How do I create Mexican hat potential figure in Illustrator using 3D Map Art?

Question

This is what I'm trying to create in Illustrator. (for those interested, it's a graph of a Mexican hat potential function)

Here's what I've done so far, using Illustrator's 3D->Revolve and Map Art.

I drew a curve simliar to the shape first. (I know this is not the exact shape, the top should be curve a bit. I'm just using this curve for a proof of concept for now.)

then I drew a grid(expanded) and made it to a symbol.

While having the curve selected, I picked 3D->Revolve. I checked the "Invisible Geometry" option. I used the grid symbol I made to map the 2 surfaces. Illustrator divided the 3d shape to 6 surfaces. But only 2 are needed. Basically the "cone" and the "bowl".

This is what I got.(I'm showing the perspective at a different angle from the original to better present my problems) At the first glance it wasn't too bad. But looking closely:

1. The grid lines don't connect seamlessly between the cone and the bowl surfaces.
2. The lines at the edge of the bowl are too thick. I want all the lines to be the same width.
3. I need to hide the lines of the bowl behind the cone, so the cone doesn't look like it's transparent.

For problem (1) my guess is that I need to create different grid symbols and scale them differently. It may be a painful process through trial and error. I wish I could somehow make Illustrator see this whole thing as 2 surfaces(top and bottom), then I'd only have to map art to the top surface and ignore the bottom surface.

For problem (2) I'm not sure if there's anyway to get around this at all, since mapping to a circular surface will no doubt skew the mapped pattern.

For problem (3) I guess a work-around would be to expand the 3D shape and then manually clip off those paths. Still this will be painful.

Answer 1

(1) torn textures

Further to my comment, your problem is with the number of points you're using and seems to be a bizarre Illustrator quirk.

Depending on where you put your anchor point, Illustrator is calculating a different number of surfaces for your object.

1. Make the shape with just 2 points...

   

2. Map your art...

   

3. Bingo!

   

4. Test with third point just for kicks...

   

5. That doesn't look right!?

   

6. Investigating the texture mapping reveals a 5th face?

   

I'm not sure why Illustrator does this. It is worth noting that it only happens when the mid-point is in certain areas of the bowl. Trying to reproduce the quirk with the point further up the side of the hat does not work.

(Tested on CS3 Win)

(2) Texture Warping

I don't think there's anything you can do about this in Illustrator

Edit: Instead of expanding the shape and manually altering all your lines, you may want to do a little more "pre-production" on your grid symbol. Make the symbol out of a repetition of another symbol so you can make as few edits as possible to achieve the effect. May still need some tweaking after expanding the 3D revolve, but could save you some time.
Something like this:

(3) "Behind" Texture Showing Through

Give your grid pattern an opaque white background.

Edit: Forgot to mention. My hat still has minor texture problem (visible on the right-hand side where the 360° revolve starts & ends) but this is due to my hastiness and skipping proper texture scaling.