CS 660: 3D Computer Graphic

Student Name: CS 660: 3D Computer Graphic Final Exam, Fall 2007

2 pts 1. Shown below is the stencil for Modified Butterfly subdivision surfaces for regular vertices. Figure 1(a) shows a portion of a triangulated subdivision surface after i iterations of the Modified Butterfly algorithm.

 In Figure 1(b), mark in the shaded region the approximate location of all vertices that would exist on the surface after (i+1) iterations of the Modified Butterfly algorithm. Do not mark any vertices that would likely fall outside the shaded region.

 For each vertex you mark, assign a label and write an expression for the vertex’s (x,y,z) coordinates in terms of the coordinates of the labeled vertices in Fig 1(a). p Fig. 1(a) 4 p p 3 1 p 2 p 9 p p 7 8 p p 13 6 p p 18 5 p 12 p p 17 11 p 10 p 16 p p 15 14 Fig. 1(b) 2. Given a bi-linear patch defined by vertices (a, b, c, d), where the vertices have the (x, y, z) coordinates and (u,v) coordinates shown in the table, what are the (x,y,z) coordinates of the point that corresponds to (u = 0.2, v = 0.6)? c 1 pt d x y z u v a 1 3 1 0 0 b 4 3 1 1 0 c 5 7 2 1 1 a b d 2 6 4 0 1 3. p1 p4

The cubic Bezier matrix equation is:

Write a formula for computing the (x,y) coordinates of the t = 0.25 point on the cubic Bezier curve that is governed by control points p1, p2, p3, and p4. Write the formula in terms of the control points’ (x,y) coordinates and the parameter t..

1 pt 4. Describe how form factors are computed for radiosity rendering algorithms using a) hemi-cubes and b) hemispheres

What are the relative advantages and disadvantages of the two approaches?

2 pts 5. (a) How many texels would be required to store a MIP mapped texture image that has a full- resolution size of (64 x 64) ?

(b) What MIP map level would be used to apply texture in the scene shown below? Level 0 is the highest resolution texture in the MIP map and lower resolution levels are indexed with increasing level numbers.

1 pt 2 pts 6. Assume you are writing rendering software that uses a ray tracing approach. You know that the most computationally expensive part of the ray tracing algorithm is the intersection test, so you decide to use scene graphs in your software in an effort to improve your renderer’s efficiency.

In class, we looked at pseudocode for a ray tracer, but in that pseudocode we left the call to “intersection test” undefined, because the calculations for an intersection depend on the type of model (triangular facets, spheres, subdivision surfaces, meta-balls, etc.) being rendered.

For the sake of simplicity, assume that the only models you need to render are perfect spheres. For each sphere, your scene graph will store - a center point (x,y,z) - scalar valued radius, r - id number (could be used to lookup other information: color, material, etc.)

Based on these assumptions, write pseudocode for an intersection test function that could be used by your ray tracer. The inputs to the function are: - a scene graph tree that stores a collection of sphere models and bounding volumes - a ray’s o point of origin (x,y,z) and o direction vector (x,y,z) The output of the function is: - intersection point (x,y,z) - id number of the sphere that is intersected

Be sure to make use of the scene graph and associated bounding volumesto get an efficient algorithm. diffuse sphere center = (0, 4, 0) radius = 3 diffuse & specular color = (.5, .5, .5) light position = (0, 10, 5) ambient & emissive color = (0,0,0) color = (1, 1, 1) shininess coefficient = 1 opaque (reflection only, no refraction)

specular sphere center = (0, -3, 10) radius = 4 diffuse & specular color = (.5, .5, .5) ambient & emissive color = (0, 0, 0) shininess coefficient = 1 opaque (reflection only, no refraction)

7. In the diagram above, the camera is located at the origin and ray R points in the positive z direction. What color is rendered for ray R?

2 pts 8. Compare and contrast the computational efficiency and visual realism of rendering with each of the rendering methods listed below. Name specific visual effects that each method can or cannot render, and explain what characteristics of a scene or image primarily determine the amount of time required for rendering with each of the methods. a) Ray tracing b) Radiosity c) Light Cut algorithm