Exploring the Frenet Frame Equations for Curves in R_3

[factor]

I'm trying to show that the Frenet frame structural equation for a curve in R_3 can be written in the following form for a vector W(s):

T'(s) = W(s) x T(s)
N'(s) = W(s) x N(s)
B'(s) = W(s) x B(s)

The problem I'm having here is that I define first that T(s) should be the unit tangent at the point s. I assume that my curve is parametrized by arc length so T'(s) is certainly orthogonal to T(s). The problem I have now, is that N(s) is parallel to T'(s). So my only possible choice for W(s) at this point is +/- B(s), as it must be orthogonal to both T(s) and N(s). But this gives that B'(s) is identically 0 for all s, which is fine for a plane curve I suppose because then the osculating plane isn't rolling and B'(s) is supposed to vanish because the torsion is identically 0. But this is supposed to be for a general curve in R_3.

Can anyone perhaps just point out a serious flaw in my logic so that I might continue on with this, or am I not mistaken in what I'm saying here?

 

[quasar987]

I do not see any mistake in what you said. Do u have reasons to think such a W exists?

 

[lavinia]

I'm trying to show that the Frenet frame structural equation for a curve in R_3 can be written in the following form for a vector W(s):

T'(s) = W(s) x T(s)
N'(s) = W(s) x N(s)
B'(s) = W(s) x B(s)

The problem I'm having here is that I define first that T(s) should be the unit tangent at the point s. I assume that my curve is parametrized by arc length so T'(s) is certainly orthogonal to T(s). The problem I have now, is that N(s) is parallel to T'(s). So my only possible choice for W(s) at this point is +/- B(s), as it must be orthogonal to both T(s) and N(s). But this gives that B'(s) is identically 0 for all s, which is fine for a plane curve I suppose because then the osculating plane isn't rolling and B'(s) is supposed to vanish because the torsion is identically 0. But this is supposed to be for a general curve in R_3.

Can anyone perhaps just point out a serious flaw in my logic so that I might continue on with this, or am I not mistaken in what I'm saying here?

W must be a linear combination of T and B. It does not have to be orthogonal to T. I think it will be orthogonal only when the torsion is zero.