Studies in Motion
If a picture's worth a thousand words, then a movie—displaying 34 or 36 pictures per second—is worth some two million words per minute. Adding animation—motion or any form of change over time—to computer-generated images greatly enhances their value. In scientific visualization, time can model a new dimension of the data set or structure under investigation. In industrial applications, animation allows computer graphics to represent the behavior of a simulated real-world system as it adapts or responds to changes in its environment (as, for instance, a car's rate of fuel consumption varies with its acceleration and velocity). In television advertisements and commercial films, animation allows computer-generated objects or characters to interact with human actors, and generally adds realism, drama, or humor to graphical images.
From a mechanical perspective, animation is accomplished simply by showing a viewer still images at a rate fast enough to suggest motion. But determining how this sequence of physical images differs from each other over time introduces a variety of issues beyond the task of generating individual images. In the present era, for instance, computer graphics are quite capable of generating photorealistic pictures of humans (given sufficiently complicated geometric models, and sufficiently powerful computers to render them), but the ability of machines to calculate or model human motion—the anatomical kinematics of walking, smiling, and flying into a rage—remains in its infancy. (It's no surprise that, given this state of the affairs, the computer-generated transformations of human actors in Hollywood films like Terminator II or The Mask are frequently to alter egos who are cartoon characters, robots, or other creatures with simpler motions than ours.)
Perhaps the single greatest consequence of insight into the geometry of a system in motion was that of Copernicus, whose revolutionary ideas about the solar system can be effectively demonstrated with simple two-dimensional graphics. Up to the sixteenth century, the orthodox view of the motion of planets—and consequently, the divine order of the cosmos—located the earth at the center of the universe, with other celestial bodies (the sun, moon, and observed planets) rotating about it in space. [Copernicus] was uneasy with this model's convoluted explanation of certain peculiar astronomical observations, writing that "... I considered that I too might well be allowed to try whether sounder demonstrations of the revolutions of the heavenly orbs might be discovered by supposing some motion of the earth."
Figure 9. Geocentric Planetary Motion
The Ptolemaic system to which Copernicus responded is depicted—in oversimplified form—in Figure 9. Earth is fixed, and Sun and (some) Planet rotate about in on orbits implied by dashed circles. One consequence of this model is that from the viewpoint of Earth, all orbiting bodies always move in the same direction across the heavens. While it's difficult to capture this motion in a static illustration, Figure 9 attempts to illustrate it by plotting the position of the orbiting Planet as if its distance from Earth were increasing over time. The resulting simple spiral shows that Planet (seen along a ray of sight from Earth) always moves in the same direction.
Copernicus observed that though this simple, ever-forward motion well describes the sun (rising in the east, setting in the west, day after day) and moon, it failed to address occasional eccentricities seen in the motion of other planets. Rejecting the Ptolemaic explanation for this effect (a complicated additional construction involving circles rolling inside circles), he proposed locating the sun at the center of the universe, positioning the earth along with other planets as a solar orbiter. From a geometric perspective, the Copernican construction in Figure 10 is accomplished by simply switching the labels of Figure 9: Sun and Earth's positions are now exchanged.
In the Copernican system, planets still travel continuously and in a single direction through space, but our perspective on them is now biased by the fact that we, on Earth, are now moving too. This bias reveals itself in Figure 10 as slight "hiccups" in the position of Planet as plotted (by the sight ray) from Earth. These hiccups are precisely the observed phenomena that Copernicus found unaccounted for by the Ptolemaic model: while planets usually follow the sun across the sky, occasionally they fall into retrograde, marching in precisely the opposite direction. Though a simple account of the observed motion of an elementary geometric system, Copernicus' idea—and the Copernican revolution—deeply influenced the later discoveries of Galileo and Newton, and profoundly contributed to the end of the central social role played throughout the Middle Ages by the Catholic Church.
Figure 10. Heliocentric Planetary Motion
- Imagine there were no other observable planets or stars. From a viewpoint positioned on earth, could you still determine geometrically whether the earth rotated the sun or the sun rotated the earth?
- Construct a moon that orbits the earth as the earth orbits the sun. What's the locus of the moon (when viewed, as in the previous figures, from above the plane of the solar system)?