g) Consider 2 raster systems with the resolutions of 640 x 480 and
1280 x 1024.
a) How many pixels could be accessed per second in each of these systems by a display controller that refreshes the screen at a rate of 60 frames per second? (3 marks).
b) What is the access time per pixel in each system?(2 marks)
a) Using homogeneous coordinates,
i) Describe the matrix formula for 2D translation, scaling and rotation. 2mks)
ii) For the Triangle A(1,0),B(0,1),C(1,1), find its transformation by rotating 450 about the origin then translating one unit in X and Y direction.
a) The number of pixels that could be accessed per second can be calculated by multiplying the resolution (number of pixels) by the refresh rate (frames per second).
For the system with a resolution of 640 x 480:
Number of pixels = 640 x 480 = 307,200 pixels
Pixels accessed per second = 307,200 x 60 = 18,432,000 pixels
For the system with a resolution of 1280 x 1024:
Number of pixels = 1280 x 1024 = 1,310,720 pixels
Pixels accessed per second = 1,310,720 x 60 = 78,643,200 pixels
b) The access time per pixel can be calculated by dividing the time (1 second) by the number of pixels accessed per second.
For the system with a resolution of 640 x 480:
Access time per pixel = 1 second / 18,432,000 pixels = 5.42 x 10^-8 seconds
For the system with a resolution of 1280 x 1024:
Access time per pixel = 1 second / 78,643,200 pixels = 1.27 x 10^-8 seconds
a) i) The matrix formula for 2D translation, scaling, and rotation are as follows:
Translation:
[ 1 0 tx ] [ x ] [ x + tx ]
[ 0 1 ty ] [ y ] = [ y + ty ]
[ 0 0 1 ] [ 1 ] [ 1 ]
Scaling:
[ sx 0 0 ] [ x ] [ sx * x ]
[ 0 sy 0 ] [ y ] = [ sy * y ]
[ 0 0 1 ] [ 1 ] [ 1 ]
Rotation:
[ cos(theta) -sin(theta) 0 ] [ x ] [ x * cos(theta) - y * sin(theta) ]
[ sin(theta) cos(theta) 0 ] [ y ] = [ x * sin(theta) + y * cos(theta) ]
[ 0 0 1 ] [ 1 ] [ 1 ]
ii) For the Triangle A(1,0), B(0,1), C(1,1), the transformation by rotating 450 about the origin then translating one unit in X and Y direction can be calculated as follows:
Rotation by 450:
[ cos(45) -sin(45) 0 ] [ 1 ] [ 1 * cos(45) - 0 * sin(45) ] [ sqrt(2)/2 ]
[ sin(45) cos(45) 0 ] [ 0 ] = [ 1 * sin(45) + 0 * cos(45) ] = [ sqrt(2)/2 ]
[ 0 0 1 ] [ 1 ] [ 1 ] [ 1 ]
Translation by one unit in X and Y direction:
[ 1 0 1 ] [ sqrt(2)/2 ] [ sqrt(2)/2 + 1 ]
[ 0 1 1 ] [ sqrt(2)/2 ] = [ sqrt(2)/2 + 1 ]
[ 0 0 1 ] [ 1 ] [ 1 ]
a) To calculate the number of pixels that can be accessed per second in each system, we need to multiply the resolution by the refresh rate.
For the system with a resolution of 640 x 480:
Number of pixels = 640 x 480 = 307,200 pixels
Pixels accessed per second = Number of pixels x Refresh rate = 307,200 x 60 = 18,432,000 pixels/second
For the system with a resolution of 1280 x 1024:
Number of pixels = 1280 x 1024 = 1,310,720 pixels
Pixels accessed per second = Number of pixels x Refresh rate = 1,310,720 x 60 = 78,643,200 pixels/second
b) To calculate the access time per pixel in each system, we need to divide the time taken to refresh the screen by the number of pixels.
For the system with a resolution of 640 x 480:
Access time per pixel = Time to refresh the screen / Number of pixels
Since the refresh rate is 60 frames per second, the time to refresh the screen is 1/60 seconds.
Access time per pixel = (1/60) / 307,200 = 1.63 x 10^-7 seconds/pixel
For the system with a resolution of 1280 x 1024:
Access time per pixel = Time to refresh the screen / Number of pixels
Since the refresh rate is 60 frames per second, the time to refresh the screen is 1/60 seconds.
Access time per pixel = (1/60) / 1,310,720 = 1.28 x 10^-8 seconds/pixel
a) Homogeneous coordinates are used to represent geometric transformations in computer graphics. The matrix formula for 2D translation, scaling, and rotation can be described as follows:
i) Translation:
To translate a point (x, y) by (tx, ty), the matrix formula is:
[x' ] [x] [tx]
[y' ] = [y] + [ty]
[1 ] [1] [1 ]
Scaling:
To scale a point (x, y) by (sx, sy), the matrix formula is:
[x' ] [x] [0 ]
[y' ] = [y] x [0 ]
[1 ] [1] [1 ]
Rotation:
To rotate a point (x, y) by an angle θ, the matrix formula is:
[x' ] [cos(θ) -sin(θ) ] [x]
[y' ] = [sin(θ) cos(θ)] x [y]
[1 ] [0 ] [1 ]
ii) To find the transformation of the triangle ABC (A(1,0), B(0,1), C(1,1)) by rotating 450 degrees about the origin and then translating one unit in the X and Y direction:
Step 1: Rotation by 450 degrees
Using the rotation matrix formula, the transformation of each point is as follows:
A' = (1,0) x [cos(450) -sin(450)] = (1,0)
B' = (0,1) x [cos(450) -sin(450)] = (-0.71, -0.71)
C' = (1,1) x [cos(450) -sin(450)] = (0.71, -0.71)
Step 2: Translation by one unit in the X and Y direction
Adding the translation values to each point, the final transformation is:
A" = (1,0) + (1,1) = (2, 1)
B" = (-0.71, -0.71) + (1,1) = (0.29, 0.29)
C" = (0.71, -0.71) + (1,1) = (1.71, 0.29)
Thus, the transformed triangle ABC after rotating 450 degrees about the origin and then translating one unit in the X and Y direction is A"(2,1), B"(0.29,0.29), C"(1.71,0.29).