I was wondering how can I calculate the curvature of a surface?
For example:
Given a Lambert quadrilateral ABCD (see http://en.wikipedia.org/wiki/Lambert_quadrilateral ) with: $ DA \bot AB $, $ AB \bot BC $, $ BC \bot CD $ and we know the length of each segment (but not the $\angle ADC $ )
Can I calculate the curvature of the plane? (and how!)
Or do I also need to know $\angle ADC $?
Do I even have to know all four lengths, can I do it with 3?
If the segments are geodesics in your surface, then the Gauss Bonnet Theorem will tell you that (letting $Q$ denote the quadrilateral and $K$ the (Gaussian) curvature of the surface) $$\iint_Q K\,dA = \sum_{j=1}^4 \iota_j - 2\pi,$$ where $\iota_j$, $j=1,\dots,4$, are the interior angles. In your case, you'll have $\iota_4-\pi/2$ for that right-hand side. That is, if the fourth angle is a right angle for all such possible quadrilaterals, $K=0$, as in Euclidean space. (In particular, the lengths of the sides are irrelevant.)
If $K$ is not constant, you would have to take limits as your quadrilateral shrinks down to a particular point $p$, and you would have $$K(p) = \lim_{\Delta A\to 0} \frac1{\Delta A}\iint K dA = \lim_{\Delta A\to 0} \frac1{\Delta A}(\iota_4-\pi/2).$$