Graphics 2 Output Primitives Slide.pdf

Prof. Vijay M. Shekhat
9558045778
vijay.shekhat@darshan.ac.in
2160703
Computer Graphics
Unit-2
Graphics Primitives

Unit: 2 Graphics Primitives 2
Unit: 2 Graphics Primitives 2 Darshan Institute of Engineering & Technology
Darshan Institute of Engineering & Technology
Outline
 Points
 Line drawing algorithms.
 Circle drawing algorithm.
 Ellipse drawing algorithm.
 Scan-Line polygon filling algorithm.
 Inside-Outside test.
 Boundary fill algorithm.
 Flood fill algorithm.
 Character generation.
 Line attributes.
 Color and grayscale levels
 Area fill attributes.
 Character attributes.

Point
 Point plotting is done by converting a single coordinate position
furnished by an application program into appropriate operations for the
output device in use.
 Example: Plot point
 Line
 Line
 Point
 To draw the point on the screen we use function

 To draw the pixel in C language we use function

 Similarly for retrieving color of pixel we have function

0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7

0
1
2
3
4
5
1 2 3 4
Line
 Line drawing is done by calculating intermediate positions along
the line path between two specified endpoint positions.
 The output device is then directed to fill in those positions
between the end points with some color.
 For some device such as a pen plotter or random scan display, a
straight line can be drawn smoothly from one end point to other.
 Digital devices display a straight line segment by plotting discrete
points between the two endpoints.
 Discrete coordinate positions along the line path are calculated
from the equation of the line.
0
1
2
3
4
5
0 2 4 6

Contd.
 Screen locations are referenced with integer values.
 So plotted positions may only approximate actual line positions
between two specified endpoints. For example line position of
(12.36, 23.87) would be converted to pixel position (12, 24).
 This rounding of coordinate values to integers causes lines to be
displayed with a stair step appearance (“the Jaggies”).

Line Drawing Algorithms
 The Cartesian slop-intercept equation for a straight line is
•
• with ‘ ’ representing slop and ‘ ’ as the intercept.
 It is possible to draw line using this equation but for efficiency
purpose we use different line drawing algorithm.
• DDA Algorithm
• Bresenham’s Line Algorithm
 We can also use this algorithm in parallel if we have more number
of processors.

Introduction to DDA Algorithm
 Full form of DDA is Digital Differential Analyzer
 DDA is scan conversion line drawing algorithm based on
calculating either or using line equation.
 We sample the line at unit intervals in one coordinate and find
corresponding integer values nearest the line path for the other
coordinate.
 Selecting unit interval in either or direction based on way we
process line.

Unit Step Direction in DDA
Algorithm
 Processing from left to right.
 Slope is “+ve”, & Magnitude is Less than 1
 Slope is “-ve”, & Magnitude is Less than 1
 Slope is “+ve”, & Magnitude is greater than 1
 Slope is “-ve”, & Magnitude is greater than 1
 Processing from right to left.
 Slope is “+ve”, & Magnitude is Less than 1
 Slope is “-ve”, & Magnitude is Less than 1
 Slope is “+ve”, & Magnitude is greater than 1
 Slope is “-ve”, & Magnitude is greater than 1

Derivation of DDA Algorithm
 We sample at unit interval and calculate each successive
value as follow:

 [For first intermediate point]
 [For 𝑡ℎ intermediate point]
 Subtract from


 Using this equation computation becomes faster than normal line
equation.
 As is any real value calculated value must be rounded to nearest
integer.

Contd.
 We sample at unit interval and calculate each successive
value as follow:

 [For first intermediate point]
 [For intermediate point]
Subtract from


Similarly
 for ∆x = -1: we obtain
 for ∆y = -1: we obtain

Procedure for DDA line algorithm.
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));
}
}

Example DDA Algorithm
 Example: Draw line with coordinates , .

 is “+ve” and less than 1 so .
 ,
[Plot the initial point as given]
 ,
[Plot the first intermediate point by rounding it to ]
 ,
[Plot the second intermediate point by rounding it to ]
 ,
[Plot the third intermediate point by rounding it to ]
 ,
[Plot End point given]
0 1 2 3 4 5 6 7 8
4
3
2
1
0

Introduction to Bresenham’s Line
Algorithm
 An accurate and efficient raster line-generating algorithm,
developed by Bresenham.
 It scan converts line using only incremental integer calculations.
 That can be modified to display circles and other curves.
 Based on slop we take unit step in one direction and decide pixel
of other direction from two candidate pixel.
 If we sample at unit interval and vice versa.

