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?
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)
.
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.