# What is the mean curvature of a developable surface?

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?

• 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 $|g_u|^2=1$? – Upax Dec 30 '17 at 16:45