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Parametric curves

  • 02-03-2014 9:35pm
    #1
    Daniel S
    Registered Users, Registered Users 2 Posts: 3,533 ✭✭✭


    The curve,

    r(t)=<10cos(2t)+5sin(2t)−8t,10cos(2t)−10sin(2t)+4t,5cos(2t)+10sin(2t)+8t>

    Describes a helix. Find the arc length s(t) of the curve, as a function of t, starting from t=0.

    Find also the arc length paramaterisation of the helix r(s) and its constant curvature κ.
    r(s)= xi,yj,zk

    κ=

    I know this is supposed to be simple, yet tedious, but it's wrecking my head. :(

    I've worked out r'(t)={-20Sin(2t)+10Cos(2t)-8, -20Sin(2t)-20Cos(2t)+4, 20Cos(2t)-10Cos(2t)+8} but I don't know where to go from here?

    When I integrate that between 0 and t, I'm getting sqrt of 584. Is this where I'm going wrong?

    Could someone point me in the right direction? It's due for online submission by midnight and I've everything else answered... :(


Comments

  • #2
    Daniel S
    Registered Users, Registered Users 2 Posts: 3,533 ✭✭✭Daniel S


    I'm here now if anyone can help me?

    s(t)= 6sqrt(29)t

    Find also the arc length paramaterisation of the helix r(s) and its constant curvature κ.
    r(s)= 10cos(s/(3sqrt(29)))+5sin(s/(3sqrt(29)))-4s/(3sqrt(29)) i

    10cos(s/(3sqrt(29)))-10sin(s/(3sqrt(29)))+2s/(3sqrt(29)) j

    5cos(s/(3sqrt(29)))+10sin(s/(3sqrt(29)))+4s/(3sqrt(29)) k

    Just can't get the curvature (kappa)... :(


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