Do you have a specific reason for not wanting to use OpenGL? A self-imposed limitation to experiment with a technique is just fine, but if you're of the mind that OpenGL isn't worth learning to do this, you might want to reconsider.
Raytracing seems to close to what I was looking for. Because I have not done this before I did not know what to search. Now that I have done just a little bit of searching it seems that using raytracing in a video game is not a good choice unless everyone is using a server farm to play the game.
Now I'm getting into some opengl questions(yes and no are not what I'm looking for). I would like to know the api calls or a link to tutorial would be nice.
1. Can I map a png to a plain in opengl and still have transparency?
2. Can I map only a portion of a image to a plain?
You might actually be looking for ray marching. You can more easily test intersections with objects (such as your plane) and use a fisheye-like projection to curve the rays to produce the sphere effect. Then animating the plane movement is as basic as off-setting through the X and Z directions.