This slide contain description about the line, circle and ellipse drawing algorithm in computer graphics. It also deals with the filled area primitive.
Hierarchy of management that covers different levels of management
Output primitives in Computer Graphics
1. Output Primitives
Points and Lines
Line Drawing Algorithms
DDA Algorithm
Bresenham’s Line Algorithm
Midpoint Circle Algorithm
Midpoint Ellipse Algorithm
Filled Area Primitives
2. Points and Lines
• Point is the fundamental element of picture
representation.
• It is the position in the plan defined as either
pair or triplets of number depending upon the
dimension.
• Two points represent line or edge and 3 or
more points a polygon.
• Curved lines are represented by the short
straight lines.
3. Line Drawing Algorithms
• Slope-Intercept Equation
• Slope
• Intercept
• Interval Calculation
bxmy .
12
12
xx
yy
m
11 .xmyb
xmy . m
y
x
7. DDA Algorithm
Digital Differential Analyzer
– Sample the line at unit intervals in one coordinate
– Determine the corresponding integer values
nearest the line path in another co-ordinate
8. DDA Algorithm (left to right)
• Slope
• For |m|<1 (|Δy|< |Δx|)
– Sample line at unit interval in x co-ordinate
• For |m|>1 (|Δy|> |Δx|)
– Sample line at unit interval in y co-ordinate
x
y
xx
yy
m
kk
kk
1
1
myy kk 1
m
xx kk
1
1
11 kk xxx
11 kk yyy
9. DDA Algorithm (right to left)
• Slope
• For |m|<1 (|Δy|< |Δx|)
– Sample line at unit interval in x co-ordinate
• For |m|>1 (|Δy|> |Δx|)
– Sample line at unit interval in y co-ordinate
myy kk 1
m
xx kk
1
1
11 kk xxx
11 kk yyy
x
y
xx
yy
m
kk
kk
1
1
10. DDA Algorithm
1. Input the two line endpoints and store the left endpoint in (x0,y0)
2. Plot first point (x0,y0)
3. Calculate constants Δx, Δy
4. If |Δx| > |Δy| steps = |Δx| else steps = |Δy|
5. Calculate XInc = |Δx| / steps and YInc = |Δy| / steps
6. At each xk along the line, starting at k=0, Plot the next pixel at (xk + XInc, yk
+ YInc)
7. Repeat step 6 steps times
11. Pseudo Code
Void lineDDA(int xa, int ya, int xb, int yb)
{
int dx = xb – xa, dy = yb – ya, steps, k;
float xIncrement, yIncrement, x = xa, y = ya;
if( abs (dx) > abs (dy) ) steps = abs (dx);
else steps = abs (dy);
xIncrement = dx / (float) steps;
yIncrement = dy / (float) steps;
setPixel (ROUND (x), ROUND (y));
for (k=0; k<steps; k++){
x += xIncrement;
y += yIncrement;
setPixel (ROUND(x), ROUND(y));
}
}
12.
13.
14. • Use DDA algorithm for rasterizing line (0,0) to
(6,6).
• Use DDA algorithm for rasterizing line (0,0) to
(4,6).
15.
16.
17.
18. Bresenham’s Line Algorithm
• Uses only incremental integer
calculations
• Which pixel to draw ?
– (11,11) or (11,12) ?
– (51,50) or (51,49) ?
– Answered by Bresenham
19. • For |m|<1
– Start from left end point (x0,y0) step to each
successive column (x samples) and plot the pixel
whose scan line y value is closest to the line path.
– After (xk,yk) the choice could be (xk+1,yk) or
(xk+1,yk+1)
21. Defining decision parameter
[1]
Sign of pk is same as that of d1-d2 for Δx>0 (left to right sampling)
For Recursive calculation, initially
cyxxy kk .2.2
Constant=2Δy + Δx(2b-1) Which is
independent of pixel position
cyxxyp kkk 111 .2.2
)(2)(2 111 kkkkkk yyxxxypp
c eliminated here
)(22 11 kkkk yyxypp
because xk+1 = xk + 1
yk+1-yk = 0 if pk < 0
yk+1-yk = 1 if pk ≥ 0
xyp 20
Substitute b = y0 – m.x0
and m = Δy/Δx in [1]
)( 21 ddxpk
22. Algorithm Steps (|m|<1)
1. Input the two line endpoints and store the left endpoint in (x0,y0)
2. Plot first point (x0,y0)
3. Calculate constants Δx, Δy, 2Δy and 2 Δy- 2Δx, and obtain p0 = 2Δy – Δx
4. At each xk along the line, starting at k=0, perform the following test:
If pk<0, the next point plot is (xk+1,yk) and
Pk+1 = pk + 2Δy
Otherwise, the next point to plot is (xk + 1, yk+1) and
Pk+1 = pk + 2Δy - 2Δx
5. Repeat step 4 Δx times
23. What’s the advantage?
• Answer: involves only the calculation of constants Δx, Δy,
2Δy and 2Δy- 2Δx once and integer addition and
subtraction in each steps
25. • Use Bresenham’s algorithm for rasterizing the
line (5,5) to (13,9)
26.
