Picture on a ball canvas

Hello,
I'm really new in graphics and i have a task to develop a device that 
will project a picture on a white ball, but the picture has to be 
symmetrical. To achieve this, the picture should be modified/distorted 
accordingly prior it is projected. To modification will mean that the 
picture is created in a way that when it falls on the white ball will 
look fully symmetrical.

Since i'm new in technologies like GNOME that are used to create a 
graphical interface, i'll need your help in getting directions where/how 
in the GNOME framework to introduce a function that will modify the 
final picture prior it is sent to the projector.

The elements that should be projected are simple 2D components: text, 
circles, triangles...

Any help is appreciated.

Thanks,
Blagoj