Line Path & Candidate pixel
 Example |ΔX| > |ΔY|, and ΔY is “+ve”.
 Initial point (Xk, Yk)
 Line path
 Candidate pixels {(Xk+1, Yk), (Xk+1, Yk+1)}
0 1 2 3 4
4
3
2
1
0
 Now we need to decide which candidate pixel is more closer to
actual line.
 For that we use decision parameter (Pk) equation.
 Decision parameter can be derived by calculating distance of
actual line from two candidate pixel.

Derivation Bresenham’s Line
Algorithm
 [Line Equation]
 [Actual Y value at Xk position]
 [Actual Y value at Xk +1position]
Distance between actual line position and lower candidate
pixel


Distance between actual line position and upper
candidate pixel


0 1 2 3 4
4
3
2
1
0

Contd.
Calculate



 [Put ]
Decision parameter



 [Replacing single constant C for simplicity]
Similarly


Contd.
Subtract from


[where ]

[where depending on selection of previous pixel]

Initial Decision parameter
 The first decision parameter is calculated using equation of .

 [Put ]

[Substitute 0 0]

[Substitute ]


[Initial decision parameter with all terms are constant]

Bresenham’s Line Algorithm
1. Input the two line endpoints and store the left endpoint in .
2. Load into the frame buffer; that is, plot the first point.
3. Calculate constants , , , and , and obtain the
starting value for the decision parameter as
4. At each along the line, starting at , perform the following test:
If , the next point to plot is and
Otherwise, the next point to plot is and
5. Repeat step-4 times.

Discryption of Bresenham’s Line
Algorithm
 Bresenham’s algorithm is generalized to lines with arbitrary slope.
 For lines with positive slope greater than 1 we interchange the roles of
the and directions.
 Also we can revise algorithm to draw line from right endpoint to left
endpoint, both and decrease as we step from right to left.
 When 𝑘 we can choose either lower or upper pixel but same for
whole line.
 For the negative slope the procedure are similar except that now one
coordinate decreases as the other increases.
 The special case handle separately by loading directly into the frame
buffer without processing.
 Horizontal line ,
 Vertical line
 Diagonal line with

Example Bresenham’s Line
Algorithm
 Example: Draw line with coordinates , .
 Plot left end point .
 Calculate:
• ,
• ,
• ,
• ,

 Now so we select upper pixel .

 Now so we select lower pixel .

 Now so we select upper pixel .

 Now so we select lower pixel .
0 1 2 3 4 5 6 7 8
4
3
2
1
0

Parallel Execution of Line
Algorithms
 The line-generating algorithms we have discussed so far
determine pixel positions sequentially.
 With multiple processors we can calculate pixel position along a
line path simultaneously.
 One way to use multiple processors is partitioning existing
sequential algorithm into small parts and compute separately.
 Alternatively we can setup the processing so that pixel positions
can be calculated efficiently in parallel.
 Balance the load among the available processors is also important
in parallel processing.

Parallel Bresenham Line
Algorithms
 For number of processors we can set up parallel Bresenham
line algorithm by subdividing the line path into partitions and
simultaneously generating line segment in each of the
subintervals.
 For a line with slope and left endpoint coordinate
position , we partition the line along the positive
direction.
 The distance between beginning positions of adjacent partitions
can be calculated as:
 Where is the width of the line and value for partition with
is computed using integer division.

Contd.
 Numbering the partitions and the processors, as 0, 1, 2, up to
, we calculate the starting coordinate for the partition
as:
 The change in the direction over each partition is calculated
from the line slope and partition width :
 At the partition, the starting y coordinate is then

IDP for Parallel Bresenham Line
Algorithms
 The initial decision parameter(IDP) for Bresenham's algorithm at
the start of the subinterval is obtained as follow:


∆
∆


 Each processor then calculates pixel positions over its assigned
subinterval.

