What is a generally accepted definition of "curve" in mathematics?

Question

I am wondering if there is a generally accepted definition of the term curve in mathematics. If one does exist, is there any requirement of continuity, beyond what is required by the piecewise differentiable property defined and applied to the definition of a curve in what follows?

Some authors call an entire hyperbola a curve, even though its two branches nowhere share a point. That is somewhat contrary to the naive concept of curve, but isn't too difficult to accept. On the other hand, the following definitions seem to leave a lot of room to produce things which satisfy the definition of curve, but we would never call curves in real life.

This is from Thomas's Calculus and Analytic Geometry, 2nd Edition, 1953.

The cardinal principle of analytic geometry is that an equation $F(x,y)=0$ describes a curve which is the locus of all and only those points $P(x,y)$ whose coordinates satisfy the given equation.

In that context, the meaning of the term curve is closer to what Gray, et al., are calling the trace of a curve in the following taken from Modern Differential Geometry of Curves and Surfaces with Mathematica,3rd Edition:

From Thomas's definition we could produce an equation that determines a set of points, none of which are connected. I assume he was merely giving the historical definition, and not intending to be rigorous.

So, is the mathematical definition of curve really wide open, beyond piecewise differentiability?

Answer 1

There is no unified definition. Curves in differential and algebraic geometry are defined very differently, via parametric and implicit equations, respectively. While the two representations can be related under some broad assumptions (via the implicit function theorem), both subjects push the envelope beyond such relatability. Mature fields are driven by technical reach, not intuition.

While connectedness and differentiability requirements are common in the differential context, they are not in the algebraic one. Studying connected components and algebraic singularities is a big part of the job. This is why hyperbola is one curve. Even within the classical differential geometry, different authors make different conventions about how differentiable a "curve" should be, from infinitely, to twice, to once continuously, each possibly piecewise (although continuity is usually assumed). There is also an intermediate area of analytic, holomorphic and pseudoholomorphic curves that combines methods from both approaches, and has definitional variations of its own.

Even just continuous curves, once deemed "pathological", like the Peano curve filling a square, or the nowhere differentiable Koch snowflake, now have a field of their own, a part of geometric measure theory. The study of such fractal curves has a very different flavor, employing distributions and measure theory, than the classical differential or algebraic geometry.

Answer 2

Yes. And actually, if we get to such things as "space-filling curves", differentiability need not be required at all. Hence, I'd think a good definition - covering both formalism and intuitive conceptalization - would be indeed just that open-ended. The trick is in how we imagine or picture it intuitively, versus how we might imagine a function in other contexts:

In formalism, a Euclidean curve $C$ is essentially nothing more than a drawing function: it is an arbitrary function $\gamma_C : [0, 1] \rightarrow \mathbb{R}^n$, taking a single continuous parameter into $n$-dimensional Euclidean space. The semantic intuition is that we can think of this function as describing how an infinitely fine pen, writing black ink, moves to draw the curve, with discontinuous jumps leaving no connecting traces. The path of the pen is obtained by varying the input, taken as a number ranging from $0$ to $1$ and which can and should be thought of as a sort of "percentage" (to make it an actual one, we could define instead the domain interval as $[0, 100]$ - but it doesn't matter) representing how much of the curve has been drawn so far (e.g. an input value of 0.50 means 50%, or half, of the drawing is done), and the value returned by the function - a set of coordinates - is the point to fill in next. During this process, the parameter is imagined to vary smoothly throughout the interval, such that a tiny point of ink is put down for every single real number progress point.

More generally, we can replace the codomain with a manifold $M$, in which case we can talk about curves in non-Euclidean spaces, such as curves drawn on the surface of a sphere like the Earth: for example, the path of an airplane, ship, or other vehicle travelling over that surface.

ADD: And even more generally - see the comment by @Don Thousand.

Answer 3

This issue came up in a recent question. I will add some clarification in the context of Differential Geometry (and Differential Topology). Unsurprisingly, Wikipedia is also sloppy and inconsistent here.

There are several definitions of the word "curve" commonly used in Differential Geometry and Differential Topology. I will use the $C^\infty$-degree of smoothness for curves, since it is, again, most common.

Definition 1. A smooth curve in ${\mathbb R}^n$ is a smooth 1-dimensional submanifold $C$ of ${\mathbb R}^n$. In other words, $C$ is a subset of ${\mathbb R}^n$ such that for every $p\in C$ there exists a neighborhood $U$ of $p$ in ${\mathbb R}^n$ and a ($C^\infty$) diffeomorphism $f: U\to {\mathbb R}^n$ sending $U\cap C$ diffeomorphically to the 1st coordinate line $L$ in ${\mathbb R}^n$.

Here $L$ is given by: $$ \{(x_1,0,...,0): x_1\in {\mathbb R}\}\subset {\mathbb R}^n. $$

(Equivalently, one can require existence of a diffeomorphism $F: U\to V$, where $V$ is an open subset of ${\mathbb R}^n$, such that $F(U\cap C)=V\cap L$).

Definition 2. A smooth curve in ${\mathbb R}^n$ is a smooth 1-dimensional submanifold with boundary $C$ of ${\mathbb R}^n$. In other words, $C$ is a subset of ${\mathbb R}^n$ such that for every $p\in C$ there exists a neighborhood $U$ of $p$ in ${\mathbb R}^n$ and a diffeomorphism $F: U\to V$, where $V$ is an open subset of ${\mathbb R}^n$, such that $F(U\cap C)=V\cap H$), where $H$ is the closed positive half-line in a line $L$ as above:

$$ \{(x_1,0,...,0): x_1\in {\mathbb R}, x_1\ge 0\}\subset L\subset {\mathbb R}^n. $$

If $F(p)=(0,...,0)$ then $p$ is called a boundary point of $C$.

Note that some authors require $C$ to be connected in both definitions.

One also frequently encounters the notion(s) of a parameterized curve. Again, there are competing definitions here as well:

Definition 3. A smooth parameterized curve in ${\mathbb R}^n$ is a smooth map $c: I\to {\mathbb R}^n$, where $I$ is an interval in the real line. (Here an interval is allowed to be closed, open, half-open, infinite, etc).

Definition 4. A smooth regular parameterized curve in ${\mathbb R}^n$ is a smooth map $c: I\to {\mathbb R}^n$, where $I$ is an interval in the real line, such that $c'(t)\ne 0$ for all $t\in I$.

Here again an interval is allowed to be closed, open, half-open, infinite, etc and, depending on the context, the derivative is required (or not) to be nonvanishing at the end-points of $I$ (if it has any), where the derivative is understood as a 1-sided derivative.

Remark. There are also several competing notions of smoothness map at an end-point of an interval. One definition (as above) works with 1-sided derivatives (as preferred by analysts), while another requires the existence of a smooth extension to a slightly larger open interval (as preferred by topologists). This discrepancy does not affect the definition.

Lastly, one may find a definition where a smooth curve is the image of a map $c$ as in Definitions 3 and 4.