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Homework2

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

Complete the homework as a pdf, doc, or on paper (and scan), then submit it to the homework2 directory in your class repo.

1. Show that perspective projection preserves lines.
   
   Hint: the easiest argument to make is a geometric one using the 3D line, the center of projection, and the image plane.

   Prove lines remain lines after perspective

   0: No general cases

   3: Show single example

   6: Show for specific case

   10: Show for all general cases

2. For a $4 \times 4$ matrix whose top three rose are arbitrary and whose bottom row is $(0, 0, 0, 1)$, show that the points $(x, y, z, 1)$ and $(hx, hy, hz, h)$ transform to the same point after homogenization.

   Prove homogenization

   0: No general cases

   3: Show for specific cases

   10: Show for all possible points

3. For the eye position $\mathbf{e} = (0, 1, 0)$, a look vector $\mathbf{g} = (0, -1, 0)$, and an up vector $\mathbf{t} = (1, 1, 0)$, what is the resulting $\mathbf{uvw}$ basis used for coordinate rotations?

   Basis position	0: No position	1: Basis position given
   Basis vectors	0: No vectors	3: Basis u,v,w vectors given
   Basis values	0: Incorrect values	6: Basis values set according to camera equations