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Homework 7, Due Thurs Nov 10, 2011



  1. (2 pt) Give the quaternion for a 90 degree rotation about the axis (.6,0,.8).
  2. (1 pt) What does the point (3,0,4) rotate to? Use the quaternion rotation formula. Show your work.
  3. (1 pt) Name the operation that does this:
    "Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping"
  4. (1 pt) What function do you call to change the shininess of a surface (when modeling it in OpenGL)?
  5. (1 pt) What is the normal vector to the sphere {$x^2+y^2+z^2=4$} at the surface point {$(x,y,z)$}?
  6. (4pt, 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 1/d2 .
    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 10%.
    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 5%.
    E.g., the brightness at p is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}