Studies in Light
During the early 1980s, as the field of computer graphics matured, many practitioners moved beyond issues of simple geometric shape and began addressing the more fundamental issue of light as well. It is light's interaction with shape, after all, that gives rise to vision, and in photorealistic images or scenes, the physics of light and the geometry of vision determine the scene's overall composition, coloration, luminance, and shadows. This section looks at a few models of light and vision drawn from perspective and optics.
Figure 5: Nathan's Windmill
|Figure 6: Shadowfall on Nathan's Highway (dawn through mid-afternoon)|
Figure 7: Shadowcasting, simplified
As an accurate model of the physics of such a situation, Nathan's construction is not without limitations. Our own sun, for instance, is sufficiently far away, and planet Earth so small in comparison, that the sun's light rays are effectively parallel. Perhaps Nathan's world has a much smaller, closer sun, however, which acts more like a spotlight than as a general source of diffuse illumination.
More gravely, though, this geometric construction lacks any interpretation of how far behind the monolith ABCD the sun stands. Shadows exist in three-dimensional configurations of objects, light sources, and ground, but this is fundamentally a two-dimensional construction. Thus, the "height" of the shadow—measured as the vertical distance from AD to B' or C'—is arbitrary, rather than determined by the geometry of the scene. In Nathan's example, this height is fixed (as can be seen in Figure 6), where in a more realistic scene, it would vary with the height of the sun in the sky.
These limitations do not detract from Nathan's accomplishment, however. In fact, by choosing to approximate the geometric behavior of a complicated dynamic construction with that of a much simpler one, Nathan adopted a strategy common to computer graphics' practitioners: only devote as many resources to a modeling task as required to obtain the desired degree of realism. Again, where cost is a central issue—here "cost" being the amount of time Nathan had to complete his geometry assignment—figuring out ways to minimize it becomes a creative part of the problem-solving process.
- Add a second sun to Figure 7, as might be found in a binary star system. Cast a second shadow or set of shadows. Animate the two suns along different paths through the sky.
- Experiment with shadows cast by more complicated objects than simple quadrilaterals (like Nathan's highway sign or Figure 7's monolith ABCD). Can you model the shadows cast by an object with non-vertical sides, or by an object with a hole in it?
- Create a model of light projection similar to Figure 7's, but in which the sun is infinitely far away, and thus casts rays which are all parallel. How are shadows different from Nathan's in such a model?
- Describe the components of a more complicated model of shadow casting which accounts for the height of the sun (and therefore the height of shadows) geometrically. Can you create such a model in Sketchpad?