Smooth contours with grid lines Total noob here.

My goal is to be able to create figures in the following style (source):

with a workflow consisting of Blender and Inkscape. I'm sufficiently skilled with Inkscape, but I'd really like to get the mathematically defined shapes done to perfection in Blender.

Currently, my challenge is to draw the surfaces (open or closed) smoothly, but with grid lines. I'm using the Freestyle renderer with the SVG exporter addon.

I'm also using the addon for extra objects which includes meshes generated by equations of the coordinates.

My current best effort is illustrated here:

where I am rendering all of the edges as "Freestyle Edges".

I would like the nodes in the grid to be connected smoothly (i.e., I want the underlying mesh to have a higher number of samples than the selection of Freestyle edges). This might be as simple as me learning how to select ever 10th node in a row or something, but I don't have the Blender experience to make that a trivial task.

Any advice is appreciated.
To connect the nodes smoothly, all that's needed is to (CtrlE > Edge Data) mark Freestyle edges on a low-poly version of your curved surface, as you appear to have done, and then assign a Subdivision Surface modifier with Catmull-Clark interpolation. (The newly generated faces and their sides are not visible to Freestyle, unless you apply the modifier, and mark them afterwards). Then you set the Freestyle Line Style to select only by Edge Mark. 

This is assuming that the result of CC subdivision is mathematically accurate enough for your purposes. If not, it is possible to select every n'th loop on a more densely divided grid. 


Select 2 parallel edges, n apart, on the loops you will eventually want to mark.
Hitting CtrlShiftNumpad + will select successive edges at the same interval, across the grid.
Using the Select menu in the header, select 'Loops' > 'Edge Loops'.


Loops will be selected at that interval, to be marked.