15-462 Written Assignment 2

Due in class Thursday 10/17 or in Jessica Hodgins's mailbox in NSH 4th floor by 9am on Friday 10/18

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

Justify your answers, perhaps including a picture.

b. Velvet is poorly approximated by the lighting models we discussed in class. What observations tell you that this is true?

4. Clipping

You have a 6-sided convex polygon that lies partially outside the viewing volume. After clipping is performed, what are the minimum and maximum number of vertices of the resulting polygon?

What if the polygon is concave? Justify your answers.