Thread: 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?

Originally Posted by Valer

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

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

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

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.

And how can i do that? Could you show me?

Originally Posted by Valer

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

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