Other Ways for Parallel Execution of Line
Algorithms
 For line with slope greater than 1 we partitioning the line in the
direction and calculating beginning values for the positions.
 For negative slopes, we increment coordinate values in one
direction and decrement in the other.
 Another way to set up parallel algorithms on raster system is to
assign each processor to a particular group of screen pixels.
 With sufficient number of processor we can assign each processor
to one pixel within some screen region.
 This can be done by assigning one processor to each of the pixels
within the limit of the bounding rectangle.

Contd.
 Perpendicular distance d from line to a particular pixel is
calculated by:

Where



With

 , , and are constant we need to compute only once.

Contd.
 For the line each processors need to perform two multiplications
and two additions to compute the pixel distance .
 A pixel is plotted if is less than a specified line thickness
parameter.
 Instead of partitioning the screen into single pixels, we can assign
to each processor either a scan line or a column.
 Each processor calculates line intersection with horizontal row or
vertical column of pixels assigned to that processor.

Contd.
 If vertical column is assign to processor then is fix and it will
calculate .
 Similarly if horizontal row is assign to processor then is fix and
will be calculated.
 Such direct methods are slow in sequential machine but we can
perform very efficiently using multiple processors.

Circle
 A circle is defined as the set of points that are all at a given
distance from a center position say .
Center
r
Radius

Properties of Circle- Cartesion
Coordinate
 Cartesian coordinates equation :

 We could use this equation to calculate circular boundary points.
 We increment 1 in direction in every steps from to
and calculate corresponding values at each position as:





Contd.
 But this is not best method as it requires more number of
calculations which take more time to execute.
 And also spacing between the plotted pixel positions is not
uniform.
 We can adjust spacing by stepping through y values and
calculating x values whenever the absolute value of the slop of the
circle is greater than 1.
 But it will increases computation time.

Properties of Circle- Polar
Coordinate
 Another way to eliminate the non-uniform spacing is to draw circle
using polar coordinates ‘r’ and ‘’.
 Calculating circle boundary using polar equation is given by pair of
equations which is as follows.


 When display is produce using these equations using fixed angular
step size circle is plotted with uniform spacing.
 The step size ‘’ is chosen according to application and display
device.
 For a more continuous boundary on a raster display we can set the
step size at .

Properties of Circle- Symmetry
 Computation can be reduced by considering symmetry city
property of circles.
 The shape of circle is similar in each octant.
𝑂
(X, Y)
(Y, X)
(X, -Y)
(Y, -X)
(-X, -Y)
(-Y, -X)
(-Y, X)
(-X, Y)
(4, 3)
(3, 4)
(4, -3)
(3, -4)
(-4, -3)
(-3, -4)
(-3, 4)
(-4, 3)

Circle Algorithm
 Taking advantage of this symmetry property of circle we can
generate all pixel position on boundary of circle by calculating only
one sector from to .
 Determining pixel position along circumference of circle using any
of two equations shown above still required large computation.
 More efficient circle algorithm are based on incremental
calculation of decision parameters, as in the Bresenham line
algorithm.
 Bresenham’s line algorithm can be adapted to circle generation by
setting decision parameter for finding closest pixel to the
circumference at each sampling step.

Contd.
 A method for direct distance comparison to test the midpoint
between two pixels to determine if this midpoint is inside or
outside the circle boundary.
 This method is easily applied to other conics also.
 Midpoint approach generates same pixel position as generated by
bresenham’s circle algorithm.
 The error involve in locating pixel positions along any conic section
using midpoint test is limited to one-half the pixel separation.

Introduction to Midpoint Circle
Algorithm
 In this we sample at unit interval and determine the closest pixel
position to the specified circle path at each step.
 Given radius and center
 We first setup our algorithm to calculate circular path coordinates
for center .
 And then we will transfer calculated pixel position to center
by adding to and to .
 Along the circle section from to in the first quadrant,
the slope of the curve varies from to .
 So we can step unit step in positive direction over this octant
and use a decision parameter to determine which of the two
possible position is closer to the circular path.

