Computer Graphics Fall 2010 group home

# Homework 7, Due by email to TA Nov 4

*(1 pt)*What is the normal vector to the sphere {$x^2+y^2+z^2=4$} at the surface point {$(x,y,z)$}?*(1 pt)*What is the normal vector to the paraboloid {$z=2x^2+y^2$} at the surface point {$(x,y,2x^2+y^2)$}?*(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/d**. Each point on the floor is illuminated by all the lights, but more brightly by the closer lights. A point^{2}**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 10%. E.g., the brightness at**p**is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}