Arc Length Parameter

Topics: Arc Length - Vector-Valued Function

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(definition)

Let $\sigma$ be the curve traced out by the following vector-valued function in the interval $a\le t\le b$:

$$r(t)=⟨x_1(t),x_2(t),\dots ,x_n(t)⟩$$

We define the arc length parameter, denoted $s$, to be the arc length of the portion of the curve that goes from $u=a$ to $u=b$. That is:

$$s(t):={\int }_a^t\sqrt{[x_1^{\prime }(u){]}^2+[x_2^{\prime }(u){]}^2+\cdots +[x_n^{\prime }(u){]}^2}du$$

…or, more simply, with its derivative and norm, the arc length parameter $s$ is given by:

$$s(t)={\int }_a^t\Vert r^{\prime }(u)\Vert du$$

Example

For example, let’s find the arc length parametrisation of the circle of radius 4 centred at the origin. We can parametrise such a circle with the following function in the interval $0\le 0\le 2\pi$:

$$r(t)=⟨r_1(t),r_2(t)⟩=⟨4\cos (t),4\sin (t)⟩$$

For such a function, the arc length parameter $s$ is given by:

$$\begin{array}{rl}s(t)& ={\int }_0^t\sqrt{[r^{\prime }(u){]}^2+[g^{\prime }(u){]}^2}du\\ & ={\int }_0^t\sqrt{-4\sin (u){]}^2+[4\cos (u){]}^2}du\\ & =4{\int }_0^{t}1du\\ & =4t\end{array}$$

Notice that we are just calculating the arc length from $u=0$ to $u=t$. In this case, we found that $s=4t$, so $t=s/4$.

Then, this same function, now with the arc length parameter, becomes:

$$r(s)=⟨r_1\left(\frac{s}{4}\right),r_2\left(\frac{s}{4}\right)⟩=⟨4\cos \left(\frac{s}{4}\right),4\sin \left(\frac{s}{4}\right)⟩$$

Effectively drawing the same curve (the circle of radius 4 centred at the origin) now in the interval $0\le s\le 8\pi$.

Arc Length Parameter Intuitively

Intuitively, we can see the arc length parameter as the parameter that makes the function draw the curve at a constant speed that corresponds to its arc length. If this parameter draws a portion of a curve in exactly its arc length, then it’s the arc length parameter.

(theorem)

We say that a given parametrisation of a vector-valued function is an arc length parametrisation if and only if its derivative is a unit vector.

That is, a parametrisation of a vector-valued function $r(t)$ is an arc length parametrisation if and only if $\Vert r^{\prime }(t)\Vert =1$.