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CS-112: Introduction to Computer Graphics
Midterm - Fall 2023
Total Time: 70 min Total Points: 70
Name:
Pledge: I neither received nor gave any help from or to anyone in this exam.
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1) [2+3+3+2=10] Consider a 2D rectangle ABCD where A=(0,0), B= (2,0), C=(2,
1) and D=(0,1). We want to apply a 2D transformation to this rectangle which
makes it a parallelepiped AEFG where E = (0, 2), F= (-1, 2) and G=(-1,0).
a) What kind of transformation is this?
i) Scale
ii) Rotate
iii) Shear
iv) Translate
b) The 3x3 matrix M achieving this transformation is given by
i) [1 1 0 ; 0 1 0 ; 0 0 1]
ii) [0 -1 0 ; 1 0 0 ; 0 0 1]
iii) [0 1 0 ; -1 0 0 ; 0 0 1]
iv) [1 0 0 ; 0 1 0 ; 1 0 1]
c) What additional transformation N we would need to apply to AEFG to get the
parallelepiped A’E’F’G’, where A’= (0, 0), E’= (1, 2), F’= (0, 2),
and G’= (-1, 0)?
i) Rotation by 45°
ii) Scale by (2, 1)
iii) Translate by (2 ,1)
iv) Shear by (½,0)
d) What is the final concatenated matrix in terms of M and N that will
transform ABCD to A’E’F’G’?
i) MN
ii) NM
iii) M-1N
iv) N-1M
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Answer: (a) – iii, (b) – iii, (c) – ii, (d) – i, (e) – (ii)
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2) [2+2+3+2+1=10] Consider the following view setup. The eye is located at
(0,0,0) and the equation of the image plane is given by 3x+3y+3z = 18.
a) What is the distance of the near plane n?
i) 12
ii) √12
iii) √3
iv) 3
b) What is the normal to the image plane N?
i) (1, 1, 0)
ii) (1, 1, 1)
iii) (0, 1, 0)
iv) (1, 0, 1)
c) If the vertical and horizontal field-of-view are 60 and 90 degrees respectively,
what is the aspect ratio (ratio of width is to height) of the rendering window?
i) √3:1
ii) √2:1
iii) 3√3:1
iv) 3:1
d) The perspective transformation consists of which of the following.
i) Translation
ii) Scaling
iii) Rotation
iv) Shear
v) Perspective Projection
e) The perspective projection transforms the view frustum into which of the
following?
i) Truncated Cone
ii) Cuboid of size 2
iii) Cuboid of size 1
iv) Rectangular Parallelepipped
3) [3+2+2=7] Consider the default OpenGL view with the near plane (or image
plane) at a distance 4. The gaze direction is (2,6) and the size of the window in X
and Y direction in which it is centered are 16 and 12 respectively.
a) The l, r, t, and b of the view frustum is given by
i) -4, 8, 11, 1
ii) -5, 5, -6, 6
iii) 0, 12, 0, 10
iv) -6, 10, 12, 0
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v) -2, 6, -4, 6
b) The transformation required to make the gaze direction coincident with the
default OpenGL view direction is:
i) Scaling along X and Y axes
ii) Translate in X and Y direction
iii) Z-Shear
iv) X and Y Shear
v) Rotation about the Z axis
c) Once the view and gaze directions coincide, the transformation required
normalize the X and Y coordinates are given by:
i) Scaling along X and Y axes
ii) Translate in X and Y direction
iii) Z-Shear
iv) X and Y Shear
v) Rotation about the Z axis
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4) [2+1+2+3+4+3+1=16] Consider the triangle ABC being clipped at E & F.
a) If E is the middle point of BC, its coordinate is:
i) 200,100
ii) 225, 150
iii) 225, 200
iv) 200, 150
b) Therefore, the x-max for the window is:
i) 225
ii) 200
iii) 250
c) And the coordinates of F are:
i) 250,166.67
ii) 225,133.33
iii) 200, 166.67
iv) 125,133.33
d) The interpolation coefficients of A & C in order to compute F is:
i) (3/4, 1/4)
ii) (2/3, 1/3)
iii) (1/3, 2/3)
iv) (1/4, 3/4)
e) If the grayscale colors at B & C are 120 & 40 respectively, the
interpolated grayscale value at E is:
i) 50
ii) 70
iii) 80
iv) 100
Question 8 continues to the next page
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f) If the Z-value at B and C are 12 & 36 respectively, the interpolated depth
at E is:
i) 16
ii) 18
iii) 20
iv) 22
g) When rendering E, you find that the depth buffer at that point is set at
40. Would E pass the depth buffer test?
i. Yes
ii. No
5) [4+2=6] Consider the following [2 0 0 10; 0 4 0 16; 0 0 3 -15; 0 0 0 1]
a. This matrix achieves the following in the global coordinate system:
i. S of (2, 4, 3) followed by T of (10,16,-15)
ii. T of (5, 4, -5) followed by S of (2, 4, 3)
iii. T of (10,16,-15) followed by S of (2, 4, 3)
iv. S of (2, 4, 3) f followed by T(5, 4, -5)
b. Consider a translation R of the local coordinate system following these
transformations. The resultant transformation from a will be:
i. Pre-multiplied by R
ii. Post-multiplied by R
6) [2+2+2+2=8] Consider an octree data structure for spatial subdivision being
used for view frustum culling.
a) Each node of an octree has:
i) 2 children
ii) 4 children
iii) 8 children
iv) 16 children
b) If the bounding volume of the root of the octree is known, one can
calculate the bounding volume of:
i) Only the level 1 nodes
ii) All the other nodes
iii) Only the leaf nodes
c) View frustum culling is effectively finding:
i) A cut of the octree
ii) Height of the octree
iii) Ordered traversal of the octree
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d) Is the octree an optimal spatial subdivision method?
i) Yes
ii) No
