# A regularly parametrized surface

• Dec 28th 2012, 08:51 AM
Plato13
A regularly parametrized surface
Let
γ : R2 R3 be a unit speed curve of nonvanishing curvature and let its scalar curvature, torsion and Frenet frame be denoted κ, τ and [u, n, b], as usual. Consider the mapping M : R2 R3, M(s, t) = γ(s) + tb(s).
1)Show that M is a regularly parametrized surface. (You may assume that M is one-to-one.)
You may assume the frenet formulae.

I got (u(s)-tτn(s)) x b(s) for the cross product of the partial derivatives (which I need to show is non-zero for all s,t), but how do I simplify further and is this correct?

• Dec 28th 2012, 09:23 AM
xxp9
Re: A regularly parametrized surface
b = u x n, so n x b = u, b x u = n,
( u - tτn) x b = uxb - tτ nxb = -n -tτ u
It's length is 1+(tτ)^2, never vanishes.
• Dec 28th 2012, 10:26 AM
Plato13
Re: A regularly parametrized surface
Can you explain your first line please. That's surely not true in general is it?
• Dec 28th 2012, 11:03 AM
xxp9
Re: A regularly parametrized surface
The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
u, b, n are all unit vectors that are perpendicular to each other.
• Dec 28th 2012, 11:24 AM
Plato13
Re: A regularly parametrized surface
thanks mate