104 lines
5.1 KiB
Markdown
104 lines
5.1 KiB
Markdown
Chapter 61\
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Frames of Reference {#Heading1}
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### The Fundamentals of the Math behind 3-D Graphics {#Heading2}
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Several years ago, I opened a column in *Dr. Dobb's Journal* with a
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story about singing my daughter to sleep with Beatles' songs. Beatles'
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songs, at least the earlier ones, tend to be bouncy and pleasant, which
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makes them suitable goodnight fodder—and there are a *lot* of them, a
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useful hedge against terminal boredom. So for many good reasons, "Can't
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Buy Me Love" and "A Hard Day's Night" and "Help!" and the rest were
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evening staples for years.
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No longer, though. You see, I got my wife some Beatles tapes for
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Christmas, and we've all been listening to them in the car, and now that
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my daughter has heard the real thing, she can barely stand to be in the
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same room, much less fall asleep, when I sing those songs.
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What's noteworthy is that the only variable involved in this change was
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my daughter's frame of reference. My singing hasn't gotten any worse
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over the last four years. (I'm not sure it's *possible* for my singing
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to get worse.) All that changed was my daughter's frame of reference for
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those songs. The rest of the universe stayed the same; the change was in
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her mind, lock, stock, and barrel.
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Often, the key to solving a problem, or to working on a problem
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efficiently, is having a proper frame of reference. The model you have
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of a problem you're tackling often determines how deeply you can
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understand the problem, and how flexible and innovative you'll be able
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to be in solving it.
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An excellent example of this, and one that I'll discuss toward the end
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of this chapter, is that of *3-D transformation*—the process of
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converting coordinates from one coordinate space to another, for example
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from worldspace to viewspace. The way this is traditionally explained is
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functional, but not particularly intuitive, and fairly hard to
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visualize. Recently, I've come across another way of looking at
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transforms that seems to me to be far easier to grasp. The two
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approaches are technically equivalent, so the difference is purely a
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matter of how we choose to view things—but sometimes that's the most
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important sort of difference.
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Before we can talk about transforming between coordinate spaces,
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however, we need two building blocks: dot products and cross products.
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#### 3-D Math {#Heading3}
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At this point in the book, I was originally going to present a BSP-based
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renderer, to complement the BSP compiler I presented in the previous
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chapter. What changed my plans was the considerable amount of mail about
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3-D math that I've gotten in recent months. In every case, the writer
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has bemoaned his/her lack of expertise with 3-D math, and has asked what
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books about 3-D math I'd recommend, and how else he/she could learn
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more.
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That's a commendable attitude, but the truth is, there's not all that
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much to 3-D math, at least not when it comes to the sort of
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polygon-based, realtime 3-D that's done on PCs. You really need only two
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basic math tools beyond simple arithmetic: dot products and cross
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products, and really mostly just the former. My friend Chris Hecker
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points out that this is an oversimplification; he notes that lots more
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math-related stuff, like BSP trees, graphs, discrete math for edge
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stepping, and affine and perspective texture mappings, goes into a
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production-quality game. While that's surely true, dot and cross
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products, together with matrix math and perspective projection,
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constitute the bulk of what most people are asking about when they
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inquire about "3-D math," and, as we'll see, are key tools for a lot of
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useful 3-D operations.
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The other thing the mail made clear was that there are a lot of people
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out there who don't understand either type of product, at least insofar
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as they apply to 3-D. Since much or even most advanced 3-D graphics
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machinery relies to a greater or lesser extent on dot products and cross
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products (even the line intersection formula I discussed in the last
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chapter is actually a quotient of dot products), I'm going to spend this
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chapter examining these basic tools and some of their 3-D applications.
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If this is old hat to you, my apologies, and I'll return to BSP-based
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rendering in the next chapter.
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#### Foundation Definitions {#Heading4}
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The dot and cross products themselves are straightforward and require
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almost no context to understand, but I need to define some terms I'll
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use when describing applications of the products, so I'll do that now,
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and then get started with dot products.
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I'm going to have to assume you have *some* math background, or we'll
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never get to the good stuff. So, I'm just going to quickly define a
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*vector* as a direction and a magnitude, represented as a coordinate
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pair (in 2-D) or triplet (in 3-D), relative to the origin. That's a
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pretty sloppy definition, but it'll do for our purposes; if you want the
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Real McCoy, I suggest you check out *Calculus and Analytic Geometry*, by
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Thomas and Finney (Addison-Wesley: ISBN 0-201-52929-7).
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So, for example, in 3-D, the vector **V** = [5 0 5] has a length, or
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magnitude, by the Pythagorean theorem, of
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(eq. 1)
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(where vertical double bars denote vector length), and a direction in
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the plane of the x and z axes, exactly halfway between those two axes.
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