# All numbered problems have equal weight although the sub-problems are weighted approximately according to difficulty.

## 1. Splines

Suppose you have four points P0, P1, P2, P3 (either in 2D or 3D). First, connect successive points with parametric line segments where u ranges from 0 to 1 for each. This way, you obtain three line segments. Now, linearly interpolate between succesive pairs of line segments by connecting points for the same value of u with a line segment and then using the same value for u to obtain a point along this new line segment. This way, you obtain two curves, each parameterized with u ranging from 0 to 1. Now, repeat the same process for this pair of curves: for every u, find the two points corresponding to u. Then draw a line segment connecting the two points, and use the same value u to find a point along this segment. The result is some curve, parameterized for u ranging from 0 to 1. Provide a mathematical derivation to show what spline this construction creates. Hint: draw a picture to see what curve is created by this construction.

## 2. Implicit Surfaces and CSG

a. Describe an algorithm for converting the implicit representation of a cylinder to a polygonal representation. What properties should the polygonal representation have to be useful?

b. Describe how you would model a Luxo lab using constructive solid geometry (a Luxo lab is a desk lamp--like the hopping one we saw in Luxo Jr.). Sketch each of the main parts and describe how you would model them.

## 3. Lighting and Illumination

a. Assume the lighting models we discussed in class (ambient, diffuse, and specular). Given a point P on the surface, a position of the light source, and the position of the viewer, how must the surface at the point P be oriented, so that

• a maximum amount of reflected specular light reaches the viewer?
• a maximum amount of reflected diffuse light reaches the viewer?