Homework 7, due Thurs Nov 1

Hand in your solution on RPILMS. Each team should submit their solution under only 1 student's name. The other student's submission should just name the lead student. (This makes it easier for us to avoid grading it twice.) For programming exercises, hand in code and screen dumps. We won't run your code, but will use the screendumps to judge how it worked.

This homework has no programming. Use any tool that you wish for the multiplications, e.g., Maple or Mathematica. Include a copy of the relevant part of the session.

  1. (2 pts) On the following CIE chromaticity diagram what are the coordinates of
    1. spectrally pure yellow?
    2. spectrally pure 5000 Angstroms?
    CIE chromaticity diagram
    (.45,.54). (.02,.54)
  2. (4) You have a perspective projection with the camera at (0,0,2) and the projection plane at x=3.
    1. Write the Cartesian project equations.
    2. Write the homogeneous projection matrix.
    • One way is to translate the camera to the origin, project, then translate back. Or, you could just derive the equations. I did the latter.
    • {$ \begin{array}{l} x'=3\\ y'=3y/x\\ z'=3(z-2)/x+2 \end{array} $} corrected
    • {$ \left(\begin{array}{rrrr} 3&0&0&0\\ 0&3&0&0\\2&0&3&-6\\1&0&0&0 \end{array}\right) $} corrected
  3. (4pts, this question will take some thinking).
    Imagine that you have an infinitely large room illuminated by one infinitely long row of point lights. This figure shows a side view of the room.
    The lights are h above the floor and are 1 meter from each other. Assume that the ceiling above the lights is black and that no light reflects off of anything.
    An object at distance d from a light gets illuminated with a brightness {$ \frac{1}{d^2} $}.
    Each point on the floor is illuminated by all the lights, but more brightly by the closer lights.
    A point p directly below a light will be a little brighter than a point q halfway between two such points. That is the problem --- we want the floor (at least the part directly below the line of lights) to be evenly lit, at least within 1%.
    However, the higher the line of lights, the more evenly the floor will be lit.
    Your question is to tell us what is the minimum value for h so that the line of the floor below the line of lights is evenly lit within 1%.
    E.g., the brightness at p is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}
    • Using Mathematica, the brightness at p is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} = \frac{\pi \text{Coth}[h \pi ]}{h} $$}
    • The brightness at q is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+(d+1/2)^2} = \frac{\pi \text{Tanh}[h \pi ]}{h} $$}
    • One way is to plot their relative difference and see when that falls below .01. That's at about h=.96. Saying, 1, is good enough.
    • The unevenness in the brightness falls off exponentially with h.
  4. (5) Do Exercise 11.20 on page 442. postponed until homework 8.

Total: 10