Decision Parameter Midpoint Circle
Algorithm
 Position in the other seven octants are then obtain by symmetry.
 For the decision parameter we use the circle function which is:
 Above equation we calculate for the mid positions between pixels
near the circular path at each sampling step.
 And we setup incremental calculation for this function as we did in
the line algorithm.

Midpoint between Candidate
pixel
 Figure shows the midpoint between the two candidate pixels at
sampling position .
 Assuming we have just plotted the pixel at
 Next we determine whether the pixel at position or
the one at position is closer to circle boundary.
Midpoint
Candidate
Pixel

Derivation Midpoint Circle
Algorithm
 So for finding which pixel is more closer using decision parameter
evaluated at the midpoint between two candidate pixels as below:


 If , midpoint is inside the circle and the pixel on the scan
line is closer to circle boundary.
 Otherwise midpoint is outside or on the boundary and we select
the scan line .

Contd.
 Successive decision parameters are obtain using incremental
calculations as follows:


 Now we can obtain recursive calculation using equation of and
as follow



Contd.




 Now we can put


Contd.

 In above equation is either or depending on the
sign of the .
 If we select .

 If we select .





Contd.


 Now put .


IDP Midpoint Circle Algorithm
 The initial decision parameter(IDP) is obtained by evaluating the
circle function at the start position as follows.






Algorithm for Midpoint Circle Generation
1. Input radius r and circle center , and obtain the first point on the
circumference of a circle centered on the origin as
2. calculate the initial value of the decision parameter as
3. At each position, starting at , perform the following test:
If , the next point along the circle centered on is
Otherwise, the next point along the circle is
Where , & .
4. Determine symmetry points in the other seven octants.
5. Move each calculated pixel position onto the circular path centered on
and plot the coordinate values:
,
6. Repeat steps 3 through 5 until .

 Example: Draw circle with radius , and center of circle is
(Only one octant to )
 First we find pixel position for octant to for center


 Now we select (1, 10)


Example Midpoint Circle
Algorithm
𝒌 𝟏 𝒌 𝟏
𝒌
(1, 10)
-9
0
(2, 10)
-6
1
(3, 10)
-1
2
(4, 9)
6
3
(5, 9)
-3
4
(6, 8)
8
5
(7, 7)
5
6







Contd.
𝒌 𝟏 𝒌 𝟏
𝒌
(1, 10)
-9
0
(2, 10)
-6
1
(3, 10)
-1
2
(4, 9)
6
3
(5, 9)
-3
4
(6, 8)
8
5
(7, 7)
5
6





 Now Loop exit as , as
 in our case
Contd.
𝒌 𝟏 𝒌 𝟏
𝒌
(1, 10)
-9
0
(2, 10)
-6
1
(3, 10)
-1
2
(4, 9)
6
3
(5, 9)
-3
4
(6, 8)
8
5
(7, 7)
5
6

Contd.
 Than we calculate pixel position for given center using
equations:


Center (1, 1)
Center (0, 0)
(2, 11)
(1, 10)
(3, 11)
(2, 10)
(4, 11)
(3, 10)
(5, 10)
(4, 9)
(6, 10)
(5, 9)
(7, 9)
(6, 8)
(8, 8)
(7, 7)

Contd.
 Plot the pixel.
 First plot initial point.
(1, 11)
Center (1, 1)
(2, 11)
(3, 11)
(4, 11)
(5, 10)
(6, 10)
(7, 9)
(8, 8)
12
11
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11
Center
(1, 1)

Ellipse
 AN ellipse is defined as the set of points such that the sum of the
distances from two fixed positions (foci) is same for all points.

Contd.
 If we labeled distance from two foci to any point on ellipse
boundary as and then the general equation of an ellipse can
be written as:
 Expressing distance in terms of focal coordinates and
we have

[Using Distance formula]

Properties of Ellipse-Specifying
Equations
 An interactive method for specifying an ellipse in an arbitrary
orientation is to input
 two foci and
 a point on the ellipse boundary.
 With this three coordinates we can evaluate constant in equation:
 We can also write this equation in the form
 Where the coefficients , and are evaluated in terms
of the focal coordinates and the dimensions of the major and
minor axes of the ellipse.

