g) Consider 2 raster systems with the resolutions of 640 x 480 and

Question

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.

Answer 1

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 ]