CS 418: Homework #2
Assigned: Tuesday April 19, 2005
Due: Tuesday April 26, 2005
Please be organized when writing your answers to these questions. Make sure that all solutions are clearly
indicated and labelled with the question they are answering. Remember to write clearly and legibly. Un-
readable answers will receive 0 credit.
1. Recall that a Gaussian blob is represented as an implicit surface by the zero set of the function:
F(x) = eσkxk2−τ= 0
(a) Given a ray x(t) = p+td, derive an equation for the value(s) of tat which the ray intersects this
surface.
(b) Give the conditions that we must check to determine whether the ray actually hits the surface or
not?
2. In class we discussed a method for generating shadows by projecting an object through a point onto a
plane. If the light source (which is the focal point of the projection) moves to infinity, then all light
rays become parallel. To generate such shadows, we must perform parallel projection onto a plane. This
provides us with a reasonable way to model shadows cast by the sun, which for all practical purposes is
infinitely far away.
d
n
Suppose that you are given an arbitrary light direction dand a plane n·x+k= 0. (You may assume
kdk=knk= 1. Now suppose you are given some triangulated surface model. We want to project every
vertex valong the direction donto the plane. Derive the 4×4 matrix which accomplishes this projection.