Contd.
 Then coefficient in can be
evaluated and used to generate pixels along the elliptical path.
 We can say ellipse is in standard position if their major and minor
axes are parallel to x-axis and y-axis.
 Ellipse equation are greatly simplified if we align major and minor
axis with coordinate axes i.e. x-axis and y-axis.
X-axis
Y-axis

Contd.
 Equation of ellipse can be written in terms of the ellipse center
coordinates and parameters and as.
 Using the polar coordinates r and θ, we can also describe the
ellipse in standard position with the parametric equations:

 Symmetry property further reduced computations.
 An ellipse in standard position is symmetric between quadrant.
Properties of Ellipse-Symmetry

Introduction to Midpoint Ellipse
Algorithm
 Given parameters .
 We determine points for an ellipse in standard position
centered on the origin.
 Then we shift the points so the ellipse is centered at .
 If we want to display the ellipse in nonstandard position then we
rotate the ellipse about its center to align with required direction.
 For the present we consider only the standard position.
 We draw ellipse in first quadrant and than use symmetry property
for other three quadrant.

 In this method we divide first quadrant into two parts according to
the slope of an ellipse
 Boundary divides region at
• slope = -1.
 We take unit step in X direction
• If magnitude of ellipse slope < 1 (Region 1).
 We take unit step in Y direction
• If magnitude of ellipse slope > 1 (Region 2).
Regions in Midpoint Ellipse
Algorithm
Region 1
Region 2

Ways of Processing Midpoint Ellipse
Algorithm
 We can start from and step clockwise along the elliptical
path in the first quadrant
 Alternatively, we could start at and select points in a
counterclockwise order.
 With parallel processors, we could calculate pixel positions in the
two regions simultaneously.
 Here we consider sequential implementation of midpoint
algorithm.
 We take the start position at and steps along the elliptical
path in clockwise order through the first quadrant.

Decision Parameter Midpoint Ellipse
Algorithm
 We define ellipse function for center of ellipse at (0, 0) as follows.

 Which has the following properties:
 Thus the ellipse function serves as the decision parameter in the
midpoint ellipse algorithm.
 At each sampling position we select the next pixel from two
candidate pixel.

Processing Steps of Midpoint Ellipse
Algorithm
 Starting at we take unit step in direction until we reach
the boundary between region-1 and region-2.
 Then we switch to unit steps in direction in remaining portion on
ellipse in first quadrant.
Region 1
Region 2
 At each step we need to test the value
of the slope of the curve for deciding
the end point of the region-1.

Decide Boundary between Region
1 and 2
 The ellipse slope is calculated using following equation.

 At boundary between region 1 and 2 slope= -1 and equation
become.

 Therefore we move out of region 1 whenever following equation
is false:


Midpoint between Candidate pixel in
Region 1
 Figure shows the midpoint between the two candidate pixels at
sampling position in the first region.
 Assume we are at position and we determine the next position
along the ellipse path, by evaluating decision parameter at midpoint
between two candidate pixels.

k
k
k
k
k
y
2 2
x
2 2
x
2
y
2
Midpoint
Candidate
Pixel

Derivation for Region 1


 If , the midpoint is inside the ellipse and the pixel on scan
line is closer to ellipse boundary
 Otherwise the midpoint is outside or on the ellipse boundary and
we select the pixel .

Contd.
 At the next sampling position decision parameter for region 1 is
evaluated as.


 Now subtract from


Contd.



 Now making as subject.


Contd.

 is either or , depends on the sign of

IDP for Region 1
 Now we calculate the initial decision parameter by putting
as follow.





Midpoint between Candidate pixel in
Region 2
 Now we similarly calculate over region-2.
 Unit stepping in negative direction and the midpoint is now
taken between horizontal pixels at each step.
 For this region, the decision parameter is evaluated as follows.

k+2
k+1
k
k
k-1
y
2 2
x
2 2
x
2
y
2
Midpoint
Candidate
Pixel

Derivation for Region 2


 If the midpoint is outside the ellipse boundary, and we
select the pixel at .
 If the midpoint is inside or on the ellipse boundary and
we select .

Contd.
 At the next sampling position decision parameter for region 2 is
evaluated as.


