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Homework 6, Due Thurs Nov 3, 2011



  1. (2 pt) Write the vector equation for rotating points by {$180^\circ$} about the axis (3,4,5).
  2. (1 pt) What does the point (0,2,0) rotate to?
  3. (1 pt) If {$a=(1,5,6)$} then write {$a\cdot p a$} as a matrix depending on {$a$} times {$p$}.
  4. (1 pt) Write {$a\times p$} ditto.
  5. This {$M=\begin{pmatrix}.6&0&-.8\\0&\color{red}{0}&0\\-.8&0&-.6\end{pmatrix} $} is not a rotation matrix.
    1. (1 pt) Give 2 rules for being a rotation matrix that it violates.
    2. (1 pt) Change {$m_{2,2}$}, highlighted in red above, to make {$M$} a rotation matrix.
  6. (2 pt) This {$M=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix} $} is a rotation matrix. What are the normalized axis and angle of rotation?
  7. (2 pt) Consider the following projection:
    • The camera is at (0,0,0).
    • The projection plane is x+2y+3z=4.
    Write the projection equations to take a general point (x,y,z) to a point on the projection plane. You do not need to put this as a matrix.
  8. (4 pt) Consider this projection:
    • The camera is at (0,0,0).
    • The projection plane is z=-4.
    1. Write the view normalization equations for this projection.
    2. Consider the cube with opposite corners at (-4,-4,-2) and (4,4,-10). What does it transform to? Show the projection vertex by vertex. E.g., what does (4,4,-10) transform to?