@Kitonum Thank you for advancing my understanding.

@acer Thanks for this info, I didn't realize the effect abs would have on the plotted expression.

Does Maple Learn have a 3D capability?

@Carl Love Just before I uploaded my worksheet I made a last second change that omitted a colon. This is fixed in the worksheet below. Interestingly this error only caused a "Cannot parse..." error message on my machine under Maple 2020.

I just executed this worksheet and it consumed just under one gigabyte and 230 seconds. I hope it now works for you.

I'm afraid I could not understand how to apply the color technique you kindly explained to individual cells, so I tried one I was familiar with.

Stereographic_projection_with_colors.mw

@Carl Love Here is an effort at displaying stereographic projections of a grid on a sphere onto the plane on which the sphere rests, colored in three ways: a gradient of color, colored bands, and individual colored cells.

The worksheet defines two new coordinate systems, as you have suggested.

One of them is used to "reverse" project a grid on the plane onto the sphere.

The other projects the uncolored grid from the sphere to the plane.

However I could not find a way of projecting a colored grid from the sphere to the plane, so the colors are then applied directly to the plane grid.

Stereographic_projection_with_colors.mw

@acer Thanks for this very useful information

@Carl Love You have provided a more universally applicable answer than I expected and it will take me some time and effort to digest.

I am grateful to you for the time and effort which you have put into this.

@acer I was inspired to do this animation by viewing 

https://www.youtube.com/watch?v=0z1fIsUNhO4

This youtube presentation shows a gradient style coloring of the cells which I prefer.

However, if you have any other coloring suggestions I would be pleased to see them.

This is an example of shading between spacecurves. One of the spacecurves can be a single point. For 2D set the z spacecurve component to zero or transform 3D to 2D.

eq1 := <0.8000*cos(t), 0, sin(t)>;
eq2 := <2*cos(t), sin(t), 3>;
C1 := plots[spacecurve](eq1, t = 0 .. 2*Pi, color = blue, thickness = 3);
C2 := plots[spacecurve](eq2, t = 0 .. 2*Pi, color = blue, thickness = 3);
S := plot3d(s*eq1 + (1 - s)*eq2, t = 0 .. 2*Pi, s = 0 .. 1, style = surface, color = yellow);
display([S, C1, C2], scaling = constrained, axes = boxed, axis[1] = [color = red], axis[2] = [color = blue], axis[3] = [color = green]);

@acer This looks like just what I was after. I thank you.

@Carl Love After much effort I failed to find documentation describing your ingeneous use of this restriction on parameter fnc.

Please tell me where Maple help explains this.

@Rouben Rostamian I appreciate the head start you've given me. I'll work on it. Much thanks

@Rouben Rostamian  Thank you for your reply. The uploaded worksheet below shows a chain hanging from two gears and several questions regarding this situation. I'd appreciate any answers you can give me.

ChainOnTwoGears.mw

@Carl Love Thanks. I'll look that up and see if I can continue from there.

@Carl Love Let's assume that the gear faces are all oriented vertically i.e. that gravity's direction is perpendicular to the axes of all gears, and from top to bottom in the image parallel to the image's vertical side lines.