Chapter 2: Arc-Length Parameter
Definition of arc length, reparametrization by arc length, and unit-speed curves
Introduction
The parametric representation of a curve is not unique. However, when we use the "distance along the curve" (arc length) as the parameter, the geometric properties of the curve can be expressed in the most natural way.
Definition of Arc Length
Definition: Arc Length
The arc length of a regular curve $\gamma: [a, b] \to \mathbb{R}^n$ is defined by
$$L = \int_a^b |\gamma'(t)| \, dt = \bigint_a^b \sqrt{\sum_{i=1}^n \left(\frac{d\gamma_i}{dt}\right)^2} \, dt$$Derivation: From Polygonal Approximation to the Integral
Why does the above formula represent the length of a curve? We derive it step by step from a polygonal approximation.
Step 1. Partition the interval
Divide the parameter interval $[a, b]$ into $N$ equal parts:
$$a = t_0 < t_1 < t_2 < \cdots < t_N = b, \qquad \Delta t = \frac{b - a}{N}$$Connecting adjacent points $\gamma(t_k)$ and $\gamma(t_{k+1})$ on the curve with straight lines produces an inscribed polygonal line.
Step 2. Total length of the polygonal line
The total length of the polygonal line is the sum of the Euclidean distances between adjacent points:
$$L_N = \sum_{k=0}^{N-1} |\gamma(t_{k+1}) - \gamma(t_k)|$$Writing this in component form with $\gamma = (\gamma_1, \gamma_2, \ldots, \gamma_n)$:
$$L_N = \sum_{k=0}^{N-1} \sqrt{\sum_{i=1}^n \bigl(\gamma_i(t_{k+1}) - \gamma_i(t_k)\bigr)^2}$$Step 3. First-order approximation from the definition of the derivative
The definition of the derivative is
$$\gamma_i'(t_k) = \lim_{\Delta t \to 0} \frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t}$$We remove the $\lim$ and consider a finite $\Delta t$. Adding $\gamma_i'(t_k)$ and $-\gamma_i'(t_k)$ (which sum to zero) to the right-hand side:
$$\frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t} = \gamma_i'(t_k) + \left\{\frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t} - \gamma_i'(t_k)\right\}$$Denoting the $\{\quad\}$ part by $\varepsilon(\Delta t)$:
$$\frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t} = \gamma_i'(t_k) + \varepsilon(\Delta t)$$By the definition of the derivative, $\varepsilon(\Delta t) \to 0$ as $\Delta t \to 0$:
$$\lim_{\Delta t \to 0} \varepsilon(\Delta t) = \lim_{\Delta t \to 0} \left\{\frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t} - \gamma_i'(t_k)\right\} = \gamma_i'(t_k) - \gamma_i'(t_k) = 0$$By definition of $\varepsilon(\Delta t)$, the following equality holds for any finite $\Delta t$ (where $\varepsilon(\Delta t)$ is an error that vanishes as $\Delta t \to 0$):
$$\frac{\gamma_i(t_k + \Delta t) - \gamma_i(t_k)}{\Delta t} = \gamma_i'(t_k) + \varepsilon(\Delta t)$$Multiplying both sides by $\Delta t$:
$$\gamma_i(t_k + \Delta t) - \gamma_i(t_k) = \gamma_i'(t_k) \cdot \Delta t + \varepsilon(\Delta t) \cdot \Delta t$$When $\Delta t$ is small, the second term $\varepsilon(\Delta t)\cdot\Delta t$ is the product of "a quantity tending to $0$" and "a small quantity," so it is negligible compared with the first term. Therefore:
$$\gamma_i(t_{k+1}) - \gamma_i(t_k) \approx \gamma_i'(t_k) \cdot \Delta t$$Step 4. Obtaining a Riemann sum
Substituting the result of Step 3 into Step 2:
\begin{align} L_N &\approx \sum_{k=0}^{N-1} \sqrt{\sum_{i=1}^n \bigl(\gamma_i'(t_k) \cdot \Delta t\bigr)^2} \\ & \quad \Delta t \text{ does not depend on } i\text{, so it can be factored out of the square root }(\Delta t > 0)\text{.} \\ &= \sum_{k=0}^{N-1} \sqrt{\sum_{i=1}^n \gamma_i'(t_k)^2} \;\cdot\; \Delta t \\ & \quad \sqrt{\sum_i \gamma_i'(t_k)^2}\ \text{equals the Euclidean norm}\ |\gamma'(t_k)|\ \text{of}\ \gamma'(t_k) = (\gamma_1'(t_k), \ldots, \gamma_n'(t_k))\text{, so} \\ &= \sum_{k=0}^{N-1} |\gamma'(t_k)| \cdot \Delta t \end{align}This is nothing but a Riemann sum for the function $|\gamma'(t)|$.
