Curves

Curves are the simplest of the multivariate function types — a single parameter traces out a path through space. We met them briefly when surveying the common types of multivariate functions; here we develop them more fully.

A curve is a mapping

γ:I⊆R→Rm,γ(t)=(x1(t)⋮xm(t))\gamma : I \subseteq \mathbb{R} \to \mathbb{R}^m, \quad \gamma(t) = \begin{pmatrix} x_1(t) \\ \vdots \\ x_m(t) \end{pmatrix}

of an interval II with continuous component functions x1,…,xm:I→Rx_1, \ldots, x_m : I \to \mathbb{R}. The two cases that show up most often are plane curves (m=2m = 2) and space curves (m=3m = 3).

Continuous by definition, differentiable only on request. The definition requires the components xix_i to be continuous — so every curve is automatically continuous, but not automatically differentiable. A curve with corners or cusps (think of a square traced around its perimeter) still counts as a curve; differentiability is an extra property we layer on in the Smoothness section below.

Why II instead of DD?

Elsewhere we use DD for the domain of a general multivariate function, since the domain can be any subset of Rn\mathbb{R}^n. For curves, the input lives in R1\mathbb{R}^1 and the domain is by definition an interval — so we switch to II to make that built-in restriction visible at a glance. It’s a notational hint, not a different concept: II is just the curve-specific name for the same domain slot.

Both plane curves and space curves were visualized when we introduced them — see those sections for the worked parametrizations.

Image (Trace)

As tt ranges over the interval II, the curve γ\gamma visits a set of points in Rm\mathbb{R}^m. That set — the geometric “drawing” the parametrization leaves behind — has its own name.

The image (or trace) of a curve γ:I→Rm\gamma : I \to \mathbb{R}^m is the set of all points it visits as tt runs over II:

γ(I)={ (x1(t),…,xm(t))⊤∣t∈I }.\gamma(I) = \{\, (x_1(t), \ldots, x_m(t))^\top \mid t \in I \,\}.

Different parametrizations can produce the same image. For example, t↦(cos⁡t,sin⁡t)⊤t \mapsto (\cos t, \sin t)^\top on [0,2π][0, 2\pi] and t↦(cos⁡2t,sin⁡2t)⊤t \mapsto (\cos 2t, \sin 2t)^\top on [0,π][0, \pi] both trace the unit circle — only the speed (and the parameter range) differ. The image cares about where the curve goes, not how it gets there.

Start, End, and Closed Curves

When a curve is defined on a closed interval [a,b][a, b], its two endpoints get standard names.

For a curve γ:[a,b]→Rm\gamma : [a, b] \to \mathbb{R}^m, the value γ(a)\gamma(a) is called the start (or starting) point and γ(b)\gamma(b) the end (or ending) point. The curve is called a closed curve if these two points coincide:

γ(a)=γ(b).\gamma(a) = \gamma(b).

A closed curve loops back on itself. The plane curve example we introduced earlier — the ellipse γ(t)=(2cos⁡t,sin⁡t)⊤\gamma(t) = (2\cos t, \sin t)^\top on [0,2π][0, 2\pi] — is closed: γ(0)=(2,0)⊤=γ(2π)\gamma(0) = (2, 0)^\top = \gamma(2\pi). The space curve spiral, by contrast, climbs in zz as tt grows, so it never returns to its starting point and is not closed.

Smoothness

A curve inherits its smoothness from its component functions: if every xi:I→Rx_i : I \to \mathbb{R} is differentiable as a 1D function, γ\gamma is differentiable as a curve. The standard CkC^k classes carry over with curve-flavored names.

A curve γ:I→Rm\gamma : I \to \mathbb{R}^m with component functions x1,…,xm:I→Rx_1, \ldots, x_m : I \to \mathbb{R} is called a CkC^k-curve (or kk-times continuously differentiable) if every component xix_i is of class CkC^k on II. The two cases that come up most often:

  • C1C^1-curve (continuously differentiable) — each xix_i has a continuous first derivative.
  • C2C^2-curve (twice continuously differentiable) — each xix_i has continuous first and second derivatives.

A curve γ:I→Rm\gamma : I \to \mathbb{R}^m is called a piecewise C1C^1-curve (or piecewise continuously differentiable) if the interval II can be partitioned into finitely many subintervals on each of which the component functions x1,…,xmx_1, \ldots, x_m are continuously differentiable.

The piecewise variant is what lets us treat curves with corners — think of a square traced around its perimeter, where the velocity flips direction abruptly at each vertex but is well-defined on the four open edges. The curve as a whole isn’t C1C^1 (the derivative jumps at the corners), but it’s piecewise C1C^1 because we can split the parameter interval into four pieces on each of which it is.