Computer Graphics CSC418/2504F, Fall 1999: Assignment 2

Due date: in class, 1 Nov. 1999.

Write your name, ID and cdf login on your written and electronic submissions.

Solutions

1. Colour:

   A light source has the following spectral distribution:

   

   in other words,

   P(L)=0.5,	400nm < L < 500nm
   P(L)=1.0,	500nm < L < 600nm
   P(L)=0.5,	600nm < L < 700nm
   P(L)=0,	elsewhere

   1. [4 marks] Compute X,Y,Z, the CIE colour components. The functions x(L), y(L) and z(L) are available online in tabular form at:
      http://www.dgp.toronto.edu/~ah/csc418/fall_1999/coursework/ciexyz.dat
      Each line contains a wavelength L, and three values for x(L), y(L) and z(L). Use the code in
      http://www.dgp.toronto.edu/~ah/csc418/fall_1999/coursework/a2q1.c
      to read the data.
   2. [1 mark] Compute x,y,z, the chromaticity coordinates for the source.
   3. [2 marks] Plot the colour on the CIE diagram (available online). Assuming white has (x,y)=(1/3,1/3), what is its dominant wavelength?
   4. [2 marks] Assuming the phosphors in a monitor have the following chromaticity coordinates:

      	x	y
      R	0.64	0.33
      G	0.30	0.60
      B	0.15	0.06

      what are the R,G,B components of the source?
   5. [1 mark] What are the Y,I,Q components?
   
   
   
2. Coordinate Transformations:

   Consider the point P and the three coordinate frames O, A, and B below:

   

   Notation: M_RS is a 3x3 homogeneous matrix, and P_R=(x,y,1) is a homogeneous point in coordinate frame S, such that

   P_S = M_RS P_R
   1. [2 marks] Find the coordinates of P in the three frames.
   2. [3 marks] Find M_OA, M_OB, M_AB and M_BA.
   3. [2 marks] Succintly express M_AB in terms of M_OA and M_OB.
      Succintly express M_AB in terms of M_BA.
   
   
   
3. [3 marks] A computer can execute 10 MFLOPS. How many triangles per second can it transform, without hardware assistance? Assume:
   • each vertex must be transformed to viewing coordinates, projected, and then transformed to the viewport.
   • geometric transformations dominate the cost, and scan conversion is handled by hardware.
   • lighting and clipping calculations are free.
   • triangles are specified in strips (see GL_TRIANGLE_STRIP).
   
   
   
4. Projection

   Given the following camera parameters:

   • P_eye = (1.5, 1.7, 2.0)
   • P_ref = (0.5, 0.5, 0.5)
   • V_up = (0.0, 0.0, 1.0)
   1. [3 marks] Construct the viewing matrix M_world-cam.
   2. [2 marks] Assume the following projection matrix,

      [1  0  0  0]
      [0  1  0  0]
      [0  0  1  0]
      [0  0 -1  0]
      

      and project onto the image plane the eight vertices of an axis-aligned unit cube, with corners at (0,0,0) and (1,1,1).
   3. [2 marks] After projection, lines which were parallel will appear to converge to a "vanishing point". The cube has three sets of parallel edges. Determine the coordinates of the vanishing points, on the image plane.
   
   
   
5. Object Hierarchies

   [4 marks] Write an openGL program to draw a hierarchical object with at least 5 degrees of freedom. Sample code is available at:
   http://www.dgp.toronto.edu/~ah/csc418/fall_1999/coursework/a2q5
   [Bonus 1 mark]: Add degrees of freedom for the camera orientation.
   
   

6. Ray Tracing

   [6 marks] The parametric equation for a ray starting at P and in direction V is:
   (x,y,z) = P + t V, 0 < t < infinity
   Assuming:

   • All objects are in viewing (camera) coordinates.
   • The image plane is on the front clipping plane, and is bounded by a rectangle with corners at (-1,-1,-1) and (1,1,-1).
   • The clipped image plane is displayed on an n x n raster of pixels,
   write a program to intersect an eye ray with a polygon, as follows:
   1. determine P and V for a ray from the eye through pixel (x,y).
   2. given a polygon whose vertices are p₀, p₁, ... p_m, determine the parameters A,B,C,D of the equation Ax+By+Cz+D=0 for the plane containing it.
   3. determine the intersection T between the ray in (a) and the plane in (b).
   4. determine if the intersection T is inside the polygon:
      • project T and all p_i to one of the 3 (xy, yz, or xz) coordinate planes (which one is best?).
      • perform a 2D point-in-polygon test.
   Use the sample code in
   http://www.dgp.toronto.edu/~ah/csc418/fall_1999/coursework/a2q6/a2q6.c