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+#### Notes on the 3-D Animation Example {#Heading9}
+
+The sample program transforms the polygon’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—we’re looking right
+down the negative Z axis of world space—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.
+
+The maximum translation along the Z axis is limited to -40; this keeps
+the polygon from extending past the viewpoint to positive Z coordinates.
+This would wreak havoc with the projection and 2-D clipping, and would
+require 3-D clipping, which is far more complicated than 2-D. We’ll get
+to 3-D clipping at some point, but, for now, it’s much simpler just to
+limit all vertices to negative Z coordinates. The polygon does get
+mighty close to the viewpoint, though; run the program and use the “T”
+key to move the polygon as close as possible—the near vertex swinging
+past provides a striking sense of perspective.
+
+The performance of Listing 50.5 is, perhaps, surprisingly good, clocking
+in at 16 frames per second on a 20 MHz 386 with a VGA of average speed
+and no 387, although there is, of course, only one polygon being drawn,
+rather than the hundreds or thousands we’d ultimately like. What’s far
+more interesting is where the execution time goes. Even though the
+program is working with only one polygon, 73 percent of the time goes
+for transformation and projection. An additional 7 percent is spent
+waiting to flip the screen. Only 20 percent of the total time is spent
+in all other activity—and only 2 percent is spent actually drawing
+polygons. Clearly, we’ll want to tackle transformation and projection
+first when we look to speed things up. (Note, however, that a math
+coprocessor would considerably decrease the time taken by floating-point
+calculations.)
+
+In Listing 50.3, when the extent of the bounding rectangle is calculated
+for later erasure purposes, that extent is clipped to the screen. This
+is due to the lack of clipping in the rectangle fill code from Listing
+47.5 in Chapter 47; the problem would more appropriately be addressed by
+putting clipping into the fill code, but, unfortunately, I lack the
+space to do that here.
+
+Finally, observe the jaggies crawling along the edges of the polygon as
+it rotates. This is temporal aliasing at its finest! We won’t address
+antialiasing further, realtime antialiasing being decidedly nontrivial,
+but this should give you an idea of why antialiasing is so desirable.
+
+### An Ongoing Journey {#Heading10}
+
+In the next chapter, we’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’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’ll use fixed-point arithmetic to speed things up, and
+do some shading and texture mapping. The journey has only begun; we’ll
+get to all that and more soon.
+
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+* * * * *
+
+Graphics Programming Black Book © 2001 Michael Abrash