Those "grid lines" are just generated haphazardly by plotting the sqrt function for polar rays, becaues they extended past the standard plot it looked funky so I used min/max to limit them and the effect is those corner pieces. They are effectively just "glitches" but they ended up looking nice(I wouldn't mind if they wrapped all the way around).
As I said, the standard rectangular grid lines do not give a meaningful representation of the surface. The surface is generated from a revolution of the sqrt function. The grid lines I ploted are just "sections" of to give a refernce to the way the surface evolves. Rectangular grid lines do nothing to expose how the surface is formed or it's innate curvature.
sqrt(z) = sqrt(r)*e^(i*t/2)
I'm just ploting those rays for various angles t. using min/max on it to limit it to the range of the plot(an artifact of how plotting the spacecurves as r goes from 0 to x(I didn't want to figure out x, which depends on the angle so I just used min/max).
In fact, it would be nice to plot for fix r too, to get a proper "grid". I guess I can do that.
The issue is coloring those space curves properly(rather than all just being purple or solid). I want to color them based on the property of the surface.
Not sure if this is all the code or not but it will give you an idea:
Cl := x->max(-1,min(1, x)):
M := 25:
f := (x,y)->Re(sqrt(x+_i*y)):
P1 := plot3d([seq([x, y, k*f(x,y)], k=[-1,1])
], x=-1..1, y=-1..1, labels=[x,y,Rew],style=surface, grid=[200,200], thickness=2, colorscheme=["xyzcoloring", [(x,y,z)->y^2,(x,y,z)->x*y,(x,y,z)->x^2]]):
P2 := seq(spacecurve([seq([Cl(t*cos(2*Pi/M*j)), Cl(t*sin(2*Pi/M*j)), sign(k)*(abs(k) - 1.5)/140 + sign(k)*f(Cl(t*cos(2*Pi/M*j)),Cl(t*sin(2*Pi/M*j))),color=[RGB(1,t^2*sin(2*Pi/M*j)^2,1)]], k=[-2,-1,1,2])], t=0..13, thickness=4, transparency=0.7, numpoints=1500),j=1..M):
Oh, damn, I already posted that code ;/ lol
The issue is not the grid lines, I already generated them, it is the coloring and transparency. I can create the lines how I want but I can't color them how I want. some of the issues I mentioned were because when I have to manually create these gridlines(maybe better to call them surface lines) the rendering would be funky. I have to double them up and put them slightly above and below the surface so they are not partially obsucured by the "infinitely thin" surface(since it's not infinitely thin it inserects the gridlines and cover them up).