I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$ where $\lVert \cdot\rVert$ is the Euclidean norm.
Bizarre formula for arc length
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$\begingroup$
integration
norm
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0Also, the absolute value outside isn't needed; things are set up such that the integral's value is nonnegative... – 2012-09-01
1 Answers
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Yes that's the usual definition. It depends on your setting, for simplicity let's assume you are in $\mathbb R^n$. You need a scalar product (dot product) $< . >$ on $\mathbb R^n$ (seen as the tangent space at each point of the first $\mathbb R^n$). Then if $f : [a,b] \longrightarrow \mathbb R^n$ is a piecewise $\mathcal C^1$ curve, its length is defined by:
$ \int_a^b \sqrt{ \Big< \frac{df}{dx}(x), \frac{df}{dx}(x) \Big>} dx $
Notice that we ca take the square-root since $
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0Ok, so forget it. In the case of $\mathbb R^n$ it's not really needed. This notion is useful in more general settings: Manifolds. Given a point $p$ on the image of your curve $\mathcal C$ (the set $f([a,b]) \subset \mathbb R^2$, the tangent space $T_p \mathcal C$ of $\mathcal C$ at $p$ is the set of all possible speeds a point moving along the curve can have at the point $p$ (all tangent vectors at $p$). So this space is juste a line tangent to $\mathcal C$ at $p$, a copy of $\mathbb R$. – 2012-09-01