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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="50-06.html">Previous</a></td>
<td>
<a href="50-06.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="51-01.html">Next</a></td>
<td>
<a href="51-01.html">Next</a>
</td>
</tr>
</table>
</center>
<p><br /></p>
<h4 align="left" id="Heading9">Notes on the 3-D Animation Example</h4>
<h4 id="Heading9">Notes on the 3-D Animation Example</h4>
<p>The sample program transforms the polygon&rsquo;s vertices from object space to world space to view space to the screen, as described earlier. In this case, world space and view space are congruent&mdash;we&rsquo;re looking right down the negative Z axis of world space&mdash;so the transformation matrix from world to view is the identity matrix; you might want to experiment with changing this matrix to change the viewpoint. The sample program uses 4x4 homogeneous coordinate matrices to perform transformations, as described above. Floating-point arithmetic is used for all 3-D calculations. Setting the translation from object space to world space is a simple matter of changing the appropriate entry in the fourth column of the object-to-world transformation matrix. Setting the rotation around the Y axis is almost as simple, requiring only the setting of the four matrix entries that control the Y rotation to the sines and cosines of the desired rotation. However, rotations involving more than one axis require multiple rotation matrices, one for each axis rotated around; those matrices are then concatenated together to produce the object-to-world transformation. This area is trickier than it might initially appear to be; more in the near future.</p>
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<p>In the next chapter, we&rsquo;ll assign fronts and backs to polygons, and start drawing only those that are facing the viewer. That will enable us to handle convex polyhedrons, such as tetrahedrons and cubes. We&rsquo;ll also look at interactively controllable rotation, and at more complex rotations than the simple rotation around the Y axis that we did this time. In time, we&rsquo;ll use fixed-point arithmetic to speed things up, and do some shading and texture mapping. The journey has only begun; we&rsquo;ll get to all that and more soon.</p>
<p><br /></p>
<center>
<table border="1">
<tr>
<td><a href="50-06.html">Previous</a></td>
<td>
<a href="50-06.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="51-01.html">Next</a></td>
<td>
<a href="51-01.html">Next</a>
</td>
</tr>
</table>
</center>