Geometry Lessons

On the screen, 3-D objects consist of hundreds or even thousands of polygons, usually triangles. The smaller and more numerous the polygons, the more detailed the object.

When an object moves on the screen, rotates to a different viewing angle, or moves forward or backward in virtual space (e.g., scales to a different size), the program must recalculate every vertex (corner) of every polygon. This is a geometry transformation, and it requires heavy-duty matrix multiplication. Typically, the program must multiply a 1x4 matrix of coordinates against a 4x4 transform matrix. It requires up to 16 multiplies and 12 additions for every vertex. (See the figure "The Math Behind 3-D Graphics" .)

A typical 3-D game object might have 1000 polygons. Each time it moves, that's at least 84,000 multiplies and adds. And that's just to recalculate the wire-frame mesh -- it takes even more number-crunching to wrap the skeleton with a surface texture map.

All the numbers that a geometry transformation manipulates are single-precision floating point values, 32 bits long. RISC chips are generally better at this math than x86 chips. Even the Sony PlayStation and Nintendo 64 -- home videogame machines that sell for under $150 -- have special geometry-transform hardware t hat blows most PCs into the weeds.

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The Math Behind 3-D Graphics

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