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Thread: Proof using frenet equations

  1. #1
    Jan 2010

    Proof using frenet equations

    (R-Rc).(R-Rc)=a^2 where R=R(s) lies on the surface of a sphere

    prove R(s)=Rc-pn -1/t*(dp/ds)b where p = curvature t = torsion n is the unit normal and b is the unit binormal

    getting to the proof involves differentiating the original equation and then using the Frenet equations to simplify

    Questions? (Rc) and (a) would be considered constants?

    if so the
    first derivative is 2*(dR/ds.(R-Rc))
    sencond derviative is 2*(d^2R/ds^2.(R-Rc)+dR/ds.dR/ds)
    third derivative is 2*(d^3R/ds^3.(R-Rc)+3*d^2R/ds^2.dR/ds)

    I have tried using the Frenet equations along with R'(s) = v where v is unit tangent and R''(s)=dv/ds

    any suggestions on where to go from here?
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  2. #2
    Jan 2010


    Lots of derviative taking and using the fact that unit vector cross itself = 0 and unit vector dot itself =1
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