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Math Help - Curvature

  1. #1
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    Question Curvature

    r(t) = < e^t * cost, e^t*sint, 5>
    Find the curvature at the point (x, y, z) = (-e^π, 0, 5)
    explain briefly how you come up with answer and also express answer as a decimal.
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  2. #2
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    Re: Curvature

    If a point T(x,y,z) belongs to a curve \vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}+z(t)\vec{k} then exists a parameter value t\in \mathbb{R} such that
    x=x(t), \, y=y(t), \, z=z(t). In case of a given point t=-\pi.

    Wikipedia says that for a space curve its curvature is given by
    .

    All derivatives are with respect to variable t. Here \kappa is a function in variable t, \kappa=\kappa (t). Find the expression for \kappa(t) and evaluate it for t=-\pi.
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  3. #3
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    Re: Curvature

    Also, notice that as z is constant you are really looking at,



    ... so you only need the 2-D version (Curvature - Wikipedia, the free encyclopedia).

    Just in case a picture helps differentiate...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to t. And,




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