alpha(u) is a unit speed space curve, the chart (x,U) where
x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U
i want to show that this surface is regular if the curvature k(u): = || alpha''(u)|| of the unit speed curve alpha(u) is non zero
does anyone know how i would go about this?
alpha(u) is a unit speed space curve, and use this curve to construct a tangent developable surface with where
where
i want to show that this surface is regular if the curvature k(u): = || alpha''(u)|| of the unit speed curve alpha(u) is non zero
so it is really alpha' in the equation. And i have re-written the question to specify what U is. Thank you.
the solution i got was
where i used as from the frenet serret equations.
i replaced with the curvature, wasnt too sure how else to do it
and so as must be non zero for the chart to be regular.