Rectifiable Curve

Curve

A curve can be conceptualized as the trace created by a moving pen on a piece of paper. It is considered a one-dimensional object, which implies that a single number is sufficient to describe the position of a point on the curve. More generally, in $\mathbb{R}^k$ , we define a curve rigorously thus.

Definition. A continuous mapping $\gamma$ of an interval $[a, b]$ into $\mathbb{R}^k$ is called a curve in $\mathbb{R}^k$ .

It is important to note that, under this definition, different curves may specify the same set of points, i.e. they have the same range.

Curve Length

Given a partition $P = \{x_0 , \ldots , x_n \}$ of $[a, b]$ , define

\Lambda(P, \gamma) \doteq \sum_{i \in [n]} \Vert \gamma(x_i) - \gamma(x_{i - 1}) \Vert_2,

where the $i^{(th)}$ term in the sum is the Euclidean distance between $\gamma(x_{i - 1})$ and $\gamma(x_i)$ .

Therefore, $\Lambda(P, \gamma)$ represents the length of a polygonal path with vertices located at $\gamma(x_0), \gamma(x_1), \ldots , \gamma(x_n)$ in order. As the partition becomes increasingly finer, the polygon approaches the range of $\gamma$ more closely. Hence, it is reasonable to define the length of $\gamma$ as

\Lambda(\gamma) \doteq \sup \Lambda(P, \gamma),

where the supremum is taken over all partitions of $[a, b]$ . If $\Lambda(\gamma) < \infty$ , we say that $\gamma$ is rectifiable.

Riemann Integral

In certain cases, $A( \gamma)$ is given by a Riemann integral. We shall prove this for continuously differentiable curves, i.e., for curves $\gamma$ whose derivative $\gamma'$ is continuous.

Theorem. If $\gamma'$ is continuous on $[a, b]$ , then $\gamma$ is rectifiable, and
$\Lambda(\gamma) = \int_{x_{i - 1}}^{x_i} \left\Vert \gamma'(t) \right\Vert_2 \, dx.$

Proof. For each partition $P = \{x_0 , \ldots , x_n \}$ of $[a, b]$ , we have

\begin{aligned} \Lambda(P, \gamma) &= \sum_{i \in [n]} \Vert \gamma(x_i) - \gamma(x_{i - 1}) \Vert_2 \\ &= \sum_{i \in [n]} \left\Vert \int_{x_{i - 1}}^{x_i} \gamma'(t) \, dx \right\Vert_2 \le \sum_{i \in [n]} \int_{x_{i - 1}}^{x_i} \left\Vert \gamma'(t) \right\Vert_2 \, dx = \int_{a}^{b} \left\Vert \gamma'(t) \right\Vert_2 \, dx . \end{aligned}

It follows that $\Lambda(\gamma) \le \int_{a}^b \left \Vert \gamma'(t) \right \Vert_2 \, dx$ and $\gamma$ is rectifiable.

Further, since $\gamma'$ is uniformly continuous on $[a, b]$ , for each $\varepsilon > 0$ , there exists $\delta > 0$ , such that for each partition $P$ with $\max_{i \in [n]} (x_i - x_{i - 1}) \le \delta$ , we have

\Vert \gamma'(s) - \gamma'(t) \Vert_2 \le \varepsilon, \quad \forall s, t \in [x_{i - 1}, x_i].

Hence for each $i \in [n]$ ,

\begin{aligned} \int_{x_{i - 1}}^{x_i} \Vert \gamma'(t) \Vert_2 \, dx &\le \int_{x_{i - 1}}^{x_i} \Big( \Vert \gamma'(x_i) \Vert_2 + \varepsilon \Big) \, dx \\ &= \left\Vert \int_{x_{i - 1}}^{x_i} \gamma'(x_i) \, dx \right\Vert_2 + \int_{x_{i - 1}}^{x_i} \varepsilon \, dx \\ &\le \left\Vert \int_{x_{i - 1}}^{x_i} \gamma'(x_i) - \gamma'(t) \, dx \right\Vert_2 + \left\Vert \int_{x_{i - 1}}^{x_i} \gamma'(t) \, dx \right\Vert_2 + \int_{x_{i - 1}}^{x_i} \varepsilon \, dx \\ &\le \varepsilon \cdot (x_i - x_{i - 1}) + \left\Vert \int_{x_{i - 1}}^{x_i} \gamma'(t) \, dx \right\Vert_2 + \varepsilon \cdot (x_i - x_{i - 1}) \\ &= \Vert \gamma(x_i) - \gamma(x_{i - 1}) \Vert_2 + 2 \varepsilon \cdot (x_i - x_{i - 1}). \end{aligned}

Summing over all $i \in [n]$ , we have

\int_a^b \Vert \gamma'(t) \Vert_2 \, dx \le \Lambda(P, \gamma) + 2 \varepsilon (b - a) \le \Lambda(\gamma) + 2 \varepsilon (b - a).

$\square$

Reference

[1] W. Rudin, Principles of mathematical analysis, 3d ed. in International series in pure and applied mathematics. New York: McGraw-Hill, 1976.