wrapping shape around a "sphere"

[ameliab2005]

I'm still figuring some techniques in illustrator for probably a simple effect. I'm trying to emulate a logo wrapped around a sphere...I tried to map an image around a sphere using Illustrator's 3D tool, but it distorts my logo at the polar corners and the warp tool's inflate feature does a horrible job at bloating...

Is there a special technique to achieve a nice effect?

Examples of such logos are in the following attachment.

Thanks.

[Lukas Engqvist]

If you do not want to hand craft it. The Maping is the most correct. To avoid distortion try to understand how the map works, it is as "distorted" as the standard worldmap, the wire frame gives you a hint.
To distortion is inevitable but understanding that the mapping pinches at the globes allows you to either place the logo elsewhere, or Warp the object before mapping to counter the pinchiing at the poles. Alternatibeley place the logo esewhere and rotate the sphere, so that you can avoid the poles.

[eugenetyson]

Create logo

Open the symbols panel

Drag your logo in there

Draw an ellipse/circle

Use direct select tool to delete half of it.

Effect>3D>Rvolve

Map Art>Select your logo

Invisible Geomeotry

Say ok

Rotate sphere to angle necessary to see logo

Tweak to your liking

[eugenetyson]

If you're concerned about other things like stretching or something, sometimes stretching the sphere alters the way the logo is mapped.

You can also scale the artwork onto the sphere to get better results.

I really don't know how else to help you unless you post an image that you're doing and what's going wrong.

I've made several using the steps below, just this morning, and it works just fine and I don't see distortions and things.

Screenshots?