# Abstract Nonsense

## Arc Length (Pt. II)

Point of Post: This is a continuation of this post.

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What are the properties of this arc length function? Well, perhaps one of the most obvious one is that the arc length function is invariant under isometries of $\mathbb{R}^n$.  Recall that an isometry of $\mathbb{R}^n$ is a distance preserving map, i.e. a map $f:\mathbb{R}^n\to\mathbb{R}^n$ with the property that $\|f(x)-f(y)\|=\|x-y\|$. A nice way to think about isometries are that they are the composition of a rotation and a translation (i.e. a rigid motion). Or, if one likes to think more about matrices then isometries are just affine maps $x\mapsto Ax+b$ where $A$ is an orthogonal matrix.  We denote the set of all isometries of $\mathbb{R}^n$ by $\text{Iso}(\mathbb{R}^n)$. Now, as stated before we have the following result:

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Theorem: Let $\gamma:I\to\mathbb{R}^n$ be a smooth curve, $x_0\in I$, and $f\in\text{Iso}(\mathbb{R}^n)$. Then, $s_\gamma=s_{f\circ \gamma}$.

Proof: It clearly suffices to prove that for an arbitrary partition $P:x_0=t_0\leqslant\cdots\leqslant t_n$ in $\mathcal{P}[x_0,x]$ one has that $L_P(\gamma)=L_P(f\circ \gamma)$. To do this we merely note that

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\begin{aligned}L_P(\gamma) &= \|\gamma(t_1)-\gamma(t_0)\|+\cdots+\|\gamma(t_n)-\gamma(t_{n-1})\|\\ &= \|f(\gamma(t_1))-f(\gamma(t_0))\|+\cdots+\|f(\gamma(t_n))-f(\gamma(t_{n-1}))\|\\ &= L_P(f\circ \gamma)\end{aligned}

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from where the conclusion follows. $\blacksquare$

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The next obvious geometric fact should be that the curve between any two points in $\mathbb{R}^n$ which minimizes arc length is a straight line, i.e. backing up the oft  spouted ‘the shortest path between two points is a straight line’.

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Theorem: Let $p,q\in\mathbb{R}^n$, then for any curve $\gamma:I\to\mathbb{R}^n$ with $\gamma(t_0)=p$ and $\gamma(t_1)=q$, with $x_0\leqslant x_1\in I$, one has that $s_\gamma(x_1)\geqslant \|p-q\|$ where $s_\gamma$ starts from $x_0$.

Proof: We merely note that if $u\in\mathbb{R}^n$ is a unit vector then

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$\displaystyle (p-q)\cdot u=\left(\int_{x_0}^{x_1} \gamma'(t)\; dt\right)\cdot u=\int_{x_0}^{x_1} \gamma'(t)\cdot u\; dt\leqslant \int_{x_0}^{x_1} \|\gamma'(t)\|\; dt$

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Taking then $\displaystyle u=\frac{p-q}{\|p-q\|}$ gives the desired result. $\blacksquare$

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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.