The first fundamental form to compute the lenght of a curve and the angle between tw0 curves

by Sergi De la Torre   Last Updated April 25, 2018 19:20 PM

In my case, we consider an helicoid $\varphi(u,v)= (u \; sin\, v, u \; cos\, v,v).$

It's first fundamental form is $\mathit{I} = \begin{pmatrix} E & F\\ F & G \end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & u^2 + 1 \end{pmatrix}.$

Now, we have to compute the area, the lenght of the sides and the angles of the triangle, defined by: $$0< u< sinh\, v,\; 0< v< a.$$

For the area I used the formula $A = \int_{0}^{a}\int_{0}^{sinh\, v}\sqrt{EG-F^2}dudv$.

For the sides I have done the following parametrizations: $$\alpha _1(t) = (0,t);\; 0< t< a.$$ $$\alpha _2(t) = (t,0);\; 0< t< sinh\, a.$$ $$\alpha _3(t) = (sinh\, t,t);\; 0< t< a.$$ For the lenght of the curves I have that if $\alpha (t)=(u(t),v(t))$ then: $$L = \int_{a}^{b} \sqrt{E(u')^2+2Fu'v'+G(v')^2}dt.$$ And finally, if $\alpha$ and $\beta$ are two curves that intersect in the surface they form an angle $\theta$ such that: $$cos\, \theta=\frac{\mathit{I(\alpha',\beta')}}{\left | \alpha' \right |\left | \beta' \right |}$$ The problem I have is that the $G$ has an $u^2$ and the lenght of the curves and their angles depends of $u$. So there is something I'm not understanding well. I'll appreciate any help. Thanks!

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