15-462 Written Assignment 2

Due in class Thursday 3/8 or in Jessica Hodgins's mailbox in NSH 4th floor by 9am on Friday 3/9.

Please use one page/problem and put your name at the top of each page.

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

1. Textures:
Why do you need to use texture levels (mipmaps) in OpenGL? If you just use one texture level (high resolution texture) and GL_LINEAR filtering provided by OpenGL will this fix all the problems? Explain your answer.

2. Curves:
a) Shirley Chapter 15, question 6
Devise an arc-length parameterization for the curve represented by the parametric function f(u)=(u, u2)
b) Natural Cubic Splines allow for C2 continuity with a piecewise cubic curve. How is it done? Do you get local control with Natural Cubic Splines? Explain your answer. Can you get C2 continuity and local control with piecewise cubic splines? Explain your answer.
c) Find the matrix, Mb_h that converts control points for a Bezier curve into control points for a Hermite curve. Show all steps of your derivation.

3. Meshes:
a) List advantages and disadvantages of implicit, parametric and triangular mesh representations
b) Which of the modeling techniques covered in class (Triangle Meshes, Parametric Surfaces, Implicit Surfaces, Constructive Solid Geometry or any other you know) would be most applicable for modeling each of the following objects. If there are several good candidates, feel free to mention more than one technique. Explain your answer.
a. Swiss cheese
b. A nut, in the nut-and-bolt sense
c. A christmas tree
d. A bowl
e. A set of bookshelves
f. A cloud
g. Cloth

4. Viewing: Shirley Chapter 7, question 1
Show algebraically that the perspective matrix preserves order of z values within the view volume.

5. Shading:
a) Why is the moon poorly approximated by diffuse or Phong shading?
b) Why do so many computer graphics objects look like they are made from hard shiny plastic?
c) 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.