CSC418/2504F: Fall 2001 Assignment 2

Solutions:

New Due Date:

How to hand in your work:

• If you attend the MW 1pm section, hand your assignment in to the T.A. at your Fri 1pm tutorial.
• If you attend the W 7pm section, drop your assignment in SF 4306 before 5pm.

Answers to frequently-asked questions.

Questions about marking?

1. Transformations [6 marks] Consider the following 2-dimensional coordinate frames O, A and B.
   
   1. [1 mark] What are the coordinates of point P in frames A and B?
   2. [3 marks] What is the 3x3 homogeneous matrix M_AB that, when multiplied by a point in frame A, yields that point's coefficients in frame B?
   3. [2 marks] What is the 3x3 matrix M_BA that does the opposite conversion?

2. Viewing and Projection [6 marks] Given a camera with the following parameters:
   P_eye=(3,2,4) P_ref=(0.5,0.5,0.5) V_up=(0,1,0), and with an image plane at z=-1 (in Viewing coordinates),
   1. [2 marks] What is the view matrix M_WCS-VCS?
   2. [3 marks] On the image plane, what are the coordinates of vertices A,B,C,D and E of the following pyramid?
      
   3. [1 mark] There will be two vanishing points. What are their coordinates on the image plane?

3. Modeling [4 marks] Consider the following skew cube, which is built as follows: The base abcd is a square on the xy plane with corners a(0,0,0) and c(1,1,0). The top efgh is the same square, rotated 45^o around the point (0.5,0.5,0) and translated up the z-axis by (0,0,1).
   1. [1 mark] What are the coordinates of the eight vertices a,b,c,d,e,f,g and h?
   2. [3 marks] What are the outward-facing unit normal vectors for each of the ten faces of the object?

4. Colour [6 marks] Write a program that displays the CIE chromaticity diagram, as an image made up of pixels. The diagram should show the following:
   1. [3 marks] The triangular gamut of colours that can be displayed by a monitor whose RGB phosphors have the following chromaticities:

      Phosphor	x	y
      R	0.628	0.330
      G	0.258	0.590
      B	0.1507	0.060

      Colours outside this gamut should appear black.
   2. [3 marks] The curve of pure spectral colours, with a white pixel at each 5-nm wavelength increment.
   3. [Bonus 1 mark] The blackbody curve for (T=1000K to 10000K, in 500K increments):
      • P(λ) = (2πc²h/λ⁵)/(e^hc/kT-1)

      You will find skeleton code and data files on the following page:

      • www.dgp.utoronto.ca/~ah/csc418/fall_2001/a2/q4
      That page also describes the program's inputs. Submit your code electronically, using one of the following commands:
      • submit -N a2q4 csc418h cie.c
      • submit -N a2q4 csc2504h cie.c
      Submit a printed copy as well.

5. Ray Tracing (6 marks)
   In ray tracing, primary rays are shot from the eye, through pixels, and tested for intersection with objects. Write a program to intersect a primary ray with a cylinder whose axis is parallel to the y-axis. The parametric form of a ray starting at P, in direction V is
   • (x,y,z) = P + t V , t>0.
   Assume all objects are in VCS. You program must use the I/O provided in the skeleton code:
   • www.dgp.utoronto.ca/~ah/csc418/fall_2001/a2/q5
   That page also describes the program's inputs. Submit your code electronically, using one of the following commands:
   • submit -N a2q5 csc418h intersect.c
   • submit -N a2q5 csc2504h intersect.c
   Submit a printed copy as well.

6. Hierarchies [4 marks] Write a program to draw and control this robot:
   Skeleton code is provided at
   • www.dgp.utoronto.ca/~ah/csc418/fall_2001/a2/q6
   Submit your code electronically, using one of the following commands:
   • submit -N a2q6 csc418h robot.cpp
   • submit -N a2q6 csc2504h robot.cpp
   Submit a printed copy as well.