1. What pixels will be marked
in drawing a circle center (100,90) and radius 10? using Bresenham's algorithm
Ans:
Take x=0,y=r=10,d=3-2r=3-2*10=-17
Decision variable
(d)
|
x
|
y
|
Point with Center
(0,0)
|
Point with Center
(100,90)
|
0
|
10
|
(0,10)
|
(100,100)
|
|
-17
|
1
|
10
|
(1,10)
|
(101,100)
|
-11
|
2
|
10
|
(2,10)
|
(102,100)
|
-1
|
3
|
10
|
(3,10)
|
(103,100)
|
13
|
4
|
9
|
(4,9)
|
(104,99)
|
-5
|
5
|
9
|
(5,9)
|
(105,99)
|
17
|
6
|
8
|
(6,8)
|
(106,98)
|
11
|
7
|
7
|
(7,7)
|
2. Which points will be scan
converted by direct method for a line whose end points are (2,3) and (8,5)
Ans:
dx=X2-X1=8-2=6
, dy=Y2-Y1=5-3=2 m=dy/dx=2/6=1/3, c=Y1-mX1=3-2/3=7/3
(since y=mx+c)
Since |m|=1/3<1, so we sample
dx=1 and calculate y correspondingly
The various point plotted are given
below:
X
|
Y=mX+c
|
Point Plotted
|
2
|
3.00
|
(2,3)
|
3
|
3.33
|
(3,3)
|
4
|
3.66
|
(4,4)
|
5
|
4.00
|
(5,4)
|
6
|
4.33
|
(6,4)
|
7
|
4.66
|
(7,5)
|
8
|
5.00
|
(8,5)
|
3. Calculate the points using
DDA Algorithm that would be plotted for a line whose end points are (12,10) and
(20,20)
Ans :
Slope
of the line m=dy/dx=(20-10)/(20-12)=5/4=1.25
Since
|m|>1 and we start from left endpoint, we sample at dy=1 and calculate
corresponding x using formula Xk+1=Xk+1/m
The following table gives the
points to be plotted on the screen for the line. The first point plotted is
(12,10).
Xk+1=Xk+1/m
|
Y
|
Point Plotted (X,Y)
|
12.8
|
11
|
(13,11)
|
13.6
|
12
|
(14,12)
|
14.4
|
13
|
(14,13)
|
15.2
|
14
|
(15,14)
|
16.0
|
15
|
(16,15)
|
16.8
|
16
|
(17,16)
|
17.6
|
17
|
(18,17)
|
18.4
|
18
|
(18,18)
|
19.2
|
19
|
(19,19)
|
20.0
|
20
|
(20,20)
|
4. Indicate which raster
locations would be chosen by Bresenham’s Line algorithm, when scan converting a
line from pixel coordinate (1,1) to pixel coordinate (8,5)
Ans:
The
two end points are (1,1) and (8,5)
So dx=8-1=7 and dy=5-1=4 and m=dy/dx=4/7<1
The
initial decision parameter is P0=2dy-dx=8-7=1 and 2dy=8, 2dy-2dx=-6
We
plot the initial point (1,1) and determine successive pixel position alone the
line as follows:
K
|
Decision Parameter
|
Point to be
plotted Xk+1,Yk+1
|
0
|
1
|
(2,2)
|
1
|
-5
|
(3,2)
|
2
|
3
|
(4,3)
|
3
|
-3
|
(5,3)
|
4
|
5
|
(6,4)
|
5
|
-1
|
(7,4)
|
6
|
7
|
(8,5)
|
So the above points are scan
convert a line between (1,1) and (8,5)
5. Using Mid point Circle
algorithm , find out the screen coordinates of the circumference of a circle
whose center is (0,0) and radius is 10 units.
Ans:
Radius
of the circle =10, Initial decision parameter P=1-r=1-10=-9
As
the circle is centered at (0,0), the initial point plotted is (0,r) =(0,10)
Successive
screen points are circulated using midpoint method as follows:
X
|
Y
|
Decision parameter P
|
Point to be plotted
|
0
|
10
|
-9
|
(1,10)
|
1
|
10
|
-6
|
(2,10)
|
2
|
10
|
-1
|
(3,10)
|
3
|
10
|
6
|
(4,9)
|
4
|
9
|
-3
|
(5,9)
|
5
|
9
|
8
|
(6,8)
|
6
|
8
|
5
|
(7,7)
|
As we reach
the condition x=y, we generate the rest of pixel positions using 8 way
symmetric of circle.
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