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   <meta name="vspubdate" content="07/01/97" />
   <meta name="vscategory" content="Web and Software Development: Game Development,Web and Software Development: Graphics and Multimedia Development" />
 
-  <title>Michael Abrash's Graphics Programming Black Book Special Edition: Frames of Reference</title><!-- HEADER -->
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-  <!--ISBN=1576101746//-->
-  <!--TITLE=Michael Abrash's Graphics Programming Black Book Special Edition//-->
-  <!--AUTHOR=Michael Abrash//-->
-  <!--PUBLISHER=The Coriolis Group, Inc.//-->
+  <title>Michael Abrash's Graphics Programming Black Book Special Edition: Frames of Reference</title>
   <meta name="chapter" content="61" />
   <meta name="pages" content="1131-1135" />
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   <p><br /></p>
 
-  <h2><a id="Heading1"></a>Chapter 61<br />
+  <h2 id="Heading1">Chapter 61<br />
   Frames of Reference</h2>
 
-  <h3><a id="Heading2"></a>The Fundamentals of the Math behind 3-D Graphics</h3>
+  <h3 id="Heading2">The Fundamentals of the Math behind 3-D Graphics</h3>
 
   <p>Several years ago, I opened a column in <i>Dr. Dobb&rsquo;s Journal</i> with a story about singing my daughter to sleep with Beatles&rsquo; songs. Beatles&rsquo; songs, at least the earlier ones, tend to be bouncy and pleasant, which makes them suitable goodnight fodder&mdash;and there are a <i>lot</i> of them, a useful hedge against terminal boredom. So for many good reasons, &ldquo;Can&rsquo;t Buy Me Love&rdquo; and &ldquo;A Hard Day&rsquo;s Night&rdquo; and &ldquo;Help!&rdquo; and the rest were evening staples for years.</p>
 
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   <p>Before we can talk about transforming between coordinate spaces, however, we need two building blocks: dot products and cross products.</p>
 
-  <h4 align="left"><a id="Heading3"></a>3-D Math</h4>
+  <h4 align="left" id="Heading3">3-D Math</h4>
 
   <p>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&rsquo;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&rsquo;d recommend, and how else he/she could learn more.</p>
 
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   <p>The other thing the mail made clear was that there are a lot of people out there who don&rsquo;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&rsquo;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&rsquo;ll return to BSP-based rendering in the next chapter.</p>
 
-  <h4 align="left"><a id="Heading4"></a>Foundation Definitions</h4>
+  <h4 align="left" id="Heading4">Foundation Definitions</h4>
 
   <p>The dot and cross products themselves are straightforward and require almost no context to understand, but I need to define some terms I&rsquo;ll use when describing applications of the products, so I&rsquo;ll do that now, and then get started with dot products.</p>
 
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     Graphics Programming Black Book &copy; 2001 Michael Abrash
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