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Thread: Local equations of a curve

  1. #1
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    Local equations of a curve

    I need to write the local equations of the curve r(t)=(cos(t),sin(t),t) in t=pi/2. I don't even know how to start, i think i need a natural parametrization, how i can find it and how i can write the local equations with it?
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    Re: Local equations of a curve

    Quote Originally Posted by Valer View Post
    I need to write the local equations of the curve r(t)=(cos(t),sin(t),t) in t=pi/2. I don't even know how to start, i think i need a natural parameterization, how i can find it and how i can write the local equations with it?
    @Valer;911492, please give definitions of local equations and natural parameterization
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    Re: Local equations of a curve

    I also do not know what you mean by "local equation" but it is pretty clear that since x= cos(t) and sin(t), x^2+ y^2= cos^2(t)+ sin^2(t)= 1. It's graph is a circle with center at (0, 0) and radius 1. At t= pi/2, x= cos(t)= cos(pi/2)= 0 and sin(t)= sin(pi/2)= 1 so this is the point (0, 1) at the "top" of the circle. The tangent line to the circle at that point is the horizontal line y= 1. Perhaps that is what you mean
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    Re: Local equations of a curve

    Through a parameter change we get s(t) = Integral from 0 to t (||r'(e)||) = Integral from 0 to t sqrt((-sin^2(e)+cos^2(e)+1^2))=Integral form 0 to t sqrt of 2 = sqrt(2)*t ==> t=s/sqrt(2) ==>Thus, the natural parameterization of the helix is given by the equations x = cos(s/sqrt(2)) ,y=sin(s/sqrt(2)), z=s/sqrt(2)
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    http://www.cs.ubbcluj.ro/~pablaga/ge...s%20(2005).pdf ---Here are all the definitions
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    Re: Local equations of a curve

    Okay, so "natural parameterization" is using arc length as parameter. That makes sense. And the "local behavior" involves finding the Taylor's series for the coordinates at that point to write the arc as in terms of the derivatives then writing those derivatives in terms of the Frenet vectors, the position, tangent, curvature, and torsion vectors.
    Last edited by HallsofIvy; Dec 5th 2016 at 02:47 AM.
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    Re: Local equations of a curve

    And how can i do that? Could you show me?
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    Re: Local equations of a curve

    Quote Originally Posted by Valer View Post
    And how can i do that? Could you show me?
    It would probably be a better starting point for you to say what you already know. What have you been studying in class recently?

    -Dan
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    Re: Local equations of a curve

    Let's see i know how to find the Frenet vectors, curvature,torsion but how i use them in the Taylor series?
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