CSSE 351 - Computer Graphics
Rose-Hulman Institute of Technology
Computer Science and Software Engineering Department

Problem Set 3

This assignment is due 20 September 2022.

This is intended to be an INDIVIDUAL exercise. Please, no assistance from other students.

  1. (15 points) Do exercise 4.1 on page 248 of your text.

    Show that the following sequences commute:

    1. a rotation and a uniform scaling
    2. two rotations about the same axis
    3. two translations

  2. (15 points) Do exercise 4.9 on page 249 of your text.

    In two dimensions, we can specify a line by the equation $y=mx+h$ find a 3$\times$3 affine transformation to reflect 2D points around this line.

    Hint: Write down the sequence of operations and the associated matrices to perform this operation. Be sure to specify the order of the matrices. There is no need to multiply the individual matrices together.

    Extend your result to reflect around a plane in three dimensions.

  3. (10 points) Do exercise 4.12 on page 249 of your text.

    Find a homogeneous-coordinate representation for a plane.

  4. (10 points) Do exercise 4.22 on page 249 of your text.

    An incremental rotation about the z-axis can be approximated by the matrix:

    \begin{displaymath} \left[ \begin{array}{cccc} 1 & -\theta & 0 & 0 \\ \thet... ...0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]. \end{displaymath}

    What negative aspects are there if we use this matrix for a large number of steps? Can you suggest a remedy?

    Hint: Consider a point 1 unit from the origin. After a large number of steps where will this point be?