About PQ Mesh, conjugate curvature network

First to try and summarise the main advantages of torsion-free nodes/beam layouts-

Say we have some curved surface which we want to build as a structure of steel beams.
One geometrically easy way to do this is to simply extrude the grid lines all in a vertical direction:

This keeps the beams planar, and they intersect in a single line at each node, which can be simple to fabricate.

However, we might need the beams to be oriented more perpendicular to the surface, especially if we have surfaces that curve around to become more vertical.

If you simply offset the mesh by moving all the vertices along their normals, and connect corresponding edges between the original and offset mesh to form the beams, then in general without optimisation you get something like this - where the beams are all twisted, because the mesh normals at their start and end do not lie in a common plane:

To fabricate this, you’d need to either make twisted beams, or planar beams but with a complicated node where they join to resolve the twist, both of which increase the cost.

If instead you design or optimise your grid in the right way, to have certain geometric properties, it is possible to make a mesh where the faces are planar quads, which can also be offset along its vertex normals to give another mesh with planar quads, and when we connect the edges of the mesh and its offset, we get a torsion-free-beam-layout.

One particular type of mesh which does allow such a layout is a conical mesh, where all the quads around a vertex lie tangent to a common cone. These meshes also have the nice property that we can generate an offset mesh where corresponding faces are all a constant distance apart.