There's only one person I can think of that has ever done that, he made Gemini. Looking through the source of that, you could probably figure out how he did it.

At the moment, I can think of some ways to do it, but I'm not sure they would be fast.

So is this just a horizontal engine? No looking up or down?

Look for wolfenstein castle.

EDIT: http://www.ticalc.org/archives/files/fileinfo/277/27767.html

Hella awesome.

Oh you said polygon. Never mind.

Well Gemini was a raycaster, not a "real" 3D engine. Which I believe is what ben_g wants. Applying textures in raycasting is easy because of the vertical scanline rendering method and the restrictions of the camera.

Yes, It's a "real" 3D engine, and it supports both vertical and horizontal rotation, as well as letting the camera rotate along it's axis using the up vector. There's no way to use routines from raycasters for this. I've already checked gemini, and it renders by casting rays, not by calculating the location of points and drawing a texture between them.

btw: what would be the easyest? Drawing a texture on the quad or drawing a quad as two textured triangles? (Triangles are used with 3D on a computer, But I don't know which is the fastest)

Ok, here is my method. I'm literally making it up as I write this post, but in the end it should work. For this you should use triangles.

1. Convert the 3 corners to xy coordinates.
2. Calculate the relative size of the leftmost point. (3d window distance thing / distance)
3. Now calculate the relative size of each of the other two points and subtract the original from them.
4. (This is where it gets sketchy) You need to figure out the two slopes of (relative distance) / x = a and (relative distance) / y = b. Right now you have two equations: d1 = ax1 + bx1 and d2 = ax2 + bx2. So you need to solve those for a and b. I would divide the first equation by x1, then subtract the the second equation from the first. the ax's will cancel and you can get b. Then just substitute b and get a.
5. Now to apply the texture, we have to base the texture somewhere, so you should probably put the leftmost point on the left side of the texture dead centered vertically.
6. To calculate the pixel in the texture to use for a screen coordinate, here is what you do. X = (1 / (screenOffsetX * a + relativeSizeOfLeft) * screenOffsetX / totalXWidth * textureWidth) and Y is the same.
7. Then, take that pixel out of the texture and put it on the screen.

As far as I can tell, that should work. I'm sure if you actually do it, you'll find lots of optimizations. For instance, the reciprocal I forgot about at the end could probably be calculated earlier.

Ok, you can use this page and this page for pictures.

What you are trying to do is to figure out how tall the wall should be on the actual screen. The concept we are using here is that an object that is twice as far from you will look half the size. So we can see that the equation would be size = k/distance. Now the question is of course, what is k.

K is the distance to the projection screen. The way you calculate this, you can see in that first link I gave you. It is (screenWidth / 2) / tan(fieldOfView / 2). So assuming you are using a 60 degree field of view on the calculator. 48/tan(30) = 83.

Now go to the second link. What this means, is that a wall on screen that is 83 units away, will be it's actual size. 166 units away it will be half its size, 41.5 units away it will be double size.

So the formula is now (relative size) = 83/distance. Which means if your sprite is supposed to be 50 pixels tall, when it is 120 units away, it will be 83/120 * 50 = 34.5 pixels tall.