27. • Digitize a line with endpoints (15,18) and
(10,15) using BLA.
28.
29. • Digitize a line with endpoints (15,15) and
(10,18) using BLA.
30.
31. Algorithm Steps (|m|>1)
1. Input the two line endpoints and store the left endpoint in (x0,y0)
2. Plot first point (x0,y0)
3. Calculate constants Δx, Δy, 2Δx and 2 Δx- 2Δy, and obtain p0 = 2Δx – Δy
4. At each xk along the line, starting at k=0, perform the following test:
If pk<0, the next point plot is (xk, yk+1) and
Pk+1 = pk + 2Δx
Otherwise, the next point to plot is (xk + 1, yk+1) and
Pk+1 = pk + 2Δx - 2Δy
5. Repeat step 4 Δx times
32. • Use BLA algorithm for rasterizing line (0,0) to
(4,6).
34. Problem (in above method)
• Computational complexity
• Spacing:
– Non-uniform spacing of
plotted pixels
35. Adjustments (To fix problems)
• Spacing problem (2 ways):
– Interchange the role of x and y whenever the absolute value of the slope of
the circle tangent > 1
– Use polar co-ordinate:
• Equally spaced points are plotted along the circumference with fixed angular
step size.
• step size chosen for θ depends on the application and display device.
• Computation Problem:
– Use symmetry of circle; i.e calculate for one octant and use symmetry for
others.
sin
cos
ryy
rxx
c
c
37. Bresenham’s Algorithm Could Be Adapted ??
• Yes
• How ?
– Setting decision parameter for finding the closest
pixel to the circumference
• And what to do For Non-linear equation of
circle ?
– Comparison of squares of the pixel separation
distance avoids square root calculations
38. Midpoint Circle Algorithm
• Circle function defined as:
• Any point (x,y) satisfies following conditions
boundarycircletheoutsideisyxif
boundarycircletheonisyxif
boundarycircletheinsideisyxif
yxfcircle
),(,0
),(,0
),(,0
),(
222
),( ryxyxfcircle
40. yk+1 = yk if pk<0
yk+1 = yk-1 otherwise
• Thus
Pk+1 = Pk + 2xk+1+1 if pk<0
Pk+1 = Pk + 2xk+1+1-2yk+1 otherwise
• Also incremental evaluation of 2xk+1 and 2yk+1
2xk+1 = 2xk + 2
2yk+1 = 2yk – 2 if pk >0
• At start position (x0,y0) = (0,r)
2x0 = 0 and 2y0 = 2r
41. • Initial decision parameter
• For r specified as an integer, round p0 to
P0 = 1-r
(because all increments are integers)
r
rr
rfp circle
4
5
)
2
1
(1
)
2
1
,1(
22
0
42. Algorithm
1. Input radius r and circle center (xc, yc) and obtain the first point on the circumference of
a circle centered on the origin as
(x0,y0) = (0,r)
2. Calculate the initial value of the decision parameter as
P0 = 5/4 – r
3. At each xk position, starting at k = 0, perform the following test:
If pk < 0, the next point along the circle centered on (0,0) is (xk+1,yk) and
Pk+1 = pk + 2xk+1 + 1
Otherwise, the next point along the circle is (xk+1,yK-1) and
Pk+1 = pk + 2xk+1 + 1 -2yk+1
Where 2xk+1 = 2xk + 2 and 2yk+1 = 2yk-2
4. Determine the symmetry points in the other seven octants.
5. Move each calculated pixel position (x,y) onto the circular path
centered on (xc,yc) and plot the co-ordinate values:
x = x + xc, y = y+yc
6. Repeat steps 3 through 5 until x ≥ y
43. • Given radius =10 use mid-point circle drawing
algorithm to determine the pixel in the first
quadrant.
Solution:
P0 = 1-r = -9
(x0 , y0)=(0,10)
2x0 =0
2y0 =20
44.
45.
46. • Digitize the circle (x-2)^2+(y-3)^2=25 in first
quadrant.
Solution:
P0=(1-r)= -4
(x0 , y0)=(0,5)
2x0 =0
2y0 =10
47.
48.
49.
