CS 426 Exercises
  3D Primitives and Transformations

  1. What is the volume of a 3D point? a 3D ray? a 3D line? a 3D polygon? a 3D sphere?
  2. What issues must be addressed by a 3D rendering system but not by a 2D rendering system?
  3. Why does a 3D line not have a convenient implicit represenation?
  4. What is the implicit representation for a 3D plane?  What are the geometric interpretations of each parameter?
  5. What types of 3D transformations can be represented with a 3x3 matrix?
  6. What types of 3D transformation can be represented by a 4x4 matrix and 3D homogeneous coordinates?
  7. Why do we represent transformations with matrices?
  8. Which of the following 3D points with homogenous coordinates is closer to the origin: (8, 4, 2) or (4, 2, 1)?
  9. What is a linear transformation?  What are its properties?
  10. What is an affine transformation?  Which properties of linear transformation do not apply to affine transformations?
  11. What is a projective transformation?  Which properties of affine transformation do not apply to projective transformations?
  12. Write a sequence of transformation matrices that scales 3D points based on their distances from an arbitrary origin O = (Ox, Oy, Oz).
  13. Write a sequence of transformation matrices that rotates 3D points counter-clockwise by theta degrees about an arbitrary 3D line defined by P1 and P2.