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Retidy, removing align=left from headings and empty paragraphs

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James Gregory 2013-12-30 15:37:13 +11:00
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
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<center>
<table border="1">
<tr>
<td><a href="60-04.html">Previous</a></td>
<td>
<a href="60-04.html">Previous</a>
</td>
<td><a href="index.html">Table of Contents</a></td>
<td>
<a href="index.html">Table of Contents</a>
</td>
<td><a href="61-02.html">Next</a></td>
<td>
<a href="61-02.html">Next</a>
</td>
</tr>
</table>
</center>
<p><br /></p>
<h2 id="Heading1">Chapter 61<br />
Frames of Reference</h2>
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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" id="Heading3">3-D Math</h4>
<h4 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" id="Heading4">Foundation Definitions</h4>
<h4 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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<p>(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.</p>
<p><br /></p>
<center>
<table border="1">
<tr>
<td><a href="60-04.html">Previous</a></td>
<td>
<a href="60-04.html">Previous</a>
</td>
<td><a href="index.html">Table of Contents</a></td>
<td>
<a href="index.html">Table of Contents</a>
</td>
<td><a href="61-02.html">Next</a></td>
<td>
<a href="61-02.html">Next</a>
</td>
</tr>
</table>
</center>