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.
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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 |
- 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 |
- 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 |