 Now subtract from


Contd.





Contd.

 Now making as subject.

 Here is either or , depends on the sign of .

IDP for Region 2
 In region-2 initial position is selected which is last position of
region one and the initial decision parameter is calculated as
follows.


 For simplify calculation of we could also select pixel position
in counterclockwise order starting at .

Algorithm for Midpoint Ellipse Generation
1. Input and ellipse center , and obtain the first point on an ellipse
centered on the origin as
2. Calculate the initial value of the decision parameter in region 1 as
3. At each position in region 1, starting at , perform the following test:
If , than the next point along the ellipse is and
Otherwise, the next point along the ellipse is and
With
,
And continue until

Contd.
4. Calculate the initial value of the decision parameter in region 2 using the last
point calculated in region 1 as
5. At each position in region-2, starting at , perform the following test:
If , the next point along the ellipse is and
Otherwise, the next point along the ellipse is and
Using the same incremental calculations for and as in region 1.
6. Determine symmetry points in the other three quadrants.
7. Move each calculated pixel position onto the elliptical path centered on
and plot the coordinate values:
,
8. Repeat the steps for region 2 until .

Example Midpoint Ellipse
Algorithm
 Example: Calculate intermediate pixel position (For first quadrant)
for ellipse with and ellipse center is at origin
 Initial point




Contd.
(xk+1, yk+1)
p1k
K
(1, 6)
-332
0
(2, 6)
-224
1
(3, 6)
-44
2
(4, 5)
208
3
(5, 5)
-108
4
(6, 4)
288
5
(7, 3)
244
6




Contd.
(xk+1, yk+1)
p1k
K
(1, 6)
-332
0
(2, 6)
-224
1
(3, 6)
-44
2
(4, 5)
208
3
(5, 5)
-108
4
(6, 4)
288
5
(7, 3)
244
6



Contd.
(xk+1, yk+1)
p2k
K
(8, 2)
-23
0
(8, 1)
361
1
(8, 0)
297
2



Contd.
 Plot the pixel.
 Plot initial point(0, 6)
Center (0, 0)
(1, 6)
(2, 6)
(3, 6)
(4, 5)
(5, 5)
(6, 4)
(7, 3)
12
11
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11
Center
(0, 0)
Center (0, 0)
(8, 2)
(8, 1)
(8, 0)

Filled-Area Primitives
 In practical we often use polygon which are filled with some
colour or pattern inside it.
 There are two basic approaches to area filling on raster systems.
• One way to fill an area is to determine the overlap intervals for scan line
that cross the area.
• Another method is to start from a given interior position and paint outward
from this point until we encounter boundary.

Scan-Line Polygon Fill
Algorithm
 For each scan-line crossing a polygon, the algorithm locates the
intersection points are of scan line with the polygon edges.
 This intersection points are stored from left to right.
 Frame buffer positions between each pair of intersection point are
set to specified fill color.
Scan line

Contd.
 Scan line intersects at vertex are required special handling.
 For vertex we must look at the other endpoints of the two line
segments which meet at this vertex.
• If these points lie on the same (up or down) side of the scan line, then that
point is counts as two intersection points.
• If they lie on opposite sides of the scan line, then the point is counted as
single intersection.
Scan line
Scan line
Scan line

Edge Intersection Calculation with
Scan-Line
 Coherence methods often involve incremental calculations applied
along a single scan line or between successive scan lines.
 In determining edge intersections, we can set up incremental
coordinate calculations along any edge by exploiting the fact that
the slope of the edge is constant from one scan line to the next.
 For above figure we can write slope equation for polygon
boundary as follows.

 Since change in coordinates between the two scan lines is
simply


Contd.
 So slope equation can be modified as follows




 Each successive intercept can thus be calculated by adding the
inverse of the slope and rounding to the nearest integer.

Edge Intersection Calculation with Scan-Line for parallel
execution
 For parallel execution of this algorithm we assign each scan line to
separate processor in that case instead of using previous values
for calculation we use initial values by using equation as.

 Now if we put in incremental calculation equation
then we obtain equation as.

 Using this equation we can perform integer evaluation of
intercept.

