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I have some text I would like to have twist and turn as if it is on a sheet of paper. I've included an image of an example I have seen.

How would I approach this in Adobe Illustrator?

Thanks in advance

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You can use the Squeeze style under Object → Envelope Distort → Make with Warp, similar to the below and play with the slider values. Also, experiment with other styles in that panel until you get it right.

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Preset envelope distortion warp styles can make it. It's well worth to try them as already suggested.

More general,but still 2D way is to try Envelope distortion with mesh. Quite nearly the presented shape can be achieved by applying Object > Envelope Distortion > Make with Mesh > 1 Row x 1 Column. It's tried in the next image:

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1 x 1 mesh has 4 nodes and 8 handles which all must be adjusted manually with the direct selection tool. One can adjust 1 x 1 mesh quite easily, but nothing quarantees it gives enough control.

One can get more control by setting the mesh size bigger or more preferably by inserting nodes with the mesh tool when needed. Complex mesh becomes easily a nightmare, so it surely pays off to try at first some preset warp styles. They can also be tweaked with the direct selection tool.

Placing the rectangular text on 3D surface and finding a good perspective projection can sometimes create a fine result result, but Illustrator's 3D with "mapped art" is too often limited when compared to general 3D programs. Your example needs a surface which cannot be produced in Illustrator; it only revolves and extrudes.

In proper 3D programs there's unfortunately no general method to solve which surface and view is needed to create a predetermined 2D projection. Creating a 3D image which looks good in 2D is based on trial and error, but experience helps substantially.

In the next two images there's a quite elementary attempt to make something resembling in an entry level 3D program named Moi3D. There's a rectangular block of outlined text (pasted from Illustrator) and a reference planar rectangle around it. They are mapped (or Flown, as the program says) onto a curved surface.

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Here's no attempt to colorize it nor to get the geometry and projection right. It's purpose is only to show the thing is theoretically possible.

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