課程名稱︰計算機圖形
課程性質︰選修
課程教師：歐陽明
開課學院：電資學院
開課系所︰資工所、網媒所
考試日期（年月日）︰2006.11.21
考試時限（分鐘）：
試題 :
Interactive Computer Graphics Midterm Exam November 21, 2006
1. (10%) Let C be a circle in 2D with radius one, centered at the origin as
shown in Figure 1. Define a transformation matrix T that transforms the
circle to an ellipse rotated clockwise by pi/4 with major axis of length 2
and minor axis length 1, as shown in Figure 1.
(You may define T as a composition of elementary transformation matrices)
Figure 1:

2. (10%) If we have a transform matrix M which transforms object from modeling
(object) coordinates to world coordinates.
(a) What if we use this M to transform the object's normal vectors? Is it
OK or not?
(b) If not OK, how to tackle this problem?
3. (15%) It is said that Microsoft game machines X-Box (1, and 306) are
costing more than their sale prices, and then use the game software
revenue to compensate for the loss in hardware.
(1) If this is true, what advantages for Microsoft in developing the
hardware platform can be predicted from this business model?
(2) In the above case, is it easy for companies from Korea or Taiwan to
develop their own game machines or clones of these machines? Why and
why not?
4. (10%) Transformations
Figure 2:

(a) (5%) Mr. J wants to implement a program to simulate solar system for
MIB institution. please help him to design the transformation matrix,
which represents the behavior of the rotation (自轉) and the
revolution（公轉）of earth. In the above picture, where R_1 is the
average orbital radius around sun, and R_2 is the average orbital
radius around earth.
(b) (5%) Assume that moon has faced toward earth with the same side for
billions of year, please try to design and describe the transformation
matrix of moon in the solar system.
5. (15%) Visible-Surface Determination
In general polygon rendering, the Z-buffer technique is used for visible
surface determination. Some modified techniques can be applied to reduce
the rendering time in implementation.
(i) Using depth sort + Z-buffer (always display the nearest polygon first)
(ii) Only using Z-buffer.
(a) (10%) Let the screen width = W, height = H, and there are N polygons
in the building model. Based on your experience, using the teapot
model and CSIE department building model, how will you describe the
improvement (speedup) using (i) over (ii)?
(b) (5%) Compare the computational complexity of the two methods above, in
terms of worst case and average case.
6. (15%) BSP tree
Figure 3:

(a) (10%) Construct the Binary Space Partitioning (BSP) tree of the model
in Fig 3. (use polygon 1 as the root, and always use smaller number
polygon to the left of the tree)
(b) (5%) From the BSP tree in (a), derive the correct display sequence in
terms of the viewing position (eye).
7. (5%) What is your term project for this semester? What are the
technical difficulties involved in the project? (You can refer to
the project listing).
8. (10%) In 2D, please consider if a rotation transformation by angle A can
be specified as a series of shear transformation matrices. Give these
matrices, or if it can't be done, prove it.
9. (10%) If we have four control points for the Bezier curve. P1 (1, 2, 4),
P2 (3, 8, 6), P3 (6, 6, 4), P4 (8, 2, 2).
(a) (6%) In order to draw this curve, we split it by half. What are the new
control points of the two split curves? (Please give exact numbers).
(b) (4%) To design airplane wings, which kind of curve should be used?
Bezier curve or B-splines? Why? (please give short answer).