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Schedule Syllabus Resources

Homework2

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 separate 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 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.

Rubric

Prove lines remain lines after perspective	0 : No general cases	3 : Show single example	6 : Show for specific	10 : Show for all general cases

2. For a $4 \times 4$ matrix whose top three rows 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.

Rubric

Prove homogenization	0 : No general cases	1 : 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?

Rubric

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