# Abstract Nonsense

## Arc Length (Pt. I)

Point of Post: In this post we discuss the notion of arc length for curves in $\mathbb{R}^n$. $\text{ }$

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Motivation

We have discussed curves, some of the fundamental objects (at least motivationally) in the geometry we will discuss, and so we’d like to begin discussing some of their fundamental features. Being interested in geometry we ‘measure’ things about the geometric objects we are interested in. Perhaps one of the most obvious, and visually stimulating feature of a curve is its arc length. Intuitively, the arc length is just how ‘long’ the curve is. The problem is, how do we measure the length of a curve? If we were dealing with a curve in the real world in which we live we would just take the curve, ‘straighten it’ and then just use the usual formula for measuring the length of a line segment (i.e. $x,y\in\mathbb{R}^n$ then $\text{length}(\overline{xy})=\|x-y\|$ where $\overline{xy}$ is the line connecting $x,y$). The problem with this is two-fold: a) in the practical real world, this isn’t always possible (e.g. a curved stone wall)– and more importantly for us b) the operation of ‘straightening’ is (just to start with its problems) incredibly difficult to formalize, and even more difficult to actually find. Thus, we must find some other way to measure arc length which, optimally, wouldn’t involve a deforming of the curve. So, what do we do? Well, let’s go back to the practical problem about measuring the length of a stone wall. If you were charged with such a task, given only a set of straight measuring instruments (i.e. rulers of different length) you may at first dismay “this is impossible!” But, after a while it may occur to you that if you keep making successive measurements with shorter and shorter straight measuring implements you are better approximating the actual length of the curve. Visually the measuring with such a straight length implement may look something like

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It’s at least intuitively clear then that each such measurement underestimates the actual length of the curve, but that one can make this error term arbitrarily small (in modulus) by choosing the length of the straight measuring implement small enough. Thus, it seems reasonable to define the arc length of the curve to be the supremum over all such measurements. Or, perhaps it would make sense to define the arc length of a curve to be the limit of the piecewise-straight measurements as the maximum of the distances between the measured points goes to zero. Moreover, neither of these definitions seems to be practical enough for real usage. $\text{ }$ Luckily for us, not only are the two mentioned intuitive definitions of arc length equivalent, but they have a third equivalent definition in terms of an integral of $\|\gamma'(t)\|$which is easily calculatable. This last fact seems plausible, given our ‘definition’ of arclength, since for points very close $p,p+\varepsilon$ on the curve we have that $\gamma(p+\varepsilon)-\gamma(p)=\frac{\gamma(p+\delta)-\gamma(p)}{\varepsilon}\varepsilon\approx \gamma'(p)\varepsilon$ and so summing over all the points, and letting $\varepsilon$ tend towards $0$ should give you something which looks awfully close to a Riemann sum.

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Arc Length

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For a closed bounded interval $[a,b]\subseteq\mathbb{R}$ we define, just as in basic Riemann integration theory, a partition $P$ of $[a,b]$ to be a collection $\{t_0,\cdots,t_m\}$ of points in $[a,b]$ such that $a=t_0\leqslant\cdots\leqslant t_m=b$. If $\gamma:[a,b]\to\mathbb{R}^n$ is a $C^1$ curve, we define $L_P(\gamma)$ to be equal to

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$\|\gamma(t_1)-\gamma(t_0)\|+\cdots+\|\gamma(t_m)-\gamma(t_{m-1})\|$

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What we now claim is that if $\mathcal{P}[a,b]$ denotes the set of all partitions of $[a,b]$ then the set $\left\{L_P(\gamma):P\in\mathcal{P}[a,b]\right\}$ is bounded above. Indeed, since $\gamma_j'(t)$ exists and is continuous on all of $[a,b]$ we have that $\displaystyle \gamma_j'$ is bounded, and so there exists $M_j$ which serve as an upper bound for $\gamma_j'$ on $[a,b]$. We then see that if $P:t_0\leqslant\cdots\leqslant t_m$ is an element of $\mathcal{P}[a,b]$ then

