Abstract Nonsense

Crushing one theorem at a time

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.

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September 19, 2011 - Posted by | Differential Geometry | , , , , , , , ,

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