Step 5. Taking the limit
In the limit $N \to \infty$ ($\Delta t \to 0$), the Riemann sum converges to a Riemann integral:
$$L = \lim_{N \to \infty} L_N = \int_a^b |\gamma'(t)| \, dt = \bigint_a^b \sqrt{\sum_{i=1}^n \left(\frac{d\gamma_i}{dt}\right)^2}\, dt$$Thus the integral formula for arc length is obtained as the limit of the polygonal approximation. $\blacksquare$
Example: Arc length of a circle
For $\gamma(t) = (r\cos t, r\sin t)$ ($0 \leq t \leq 2\pi$),
$$\gamma'(t) = (-r\sin t, r\cos t), \quad |\gamma'(t)| = r$$ $$L = \int_0^{2\pi} r \, dt = r \Big[t\Big]_0^{2\pi} = r (2\pi - 0) = 2\pi r$$This agrees with the well-known circumference formula.
Arc-Length Function
Definition: Arc-Length Function
For a regular curve $\gamma: I \to \mathbb{R}^n$, the arc-length function measured from a base point $t_0 \in I$ is
$$s(t) = \int_{t_0}^t |\gamma'(\tau)| \, d\tau \tag{1}$$Since $\gamma$ is regular and $|\gamma'| > 0$, $s(t)$ is a strictly increasing function of $t$, so its inverse function $t(s)$ exists. The following proposition holds.
Proposition: Derivative of the arc-length function
$$\frac{ds}{dt} = |\gamma'(t)| \tag{2}$$That is, the rate of change of arc length equals the speed.
Proof
Applying the Fundamental Theorem of Calculus to equation (1): $\dfrac{ds}{dt} = \dfrac{d}{dt}\displaystyle\int_{t_0}^{t}|\gamma'(\tau)|\,d\tau = |\gamma'(t)|$. $\blacksquare$
Reparametrization by Arc Length
Definition: Arc-Length Parameter
The curve obtained by reparametrizing $\gamma$ using the arc length $s$:
$$\tilde{\gamma}(s) = \gamma(t(s))$$is called the arc-length parametrization of the curve.
Theorem: Unit-Speed Condition
A curve is parametrized by arc length if and only if
$$|\gamma'(s)| = 1$$Such a curve is called a unit-speed curve.
Proof
Let $\tilde{\gamma}(s) = \gamma(t(s))$. By the chain rule:
$$\tilde{\gamma}'(s) = \gamma'(t(s)) \cdot \frac{dt(s)}{ds}$$From equation (2), $\dfrac{ds}{dt} = |\gamma'(t)|$. Since $t(s)$ is the inverse function of $s(t)$, by the inverse function differentiation rule: $$\dfrac{dt}{ds} = \dfrac{1}{ds/dt} = \dfrac{1}{|\gamma'(t)|}$$Therefore:
$$|\tilde{\gamma}'(s)| = |\gamma'(t)| \cdot \frac{1}{|\gamma'(t)|} = 1 \quad \blacksquare$$Advantages of the Arc-Length Parameter
- The speed is always 1, allowing us to focus solely on the "shape" of the curve
- Definitions of geometric quantities such as curvature become simpler
- The parameter value directly represents the "distance along the curve"
Worked Example
Example: Arc-length parametrization of a helix
Consider the helix $\gamma(t) = (a\cos t, a\sin t, bt)$ ($a > 0$).
Step 1: Compute the velocity vector
$$\gamma'(t) = (-a\sin t, a\cos t, b)$$ \begin{align} |\gamma'(t)| &= \sqrt{a^2\sin^2 t + a^2\cos^2 t + b^2} \\ & \quad \text{Since } \sin^2 t + \cos^2 t = 1\text{:} \\ &= \sqrt{a^2 + b^2} \end{align}Step 2: Compute the arc-length function (with $t_0 = 0$)
$$s(t) = \int_0^t \sqrt{a^2 + b^2} \, d\tau = \sqrt{a^2 + b^2} \cdot t$$Step 3: Express $t$ in terms of $s$
$$t = \frac{s}{\sqrt{a^2 + b^2}}$$Step 4: Arc-length parametrization
$$\tilde{\gamma}(s) = \left(a\cos\frac{s}{c}, a\sin\frac{s}{c}, \frac{bs}{c}\right),\quad c = \sqrt{a^2 + b^2}$$Summary
- Arc length is an integral quantity representing "distance along a curve"
- The arc-length function $s(t) = \displaystyle\int |\gamma'| dt$ is strictly increasing
- When parametrized by arc length, the speed is always 1
- The arc-length parameter is optimal for studying the geometric properties of curves