600.357/600.457 Computer Graphics
 

Fall 2001 Final Exam
Saturday, Dec 15, 2001. 2:00pm - 5:00pm

Good Luck!
  1. [25] We want to render a shadow of an object on the floor. Recall that the shadow can be computed as the projection of geometry on the plane of the floor. We are interested in computing that projection matrix. We will consider both a point light source as well as a directional light source. The floor is given as a polygon with the following vertices in the world space. (Do not worry about shadow clipping):
    2. [0, 0, 0]; [100, 100, -100]; [200, -100, 0]; [300, 0, -100].
    (a) [15] Compute the projection matrix if the point light is at [100, 100, 100]
    (b) [10] Compute the projection matrix for a directional light in the direction [-100, -100, -100]
     
  2. [25] Let us trace through the polygon rasterization algorithm. Assume a 100x100 pixel window. Consider the following polygon in the normalized 2D coordinates:
    3. [-1, 0]; [0, 1]; [0, 0]; [1, 1]; [1, 0]; [-1, -1].
    Find the states of the Edge table and the Active edge table, and the span for the following seven scan lines
    y = 0, 1, 49, 50, 51, 99, and 100.
    Recall that ET stores the higher point's Y, the lower point's X and the X increment. AET stores the same values but the lower point's X is updated with the edge's current X at each scan-line. Make sure that you compute and store all values in terms of integers.
     
  3. [15] Recall the mid-point line drawing algorithm. It can sometimes generate rather thin lines. We want to apply the following additional restriction on our line drawings -- no pixel may have only corner adjacent pixels (see figure on page 2). Give the pseudo-code for the restricted mid-point line drawing algorithm.
    4. none
  4. [25]

    5. (a) [5] What does sampling theory say about why aliasing occurs?
    (b) [8] What does the theory say about ways to eliminate aliasing
    (c) [12] Explain why weighted averaging of multiple samples per pixel works. How would you generate these weights?
  5. [25] We want to render shadows in a model with transparent polygons. Combine the shadow-map based algorithm and the transparency buffer based (Mammen's) algorithm.

    6.  
  6. [25] Suppose we have one pixel-processor for each pixel on the screen. Design a triangle rendering algorithm for this architecture. (Hint1: You do not have to use rasterization. Hint2: A processor can evaluate the equation of a line.)

    7.  
  7. [25]

    8. (a) [8] Describe Mip-mapping.
    (b) [7] Describe Summed area tables.
    (c) [10] What is Shading by interpolation? What are its advantages and disadvantages?
     
  8. [20]

    9. (a) [7] How can shadows be computed in the ray-tracing based display algorithm.
    (b) [13] Recall the recursive differential equation we call the Rendering Equation. Is recursive ray-tracing a solution of the Rendering Equation? Justify your answer.