Transforms and Processors: Work, Work, Work

Looking at the number of information bits that go into the makeup of a screen only gives a partial picture of how much processing is involved. To get some inkling of the total processing load, we have to talk about a mathematical process called a transform. Transforms are used whenever we change the way we look at something. A picture of a car that moves toward us, for example, uses transforms to make the car appear larger as it moves. Another example of a transform is when the 3-D world created by a computer program has to be "flattened" into 2-D for display on a screen. Let's look at the math involved with this transform -- one that's used in every frame of a 3-D game -- to get an idea of what the computer is doing. We'll use some numbers that are made up but that give an idea of the staggering amount of mathematics involved in generating one screen. Don't worry about learning to do the math. That has become the computer's problem. This is all intended to give you some appreciation for the heavy-lifting your computer does when you run a game.

The first part of the process has several important variables:

• X = 758 -- the height of the "world" we're looking at.
• Y = 1024 -- the width of the world we're looking at
• Z = 2 -- the depth (front to back) of the world we're looking at
• Sx = height of our window into the world
• Sy - width of our window into the world
• Sz = a depth variable that determines which objects are visible in front of other, hidden objects
• D = .75 -- the distance between our eye and the window in this imaginary world.

First, we calculate the size of the windows into the imaginary world.

Now that the window size has been calculated, a perspective transform is used to move a step closer to projecting the world onto a monitor screen. In this next step, we add some more variables.

So, a point (X, Y, Z, 1.0) in the three-dimensional imaginary world would have transformed position of (X', Y', Z', W'), which we get by the following equations:

At this point, another transform must be applied before the image can be projected onto the monitor's screen, but you begin to see the level of computation involved -- and this is all for a single vector (line) in the image! Imagine the calculations in a complex scene with many objects and characters, and imagine doing all this 60 times a second. Aren't you glad someone invented computers?

In the example below, you see an animated sequence showing a walk through the new How Stuff Works office. First, notice that this sequence is much simpler than most scenes in a 3-D game. There are no opponents jumping out from behind desks, no missiles or spears sailing through the air, no tooth-gnashing demons materializing in cubicles. From the "what's-going-to-be-in-the-scene" point of view, this is simple animation. Even this simple sequence, though, deals with many of the issues we've seen so far. The walls and furniture have texture that covers wireframe structures. Rays representing lighting provide the basis for shadows. Also, as the point of view changes during the walk through the office, notice how some objects become visible around corners and appear from behind walls -- you're seeing the effects of the z-buffer calculations. As all of these elements come into play before the image can actually be rendered onto the monitor, it's pretty obvious that even a powerful modern CPU can use some help doing all the processing required for 3-D games and graphics. That's where graphics co-processor boards come in.