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+Chapter 61\
+ Frames of Reference {#Heading1}
+--------------------
+
+### The Fundamentals of the Math behind 3-D Graphics {#Heading2}
+
+Several years ago, I opened a column in *Dr. Dobb’s Journal* with a
+story about singing my daughter to sleep with Beatles’ songs. Beatles’
+songs, at least the earlier ones, tend to be bouncy and pleasant, which
+makes them suitable goodnight fodder—and there are a *lot* of them, a
+useful hedge against terminal boredom. So for many good reasons, “Can’t
+Buy Me Love” and “A Hard Day’s Night” and “Help!” and the rest were
+evening staples for years.
+
+No longer, though. You see, I got my wife some Beatles tapes for
+Christmas, and we’ve all been listening to them in the car, and now that
+my daughter has heard the real thing, she can barely stand to be in the
+same room, much less fall asleep, when I sing those songs.
+
+What’s noteworthy is that the only variable involved in this change was
+my daughter’s frame of reference. My singing hasn’t gotten any worse
+over the last four years. (I’m not sure it’s *possible* for my singing
+to get worse.) All that changed was my daughter’s frame of reference for
+those songs. The rest of the universe stayed the same; the change was in
+her mind, lock, stock, and barrel.
+
+Often, the key to solving a problem, or to working on a problem
+efficiently, is having a proper frame of reference. The model you have
+of a problem you’re tackling often determines how deeply you can
+understand the problem, and how flexible and innovative you’ll be able
+to be in solving it.
+
+An excellent example of this, and one that I’ll discuss toward the end
+of this chapter, is that of *3-D transformation*—the process of
+converting coordinates from one coordinate space to another, for example
+from worldspace to viewspace. The way this is traditionally explained is
+functional, but not particularly intuitive, and fairly hard to
+visualize. Recently, I’ve come across another way of looking at
+transforms that seems to me to be far easier to grasp. The two
+approaches are technically equivalent, so the difference is purely a
+matter of how we choose to view things—but sometimes that’s the most
+important sort of difference.
+
+Before we can talk about transforming between coordinate spaces,
+however, we need two building blocks: dot products and cross products.
+
+#### 3-D Math {#Heading3}
+
+At this point in the book, I was originally going to present a BSP-based
+renderer, to complement the BSP compiler I presented in the previous
+chapter. What changed my plans was the considerable amount of mail about
+3-D math that I’ve gotten in recent months. In every case, the writer
+has bemoaned his/her lack of expertise with 3-D math, and has asked what
+books about 3-D math I’d recommend, and how else he/she could learn
+more.
+
+That’s a commendable attitude, but the truth is, there’s not all that
+much to 3-D math, at least not when it comes to the sort of
+polygon-based, realtime 3-D that’s done on PCs. You really need only two
+basic math tools beyond simple arithmetic: dot products and cross
+products, and really mostly just the former. My friend Chris Hecker
+points out that this is an oversimplification; he notes that lots more
+math-related stuff, like BSP trees, graphs, discrete math for edge
+stepping, and affine and perspective texture mappings, goes into a
+production-quality game. While that’s surely true, dot and cross
+products, together with matrix math and perspective projection,
+constitute the bulk of what most people are asking about when they
+inquire about “3-D math,” and, as we’ll see, are key tools for a lot of
+useful 3-D operations.
+
+The other thing the mail made clear was that there are a lot of people
+out there who don’t understand either type of product, at least insofar
+as they apply to 3-D. Since much or even most advanced 3-D graphics
+machinery relies to a greater or lesser extent on dot products and cross
+products (even the line intersection formula I discussed in the last
+chapter is actually a quotient of dot products), I’m going to spend this
+chapter examining these basic tools and some of their 3-D applications.
+If this is old hat to you, my apologies, and I’ll return to BSP-based
+rendering in the next chapter.
+
+#### Foundation Definitions {#Heading4}
+
+The dot and cross products themselves are straightforward and require
+almost no context to understand, but I need to define some terms I’ll
+use when describing applications of the products, so I’ll do that now,
+and then get started with dot products.
+
+I’m going to have to assume you have *some* math background, or we’ll
+never get to the good stuff. So, I’m just going to quickly define a
+*vector* as a direction and a magnitude, represented as a coordinate
+pair (in 2-D) or triplet (in 3-D), relative to the origin. That’s a
+pretty sloppy definition, but it’ll do for our purposes; if you want the
+Real McCoy, I suggest you check out *Calculus and Analytic Geometry*, by
+Thomas and Finney (Addison-Wesley: ISBN 0-201-52929-7).
+
+So, for example, in 3-D, the vector **V** = [5 0 5] has a length, or
+magnitude, by the Pythagorean theorem, of
+
+![](images/61-01d.jpg)
+
+(eq. 1)
+
+(where vertical double bars denote vector length), and a direction in
+the plane of the x and z axes, exactly halfway between those two axes.
+
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+* * * * *
+
+Graphics Programming Black Book © 2001 Michael Abrash