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I'm not an Illustrator expert user. I'd like to draw a sort of "globe", but using a cylinder instead of a sphere. can anyone help me to find a start point? I don't know where to start placing textures

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Here's one guess.

Make the cylinder as 3D revolution:

enter image description here

The revolved shape isn't pure sharp rectangle, but a little distorted one. It must be if one wants to place the image onto the cylinder as a single image, not as several clipped pieces.

Corners B and C are rounded. They are smooth nodes. Corners A and D are sharp. To get the 3D surface as one piece the edge must turn continuously and all the time to left, when one walks along the edge route A,B,C,D. This is Illustrator's inbuilt property and we must live with it or use something else.

Here's a zoomed screenshot of the upper edge:

enter image description here

Subdialog Map Art in Revolve Options offers only one surface, if the revolved shape is simple as described. Beware extra nodes and profile edge curvature direction changes. They will split the surface.

Here's the mapping dialog:

enter image description here

The mapped image here is a screenshot from a Wikipedia article. The image is an equirectangular globe map. The resolution is surely poor, because it's a screen resolution raster image. If you use your own vector artwork, you will get a razor sharp result.

To use a drawing as mapped artwork, make a copy and drag it to the Symbols panel. The mapping dialog allows only symbols to be used. I wrote "make a copy" because you avoid linking the original to a symbol.

See how the top and bottom edges squeeze to single points in mapping. That's how the mapping works and there's no practical ways to get it done otherwise in Illustrator if the sueface isn't splitted. The singular points would be the poles if the surface was a sphere.

Here's a cople of different views. Sorry for the edge line of the map. I forgot to remove it before the image was used:

enter image description here

The rightmost version has also the mapped image shaded. Play with projection, perspective, light and shading options to get the wanted look. You can reopen the editing dialog in the Appearance panel.

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