I'm fairly sure that in that particular instance, they've basically just rendered one 90° rotation animation and one "walk one tile forward" animation of the checkerboard, de-duplicated it into a tile map to save video memory, and are just playing those two animations with palette swapping to set the colours. And because will only need to have rendered the checkboard animation once, every time the player makes a full step forward, they swap the two colours of the checkboard palette so they can continue to use the same animation frames.

If you want the same basic effect on a more complex map (e.g. an arbitrary tilemap, instead of a featureless checkerboard), you're probably best off using a single-object raytrace-like projection. You can save CPU time and reduce rounding-error-induced aliasing by pre-calculating a projection of your viewport to the on-map coordinates to sample each pixel from (giving a lookup table of 2 floats per screen pixel), and then applying translation and rotation to those relative offsets from the camera. Use the inverse of that projection function to project sprite objects into camera space.

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.

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?

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.

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.

Your camera, by the constraints you gave us, is at a fixed viewing angle, and a fixed distance from the "sphere" you want to move around over. The only object it is viewing is a single sphere, onto which you are going to project a small portion of a flat 2D map. So, you only have to do one frame worth of raytracing, and then remember the U,V coordinates that each on-screen pixel got cast to (or "none", if it cast to not intersecting with the sphere at all). To render each frame, you then walk the screen, read the U,V coordinate (or lack thereof) for each pixel, apply a translate/rotate matrix to that U,V coordinate (rotating around only the axis from the center of the sphere to your camera, so this is only a 2x2 or 3x3 matrix), and then sample the texel at that coordinate of your map. This would be very cheap as a fragment shader applied to a single full-screen quad.

It's not real-time ray-tracing, because you're not performing a full scene traversal and lighting analysis every pixel, every frame. Your "scene" is actually static; an unmoving sphere and an unmoving camera. You'll be doing only a simple U/V transform for each pixel, each frame.

However, it is also sounding like you're quite new to graphics programming in general, not just OpenGL. You'll want to master a simple flat 2D tilemap first, then try projecting a view of it that is bent down at the edges to make it look like the map is curved onto a sphere. If you are already well-versed in shader programming, then you could probably dive straight in to the technique above, though.

You want a flat plane? So, something akin to this, at 12m15s;

That's vastly easier. Look up just about any OpenGL tile map drawing. I also encourage you to learn how to use a VBO early, before you learn any bad habits that immediate mode encourages.