What is the definition of a mesh plot?

Question

Mesh plots are a common tool to visualize three dimensional surfaces. They're used in many textbooks and provided by MATLAB, Wolfram Alpha, and others. Surprisingly, I can't find a mathematical definition anywhere: the closest I've found is at Mesh Plots Scientific Visualization which states

Definition 12 (Mesh Plot) A plot which depicts a function q(x, y) by plotting a fishnetshaped lattice of horizontal and vertical lines in three-dimensional space as if the net had been draped over the surface of q.

Alternative Names: FISHNET PLOT, WIREFRAME DIAGRAM

(emphasis added) which isn't much of a definition.

What is the mathematical definition of a mesh plot? I conjecture that it is along these lines: Given a continuous function $f(x,y)$:

1. Create a series of lines in a grid on the $xy$ plane (e.g. $x = nk$ and $y = nk$ for all $n \in \mathbb N$).
2. Lift each point on the line above (or below) the $xy$ plane by $f(x,y)$
3. Choose an arbitrary camera point, and project the lines onto the camera; plot what the camera sees.

This definition, besides still being quite rough, doesn't seem to account for the various mesh graphs I see, which seem to show curving of the mesh I can't explain as coming from the above.

Note: Not to be confused with lattice graphs

Answer 1

Firstly we have a definition of "coordinate curve". Then, a mesh is a series (2 intersecting families) of coordinate curves. We could formally define "mesh" to be any (non-empty) set of coordinate curves and then talk about a regular, equidistant mesh etc.

Answer 2

It's just a piecewise affine function approximation, which interpolates the evaluated function at points on a square grid. For each 4-point rectangular cell of the grid, we plot a plane, that passes through the points of the graph of $f$ evaluated on the cell.

I guess we can formally define this as follows: suppose that we have an $N\times M$ grid with $NM$ cells. Each cell is a set that can be defined as $$C_{ij}=\{(x,y):x_i\leq x < x_{i+1}, y_j\leq y < y_{j+1}\},$$ where $\{(x_i,y_j),(x_{i+1},y_j),(x_i,y_{j+1}),(x_{i+1},y_{j+1})\}$ are points on the grid that define cell $C_{ij}$ (there are $(M+1)(N+1)$ points in total).

Now, our approximation is defined as $\tilde{f}(x,y)=\sum_{i,j}f_{ij}(x,y)I_{C_{ij}}(x,y)$, where $I_C$ is the indicator function, and $$f_{ij}(x,y)=f(x_i,y_j) + \frac{f(x_{i+1},y_j)-f(x_i,y_j)}{x_{i+1}-x_i}(x-x_i) + \frac{f(x_i,y_{j+1})-f(x_i,y_j)}{y_{j+1}-y_j}(y-y_j).$$

Don't think we can get more than that.