What is the name for a surface formed by taking the cross-products of pairs of vectors, tangent to a grid, in $(0,1)^2$ at each point and defining a new point above the base space?
For example, if $S=(0,1)^2$ with a specific chart, and 90 degree angles between gridlines, one computes the cross-products and gets a plane, $1$ unit above the surface. but if one deforms the mesh, $\psi: S \to S$ with a homeomorphism $\psi$, and then computes the cross-products between pairs of vectors at each point tangent to the grid, the plane is now deformed into a curved surface.
If $\psi$ is a continuous mapping, then you could run a computer program that lets you see how the surface deforms in real time based on the base space. So, you could associate a specific geometry to a set of warped curves on a plane.
Where can I read a little more about all this?
First of all, it seems to me that in order to have intrinsic concepts you should consider the infinitesimal level of your cross products. Otherwise you will always be tied by the discretization steps...
In this perspective, you should trade your issue formulated in terms of two families of lines into a formulation with two families $X_1, X_2$ of vector fields (their tangent vectors). And, from there, compute the cross product of these vector fields.
Sooner or later, you will be driven to what is called "exterior algebra" with so-called 2-forms : see for example p.59 of the Google book "Gauge Fields, Knots and Gravity" (authors John Baez, Javier P Muniain).