50. Ellipse Generating Algorithms
• Equation of ellipse:
• F1(x1,y1), F2(x2,y2)
• General Equation
• Simplified Form
• In polar co-ordinate
constantyyxxyyxx 2
2
2
2
2
1
2
1 )()()()(
constantdd 21
022
FEyDxCxyByAx
1
22
y
c
x
c
r
yy
r
xx
sin
cos
yc
xc
ryy
rxx
52. • From ellipse tangent slope:
• At boundary region (slope = -1):
• Start from (0,ry), take x samples to boundary
between 1 and 2
• Switch to sample y from boundary between 1
and 2
(i.e whenever )
yr
xr
dx
dy
x
y
2
2
2
2
yrxr xy
22
22
yrxr xy
22
22
Slope=-1
53. • In the region 1
• For next sample
• Thus increment
222222
)
2
1
()1(
)
2
1
,1(1
yxkxky
kkellipsek
rryrxr
yxfp
22
1
222
1
222
1
222
111
2
1
2
1
)1(211
)
2
1
(1)1(
)
2
1
,1(1
kkxykykk
yxkxky
kkellipsek
yyrrxrpp
rryrxr
yxfp
01,22
01,2
1
22
1
2
2
1
2
kkxyky
kyky
pifyrrxr
pifrxr
increment
•For increment calculation;
Initially:
•Incrementally:
Update x by adding
2ry
2 to first equation
and update y by
subtracting 2rx
2 to
second equation
yxx
y
rryr
xr
22
2
22
02
55. • In the region 2
• For next sample
• Initially
222222
)1()
2
1
(
)1,
2
1
(2
yxkxky
kkellipsek
rryrxr
yxfp
22
1
222
1
2222
2
1
2
111
2
1
2
1
)1(222
11
2
1
)1,
2
1
(2
kkyxkxkk
yxkxky
kkellipsek
xxrryrpp
rryrxr
yxfp
222
0
2
2
0
2
0
000
)1(
2
1
2
1,
2
1
2
yxxy
ellipse
rryrxrp
yxfp
For simplification calculation of
p20 can be done by selecting
pixel positions in counter
clockwise order starting at (rx,0)
and unit samples to positive y
direction until the boundary
between two regions
56. Algorithm
1. Input rx,ry, and the ellipse center(xc,yc) and obtain the first point on an ellipse centered on the origin as
(x0,y0) = (0,ry)
2. Calculate the initial value of the decision parameter in region 1 as
3. At each xk position in region 1, starting at k = 0, perform the following test: If p1k < 0, the next point along the
ellipse centered on (0,0) is (xk+1,yk) and
Otherwise, the next point along the ellipse is (xk+1,yk-1) and
With
and continue until
222
0
4
1
1 xyxy rrrrp
2
1
2
1 211 ykykk rxrpp
2
1
2
1
2
1 2211 ykxkykk ryrxrpp
22
1
222
1
2
222,222 xkxkxykyky ryryrrxrxr
yrxr xy
22
22
57. 4. Calculate the initial value of decision parameter in region 2 using the last point (x0,y0) calculated in region 1 as
5. At each yk position in region 2, starting at k = 0, perform the following test: If p2k>0, the next point along the
ellipse centered on (0,0) is (xk,yk-1) and
Otherwise the next point along the ellipse is (xk+1,yk-1) and
Using the same incremental calculations for x and y as in region 1.
6. Determine the symmetry points in the other three quadrants.
7. Move each calculated pixel position (x,y) onto the elliptical path centered on (xc,yc) and plot the co-ordinate
values:
X = x + xc, y = y+ yc
8. Repeat the steps for region 1 until
222
0
2
2
0
2
0 )1(
2
1
2 yxxy rryrxrp
2
1
2
1 222 xkxkk ryrpp
2
1
2
1
2
1 2222 xkxkykk ryrxrpp
yrxr xy
22
22
58. • Digitize the ellipse with parameter rx =8 and
ry =6 in first quadrant.
Solution:
59.
60.
61.
62. Filled Area Primitives
• Two basic approaches for area filling:
– By determining overlap intervals for scan lines
– Start from given interior position and filling
outwards
63. Scan Line Polygon Filled Algorithm
• Intersection points of scan line with the
polygon edge are calculated.
• Points are sorted from left to right.
• Corresponding frame buffer position between
each intersection pair are set by specified
color.
64.
65. • A scan line pass through vertex, intersect two
polygon edges.
66. • Scan line y’
– Intersect even number of edges
– 2 pairs of intersection points correctly find the
interior span
• Scan line y
– Intersect an odd number of edges(5)
– Must count the vertex intersection as only one
point
67. How to distinguish these cases?
• Scan line y
– Intersecting edges are on the opposite side
• Scan line y’
– Intersecting edges are on the same side.
• By tracing around the boundary,
– If the endpoint y values of two consecutive edges
monotonically increases or decreases count
middle vertex as single
70. • In determining edge intersection, we can set
up incremental set up calculation using fact
that slope of edge is constant.
71.
72.
73. Inside Outside test
• Odd Even rule:
– Odd edge means interior
• Non zero winding number rule
– Non zero winding number means interior
74.
75. Scan line fill for Curved Area
• Require more works than polygon filling due
to non-linear boundary.
• We calculate the scan line intersection and fill
all the interior points.
• Symmetries between the quadrant can be
used to reduce the calculation.
76. Boundary fill algorithm
• Fill the region starting from the interior point
until the appropriate color boundary is
reached.
• Two ways:
– 4 connected
– 8 connected