mathematical goals. – symplectomorphic Jul 19 '16 at 6:46

After perusing your Wikipedia link, "I don't know for sure", but here's the explanation that seems most likely to me (a geometer who knows next to nothing about control theory).

The state of a system under sliding mode control is modeled as a point in some phase space, a mathematical object encoding both physical configuration (position) and infinitesimal motion (velocity).

In any particular situation, a system's state traces a curve in the phase space.

The phase space itself (i.e, the set of possible states), constitutes a larger dimensional "hypersurface", which for brevity has come to be called a surface.

On the Wikipedia page, it appears the terms hypersurface and manifold are used interchangeably to speak of the locus of multiple constraints. If that's right, the meanings of those terms differs from common usage in differential geometry: In mathematics, a hypersurface is given by one constraint ("has codimension one"), and a manifold is smooth ("has a tangent space at each point").

It's certainly true that the same technical terms (particularly, curve and surface) have different definitions depending whether you ask a differential geometer or a control theorist.

Separately, a complex curve (a geometric object described locally by one complex parameter) is indeed a (special type of) real surface (described locally by two real parameters), but this appears to be a coincidence in your context.

One final take-away message: Although mathematical theorems have an absoluteness about them once notation, terminology, and logical axioms are reconciled, notation and terminology (and even logical axioms) are by no means universal.

In fact, the notational idioms in mathematics, the sciences, and engineering differ considerably. That's a fact of life, the Babel of quantitative endeavors.

I am not an expert in this domain, but as a general rule, I would usually consider a curve to be a one-dimensional surface. (Is the question why you would call it a surface instead of a curve?)

However, if I wanted to split hairs about the difference between a curve and a surface (again in general), I would say that a surface is a particular shape in space (i.e. a manifold $S\subseteq \mathbb{R}^n$), and that a curve is technically a continuous function sending $f:[0,1]\rightarrow \mathbb{R}^n$.

That would make the image of the curve—i.e. the set of points $\{f(x) : x\in [0,1]\}$— a surface, while the "curve itself" refers to a function $f$.

At a high level, a surface may be parameterized in many different ways, while a curve refers to a specific parametrization of a (one-dimensional) surface.

• Do you have any reference? Specially for the definition of a curve. – polfosol Jul 18 '16 at 4:41
• @polfosol Wikipedia (en.wikipedia.org/wiki/Curve) equivocates, saying that "Depending on the context, it is either [the curve] or its image which is called a curve." Mostly I just wanted to point out that the distinction between function and image is available as a possible answer. – user326210 Jul 18 '16 at 4:45