Originally Posted by

**dopi** **here is my original quesiton**

alpha(u) is a unit speed space curve, and use this curve to construct a tangent developable surface with $\displaystyle chart (x,U) $where

$\displaystyle x(u,v) = alpha(u) + v*(alpha'(u)), (u,v) in U$

where $\displaystyle U = {(u,v) in R^2 : -infinity < u < infinity, v>0}$

so i want to work out the guassian curvature of this surface.

**my solution just for E, F , G**

$\displaystyle E = Xu . Xu = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

$\displaystyle F = Xu . Xv = ! + v*t*alpha''(u)$

$\displaystyle G = t.t =1$

and

$\displaystyle F^2 = 1 + (v^2)*(alpha''(u)^2) +2*v*t*alpha''(u)$

so $\displaystyle EG - F^2 = 0$

where i used t = alpha'(u) from frenet serret equation, as txt =0 and t.t =1

im not what i done is right but thats how i got EG - f^2 for the guassian curvature as zero