Family Name:________________________________

Given Name:_________________________________

Student Number:_____________________________


There are 7 questions and 57 marks in total.


Answer all questions.


This exam paper should have 7 pages,
including this cover page.

1:           /10

2:           /10

3:           /8

4:           /6

5:           /6

6:           /7

7:           /10

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Total:       /57

1. Curves - 10 marks
   1. 3 marks: Consider a cubic curve defined by the following geometric constraints:

      passes through P1 at t=0 passes through P2 at t=2/3 passes through P4 at t=1 has tangent T2 at t=2/3

      Write an expression for the basis matrix of the curve.
      
      
      
      
      
      
      
      
      
      
      

   2. 2 marks: What is the convex-hull property of Bezier curves?
      
      
      
      
      
      
      
      
   3. 1 mark: When rendering a Bezier curve, how can you use this property to speed up clipping?
      
      
      
      
      
      
   4. 3 marks: What is the geometric and parametric continuity of these two curves:

      C1(t) =	[t3+t2+t, -t2+2t]
      C2(t) =	[3t3+3t2+6t+3, t2-1]

      
      
      
      
      
      
      
      
      
      
      

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2. Graphics pipeline - 10 marks
   Consider a graphics pipeline with the following view (world-to-camera) matrix and projection matrix:

   

   Final image size: 512x512

   Consider the line segment P_a P_b, with P_a=(1,1,-2) and P_b=(2,-1,-1) in WCS.

   1. 2 marks: Write a set of parametric equations for the line segment, in WCS.
      
      
      
      
      
      
      
      
   2. 2 marks: Compute the endpoints of the line segment in VCS.
      
      
      
      
      
      
      
      
   3. 2 marks: Compute the endpoints of the line segment in NDCS.
      
      
      
      
      
      
      
      
   4. 2 marks: Calculate the midpoint of P_aP_b in VCS, and transform this point to device coords.
      
      
      
      
      
      
      
      
   5. 2 marks: Will this be the midpoint of the line on the device? If yes, why? If not, which step in the graphics pipeline makes a difference?
      
      
      
      
      
      
      
      

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3. Colour - 8 marks
   1. 2 marks: Suppose three coloured dots are printed on a sheet of paper, one cyan, one magenta, and one yellow. If the paper is lit with a blue light, what will be seen, qualitatively speaking?
      
      
      
      
      
      
   2. 3 marks: Suppose a monitor has the following RGB-to-XYZ transformation matrix:
      
      What are the XYZ coordinates of the three RGB phosphors?
      
      
      
      
      
      
      
      
   3. 2 marks: Compute the monitor's colour gamut and sketch it on the CIE diagram.
      
      
      
      
      
      
   4. 1 marks: On this monitor, what is the dominant wavelength of a colour with RGB=(1 0 1)?
      
      
      
      
      
      
      
      

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4. Scan conversion - 6 marks:
   1. 3 marks: Which pixels are set when drawing a line from (7,6) to (2,3) using the midpoint algorithm?
      
      
      
      
      
      
      
      
   2. 3 marks: Build the edge table for the following polygon:

      
      
      
      
      
      
      
      

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5. Global Illumination - 6 marks
   Consider the following three illumination techniques:
   
   

   1:	Phong local illumination
   2:	Non-distribution Ray tracing
   3:	Radiosity

   
   
   and the following five lighting effects:
   
   

   A:	view-independent illumination
   B:	specular highlights
   C:	soft shadows
   D:	colour bleeding
   E:	refraction

   
   
   Place check-marks in the following table if a technique can produce the corresponding effect.
   
   

   Technique	A	B	C	D	E
   Phong local illumination
   Non-distribution Ray tracing
   Radiosity

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6. Geometric transformations - 7 marks
   Using the following openGL calls:

        glScalef (float x, float y, float z)
        glTranslatef (float x, float y, float z)
        glRotatef (float angle, float x, float y, float z)
        glPushMatrix ()
        glPopMatrix ()
   

   and the function drawSquare(), which draws a square with side length 1, centred at the origin:
   
   
   1. 3 marks: write the following function:

           drawBox(float x, float y, float angle, float xsize, float ysize);
       

      which draws a xsize × ysize rectangle, with one corner at (x,y), and rotated by angle degrees:
      
      
      
      
      
      
      
      
   2. 4 marks: using the drawBox() function you wrote, write code to draw the following object:
      
      
   
   
   
   
   
   
   
   
   

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7. Shading - 10 marks
   Consider the local-illumination model:
   
   I = I_a k_a + I_i k_diff N·L + I_i k_spec (R·V)ⁿ
   for the following polygon (represented as a line). Assume an orthographic projection.
   
   
   
   1. 2 marks: On the diagram above, sketch the vectors N, L, and V at points A, B and C on the polygon.
      
      
      
      
   2. 4 marks: At points A, B and C, calculate the ambient, diffuse, and specular contributions of the surface's colour. Assume the following parameters:
      I_i = 1.0
      I_a = 0.5
      k_a = 0.2
      k_spec = 0.4
      k_diff = 0.5
      n = 4
      
      
      
      
      
      
      
      
      
      
      
   3. 2 marks: What are T-vertices?
      
      
      
      
      
      
   4. 2 marks: Why are T-vertices a problem for shading algorithms?