Simplified Method for Edge Intersection Calculation with
Scan-Line
1. Suppose
2. Initially, set counter to , and increment to (which is ).
3. When move to next scan line, increment counter by adding
4. When counter is equal to or greater than (which is ),
increment the (in other words, the
for this scan line is one more than the previous scan line), and
decrement counter by (which is ).
Counter=0
Counter=3
Counter=6
Counter=2
Counter=5
Counter=1
Counter=4
Counter=0

Use of Sorted Edge table in Scan-Line Polygon Fill
Algorithm
 To efficiently perform a polygon fill, we can first store the polygon
boundary in a sorted edge table.
 It contains all the information necessary to process the scan lines
efficiently.
 We use bucket sort to store the edge sorted on the smallest
value of each edge in the correct scan line positions.
 Only the non-horizontal edges are entered into the sorted edge
table.

Contd.
A
B
C
D
C E
Scan Line
Scan Line
Scan Line
0
1
.
.
.
.
.
.
.
.
.
Scan Line
Number

Inside-Outside Tests
 In area filling and other graphics operation often required to find
particular point is inside or outside the polygon.
 For finding which region is inside or which region is outside most
graphics package use either
1. Odd even rule OR
2. Nonzero winding number rule

Odd Even/ Odd Parity/ Even
Odd Rule
 By conceptually drawing a line from any position to a distant
point outside the coordinate extents of the object.
 Than counting the number of edges crossing by this line.
1. If Edge count is odd, than p is an interior point.
2. Otherwise p is exterior point.
 To obtain accurate edge count we must sure that line selected is
does not pass from any vertices. Boundary of Screen
1
1
2

Nonzero Winding Number Rule
 This method counts the number of times the polygon edges wind
around a particular point in the counterclockwise direction.
 This count is called the winding number.
 We apply this rule by initializing winding number with 0.
 Then draw a line for any point to distant point beyond the
coordinate extents of the object.
Boundary of Screen
Winding number=0

Contd.
 The line we choose must not pass through vertices.
 Then we move along that line we find number of intersecting
edges.
1. If edge cross our line from right to left We add 1 to winding number
2. Otherwise subtract 1 from winding number
 IF the final value of winding number is nonzero then the point is
interior otherwise point is exterior. Boundary of Screen
-1
+1
-1
Winding number=0
Winding number=-1
Winding number=+1-1=0

Comparison between Odd Even Rule and Nonzero
Winding Rule
 For standard polygons and simple object both rule gives same
result but for more complicated shape both rule gives different
result.
Odd Even Rule Nonzero Winding Rule

Scan-Line Fill of Curved
Boundary Areas
 Scan-line fill of region with curved boundary is more time
consuming as intersection calculation now involves nonlinear
boundaries.
 For simple curve such as circle or ellipse scan line fill process is
straight forward process.
 We calculate the two scan line intersection on opposite side of the
curve.
 This is same as generating pixel position along the curve boundary
using standard equation of curve.
 Then we fill the color between two boundary intersections.
 Symmetry property is used to reduce the calculation.

Introduction to Boundary / Edge Fill
Algorithm
 In this method, edges of the polygons are drawn.
 Then starting with some seed (any point inside the polygon) we
examine the neighbouring pixels to check whether the boundary
pixel is reached.
 If boundary pixels are not reached, pixels are highlighted and the
process is continued until boundary pixels are reached.
 Selection of neighbour pixel is either 4-cormected or 8-connected.
4-Cormected Region 8-Cormected Region

Contd.
 In some cases, an 8-connected algorithm is more accurate than
the 4-connected algorithm.
 Some times 4-connected algorithm produces the partial fill.
Seed

Boundary / Edge Fill Algorithm
Procedure:
boundary-fill4(x, y, f-colour, b-colour)
{
if(getpixel (x,y) ! = b-colour && gepixel (x, y) ! = f-colour)
{
putpixel (x, y, f-colour)
boundary-fill4(x + 1, y, f-colour, b-colour);
boundary-fill4(x, y + 1, f-colour, b-colour);
boundary-fill4(x - 1, y, f-colour, b-colour);
boundary-fill4(x, y - l, f-colour, b-colour);
}
}

Problem of Staking and Efficient
Method
 Same procedure can be modified according to 8 connected region
algorithm by including four additional statements to test diagonal
positions.
 This procedure requires considerable stacking of neighbouring
points more, efficient methods are generally employed.
 Efficient method fill horizontal pixel spans across scan lines,
instead of proceeding to 4 connected or 8 connected
neighbouring points.
 Then we need only stack a beginning position for each horizontal
pixel span, instead of stacking all unprocessed neighbouring
positions around the current position.
 Starting from the initial interior point with this method, we first fill
in the contiguous span of pixels on this starting scan line.

