We have that $$\sigma (u,v)=\gamma (u)+v\delta (u)$$ and $$K=\frac{-(\dot\delta \cdot \textbf{N})^2}{EG-F^2}$$ I want to show that if $\gamma$ is a curve on a surface $S$ and $\delta$ is the unit normal of $S$, then $K = 0$ if and only if $\gamma$ is a line of curvature of $S$.
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Doesn't the part "$\delta$ is the unit normal of $S$" mean $\delta=\textbf{N}$ ?
So do we have $$K=\frac{-(\dot{\textbf{N}} \cdot \textbf{N})^2}{EG-F^2}$$ ?
Which is the relation between $\dot{\textbf{N}}$ and $\textbf{N}$ ? Are they perpendicular?
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EDIT:
The exact formulation of the exercise is:
where
Do we maybe consider only the formula of $K$ and not also the formual of $\sigma$ ?