CIS 736 (Computer Graphics)

Spring, 2001

Homework Assignment 1

Problem Set

Friday, January 26, 2001

Due: Friday, February 2, 2001 (by midnight)

This written assignment is designed to apply your theoretical understanding of the analytic geometry and linear algebra towards principles of computer graphics (CG), and to help you synthesize your own examples of CG problems and methodologies.

Refer to the course intro handout for guidelines on working with other students.  Your solutions should be produced them only from your personal notes (not common work or sources other than the textbook or properly cited references).

Note: Remember to submit your solutions in electronic form using handin (instructions shall be posted on the web board) and in hard copy in class (unless you are taking this course via distance education).

Warm-up / refresher exercises (basic linear algebra)

1.        (15 points total) Coordinate systems.

a)       (10 points) Change of coordinate system. In R3, consider the two bases {(1, 0, 0), (1, 1, 0), (1, 1, 1)} and {(1, 0 , 0), (0, 1, 0), (0, 0, 1)}.  Find the two matrices that convert representations between the two bases.  Show that they are inverses of each other.

b)       (5 points) Homogeneous coordinates. Another reason that homogeneous coordinates are attractive is that 3D points at infinity in Cartesian coordinates can be represented explicitly in homogeneous coordinates.  How can this be done?

Basic Raster Graphics

2.        (10 points) Scan Conversion.  Modify the midpoint algorithm for scan converting lines to handle lines at any angle.  (You may assume that vertical and horizontal lines, i.e., slope 0 and ¥, are still detected and handled separately.)

3.        (10 points) Transformations.  Prove that we can transform a line by transforming its endpoints an then constructing a new line between the transformed endpoints.

4.        (15 points) Analysis.  Aliasing, the appearance of artifacts due to sampling (discretization) error over any function of a continuous variable that contains sharp changes in intensity, is a serious problem in that it produces unpleasant or even misleading visual artifacts.  Discuss situations in which these artifacts matter, and those in which they do not.  Discuss various ways to minimize the effects of jaggies, and explain what the “costs” or those remedies might be.

Extra credit (5 points) Clipping.  Explain why the Sutherland-Hodgman polygon clipping algorithm works only for convex clipping regions.  (To receive credit for this problem, you must explain what happens in the case of concave regions.)

Class participation:

a)       (Required) Post your “turn-to-a-partner” exercise from Wednesday, January 17, 2001, or a follow-up containing your thoughts on this exercise, to the CIS736 web board.

b)       (Optional) Post any unclear points regarding basic scan conversion, matrix transformations, mathematical foundations of CG, or examples that you would like to see covered in lecture or explained again on the class web board.