Homework1

A reminder: you are free to collaborate, but you must understand and write up the final answers by yourself!

Write up your answers on a separte sheet. You do not need to include a cover page, but your name and mailbox must be on the first page. Staple multiple pages.

  1. Show that the following sequences commute:

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

    Rubric
    General proof for any inputs No general cases : 0 Any inputs : 1
    Show single commute permutation No transformations : 0 Single way commutes : 2
    Show commute permutations are equal Single commutativity : 0 Show both commutes : 2

  2. Show that any sequence of rotations and translations can be replaced by a single rotation about the origin followed by a translation.

    Rubric
    Prove basic finite R T cases No transform cases : 0 Proves basic cases : 4
    Prove basic infinite R T cases No transform cases : 0 Proves infinite cases : 4
    Prove complex infinite R T cases No infinite cases : 0 Proves infinite cases : 5
    General proof for any inputs No general cases : 0 Any inputs : 2

  3. Three vertices determine a triangle if they do not lie in the same line. Devise a test for collinearity of three vertices.

    Rubric
    Collinear test method No method : 0 Algorithm without justification : 3 Algorithm by example : 6
    Correct algorithm Does not work at all : 0 Correct for many inputs : 2 Correct for all inputs : 5
    General proof for any inputs No general cases : 0 Any inputs : 2