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\displaystyle \begin{aligned}L_P(\gamma) &= \sum_{j=0}^{m}\|\gamma(t_{j+1})-\gamma(t_j)\|\\ &\leqslant \sum_{k=1}^{n}\sum_{j=0}^{m}|\gamma_k(t_{j+1})-\gamma_k(t_j)|\\ &= \sum_{k=1}^{n}\sum_{j=0}^{m}|\gamma'_k(\xi_j)|(t_{j+1}-t_j)\\ &\leqslant \sum_{k=1}^{n}\sum_{j=0}^{m}M_k(t_{j+1}-t_j)\\ &= (b-a)\sum_{k=1}^{n}M_k\end{aligned}

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where we used the Mean Value Theorem at the obvious point.  It thus makes sense to speak of the supremum of the set $\left\{L_P(\gamma):P\in\mathcal{P}\right\}$. We then define, for a given curve $\gamma:I\to\mathbb{R}^n$, the arc length of $\gamma$ (on the invertval $[a,b]\subseteq I$) to be $\displaystyle \sup\left\{L_P:P\in\mathcal{P}[a,b]\right\}$. $\text{ }$ What we now wish to show is that this intuitive definition has the much more tractable definition via the integral of the speed curve of $\gamma$. Namely:

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Theorem: Let $\gamma:I\to\mathbb{R}^n$ be a smooth curve. Then, the arc length of $\gamma$ over $[a,b]\subseteq I$ is

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$\displaystyle \int_a^b\|\gamma'(t)\|\; dt\quad\mathbf{(1)}$

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Proof: It clearly suffices to show that the difference between $\mathbf{(1)}$ and $L_P(\gamma)$ can be made arbitrarily small for a suitable partition $P:t_0\leqslant\cdots\leqslant _n$ . So let $\varepsilon>0$ be given. Note that if we define $F:[a,b]^n\to\mathbb{R}$ given by $F(a_1,\cdots,a_n)=\|(\gamma_1'(a_1),\cdots,\gamma_n'(a_n))\|$ then $F$ is continuous and so uniformly continuous by the Heine-Cantor theorem. Thus, there exists some $\delta>0$ such that $\|(a_1,\cdots,a_n)-(a'_1,\cdots,a'_n)\|<\delta$ implies that $|F(a_1,\cdots,a_n)-Fa'_1,\cdots,a'_n)|<\varepsilon$. Next we note that applying the Mean Value Theorem we can find $\alpha_j^1,\cdots,\alpha_j^n$  such that $\|\gamma(t_{j+1})-\gamma(t_j)\|=F(\alpha_j^1,\cdots,\alpha_j^n)(t_{j+1}-t_j)$. Now, note then that

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$\displaystyle \int_a^b \|\gamma'(t)\|\; dt=\sum_{j=0}^{n}\int_{t_j}^{t_{j+1}}\|\gamma'(t)\|\; dt=\sum_{j=0}^{n}F(\xi_j,\cdot,\xi_j)(t_{j+1}-t_j)$

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where we’ve used the Mean Value Theorem at the obvious place. Now, since each $\alpha_j^1,\cdots,\alpha_j^n,\xi_j$ lie in $[t_{j+1}-t_j]$ if we choose $\text{max}|t_{j+1}-t_j|<\delta$ we obviously have that $|\alpha_j^k-\xi_j|<\delta$ for each $j,k$. We clearly then have that $\mathbf{(1)}$ differs in magnitude from $L_P(\gamma)$ by $(b-a)\varepsilon$ by applying the triangle inequality. The conclusion follows from previous discussion. $\blacksquare$

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With this, it makes sense that given a smooth curve $\gamma:I\to\mathbb{R}^n$ we can define the arc length from $x_0\in I$$s_{\gamma}:(a,b)\to\mathbb{R}$, (where it will be clear by context what the starting point is) defined by

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$\displaystyle s_\gamma(x)=\int_{x_0}^x \|\gamma'(t)\|\; dt$

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References:

1. Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

2.  Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.