CS465 Fall 2006
Homework 2 Solution
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1. You will get a correct but upside down image of whatever is
   behind the camera.

Comment: The ray direction is computed by subtracting the
   viewpoint from the image plane point.  When the two points are
   swapped, your ray direction will be the negative of the
   correct ray direction.

2. You will get an image with blotchy shadows where there should
   be no shadows.

Comment: Because floating point numbers are discrete, the start
   of the shadow ray will be above the surface some of the time
   and under the surface some other time.  If the start of the
   shadow ray is under the surface, the ray will intersect the
   surface and the program will think that the light source is
   occluded by the surface.

3. The surface will be incorrectly shaded -- the farther a point
   is from the light source, the brighter it will be.

Comment: Lambertian shading has the following formula:
   k_d*dot(n,v_l).  If v_l is just the vector to the light, its
   the result of the dot product is wrong by a factor of d, where
   d is the distance to the light.  If the distance is more than
   one the shading will be too bright; if it is less, it will be
   too dark.

4. You will see objects that are behind other objects.

5. You will always get normal perspective view; that is, no
   tilted image plane is possible.

6. You will see a black image.
 
Comment: A shadow ray is meant to check for intersections along
   the line segment from the shading point to the light.  If we
   just intersect the ray and accept any intersection, without
   checking that it happens between the light and the surface, we
   will see shadows from objects that are behind the light
   source, which should not cast shadows.  Since the scene is
   closed, there is always an object behind the light source and
   every point will always be in shadow.

For this one you were pretty close if you recognized that objects
that should not cast shadows would cast shadows.

7. The surface of the sphere will be incorrectly shaded -- the
   larger the sphere is, the brighter it will be.

Comment: Let's take Phong shading as an example; Phong shading
   has the following formula: k_d*dot(n,v_l) + k_s*dot(n,v_h)^p.
   Normally one computes n by normalizing the radius vector from
   the sphere's center to the shading point.  If the
   normalization is skipped both dot products are wrong by a
   factor of r, the sphere's radius.  For r > 1 the shading will
   be too bright (and the specular component will be way too
   bright), and for r < 1 the shading will be too dark (and the
   specular component will be way too dark).