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Thread: Arclength parameterisation

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    Super Member Showcase_22's Avatar
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    Arclength parameterisation

    Let $\displaystyle \underline{r} (s):=(f(s),g(s))$ be the arclength parameterisation of a plane curve $\displaystyle \Gamma \subset \mathbb{R}^2$.

    Given $\displaystyle \alpha, \beta \in \mathbb{R}$ such that $\displaystyle \alpha^2+\beta^2=1$, let $\displaystyle \underline{R}(s):=(f(\alpha s), g(\alpha s), \beta s)$ be a parameterisation of a space curve $\displaystyle C \subset \mathbb{R}^3$.
    a). Show that $\displaystyle \underline{R}$ is the arclength parameterisation of C.
    I'm not really sure how to show this. Firstly, in my book it has $\displaystyle \int_0^t v( \tau)~dt$ which I can't see how i'm supposed to apply.

    Does anyone have any pointers?
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