I am trying to calculate the mean curvature of a developable surface but having some difficulty. Any help would be greatly appreciated!

I am very new to differential geometry so please bear with me, but this is what I have so far:

I have a developable surface given by:

$$ \mathbf{X}\left(u,v\right) = \boldsymbol{\gamma}\left(u\right) + v \mathbf{g}\left(u\right)$$

where $\boldsymbol{\gamma}$ is a unit-speed curve (directrix) and $ \mathbf{g} $ is a unit vector along the generator. The generators are assumed to be inclined at an angle $\beta\left(u\right)$ with the tangent to the curve $\boldsymbol{\gamma}$.

With First fundamental form: $ Edu^2 + 2F du dv + G dv^2 $
and Second fundamental form: $ e du^2 + 2 f du dv + g dv^2 $

I get: $$ E = \left(\boldsymbol{\gamma}_u + v \mathbf{g}_u\right)^2 = 1 + 2v \sin{\beta} + v^2 $$ $$ F = \left(\boldsymbol{\gamma}_u + v \mathbf{g}_u\right) \cdot \mathbf{g} = \cos{\beta}$$ $$ G = \mathbf{g} \cdot \mathbf{g} = 1 $$ $$ g = f = 0 $$ $$ e = k_N $$ where $\left(\cdot\right)_u = \frac{d\left(\cdot\right)}{du} $ and $\left(\cdot\right)_v = \frac{d\left(\cdot\right)}{dv} $ and the mean curvature is: $$ H = \frac{eG - 2fF + gE}{2\left(EG-F^2\right)} = \frac{k_n}{2\left(\sin{\beta}+v\right)^2} $$

Which doesn't seem to be correct...

I found a reference for the mean curvature of a ruled surface here:

$$ H = \frac{k_n-v\left(2 k_n k_g + \tau_g'\right)+v^2\left[k_n \left(k_g^2+\tau_g^2\right)+\left(k_g \tau_g'-\tau_g k_g'\right)\right]}{\left[\left(1-v k_g\right)^2+v^2 \tau_g^2\right]^{\frac{3}{2}}} $$

How is this obtained? And also, how can this be simplified in the case of a developable surface with the generators oriented as described above?

  • $\begingroup$ Could you please explain the following things: 1. You have $k_n$ as well as $k_N$. Are they equals? 2. Is $k_n$ one of the principal curvatures? The other principal curvature is zero, since the Gaussian curvature is zero. 3. Why $|g_u|^2=1$? $\endgroup$ – Upax Dec 30 '17 at 16:45

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