In Problem 3(a), it should refer to the "gumdrop torus". The original version of the homework said "tangle cube". (Homework updated on 11/20/2010.)
The idea of environment mapping is that you photograph
an environment as seen through a chrome circle, in order to capture the look
of that environment. Each pixel of the camera records a color corresponding
to a reflection direction (specifically, reflected through the chrome
circle). You can turn this around and construct a lookup table that maps
each reflection direction to a color. This table is your environment map,
sometimes called a reflection map.
If you now have a synthetic object and synthetic camera, you can give the object a mirrored chrome look as follows: for each viewing ray that hits the object, compute the reflection direction, return the color found in the environment map for that reflection direction. (If your reflection direction lands between directions in the environment map, then you can use bilinear interpolation to get the desired color.) The object will look a lot like it would if it were actually in that environment. Problem 4(d) asks you to explore the question of whether the synthetically rendered object would look exactly like it would if it were a real object in that original environment.
Here is a simplified re-phrasing of the two questions found in Problem 5(g):
Given a single quadratic Bezier segment, the entire Bezier curve for that segment will always lie in a plane. Why?
Given a single cubic Bezier segment, will the entire Bezier curve for that segment generally lie in a plane? Justify your answer.