Contd.
 Then we locate and stack starting positions for spans on the
adjacent scan lines.
 Spans are defined as the contiguous horizontal string of positions
bounded by pixels displayed in the area border colour.
 At each subsequent step, we unstack the next start position and
repeat the process.
 For e.g.

Contd.
6
4
5
1
(c)
4
5
1
6
4
5
1
(d)
4
5
1
1
2
(a)
2
1
3
1
(b)
3
1

Introduction to Flood-Fill
Algorithm
 Sometimes it is required to fill in an area that is not defined within
a single colour boundary.
 In such cases we can fill areas by replacing a specified interior
colour instead of searching for a boundary colour.
 This approach is called a flood-fill algorithm. Like boundary fill
algorithm, here we start with some seed and examine the
neighbouring pixels.
 However, here pixels are checked for a specified interior colour
instead of boundary colour and they are replaced by new colour.
 Using either a 4-connected or 8-connected approach, we can step
through pixel positions until all interior point have been filled.

Flood-Fill Algorithm
Procedure :
flood-fill4(x, y, new-colour, old-colour)
{
if(getpixel (x,y) = = old-colour)
{
putpixel (x, y, new-colour)
flood-fill4 (x + 1, y, new-colour, old -colour);
flood-fill4 (x, y + 1, new -colour, old -colour);
flood-fill4 (x - 1, y, new -colour, old -colour);
flood-fill4 (x, y - l, new -colour, old-colour);
}
}

Character Generation
 We can display letters and numbers in variety of size and style.
 The overall design style for the set of character is called typeface.
 Today large numbers of typefaces are available for computer
application for example Helvetica, Arial etc.
 Originally, the term font referred to a set of cast metal character
forms in a particular size and format.
 Example: 10-point Courier Italic or 12- point Palatino Bold.

Contd.
 Now, the terms font and typeface are often used interchangeably,
since printing is no longer done with cast metal forms.
 Methods of character generation are:
 Bitmap Font/ Bitmapped Font
 Outline Font
 Stroke Method
 Starbust Method

Bitmap Font/ Bitmapped Font
 A simple method for representing the
character shapes in a particular typeface is to
use rectangular grid patterns.
 In frame buffer, the 1 bits designate which
pixel positions are to be displayed on the
monitor.
0 0 1 1
1 1 0
0 1 1 1
1 1 1 0
1 1 0 0
0 0 1 1
1 1 0 0
0 0 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 0 0
0 0 1 1
1 1 0 0
0 0 1 1
0
 Bitmap fonts are the simplest to define and display.
 Bitmap fonts require more space.
 It is possible to generate different size and other variation from
one set but this usually does not produce good result.

Outline Font
 In this method character is generated using curve
section and straight line as combine assembly.
 To display the character we need to fill interior
region of the character.
 This method requires less storage since each
variation does not required a distinct font cache.
 We can produce boldface, italic, or different sizes
by manipulating the curve definitions for the
character outlines.
 But this will take more time to process the outline
fonts, because they must be scan converted into
the frame buffer.

• It uses small line segments to generate
a character.
• The small series of line segments are
drawn like a stroke of a pen to form a
character.
Stroke Method
• We can generate our own stroke method by calling line
drawing algorithm.
• Here it is necessary to decide which line segments are
needs for each character and then draw that line to
display character.
• It support scaling by changing length of line segment.

Starbust Method
 In this method a fix pattern of lines (24 line) segments are used to
generate characters.
 We highlight those lines which are necessary to draw a particular
character.
 Pattern for particular character is stored in the form of 24 bit code.
 In which each bit represents corresponding line having that number.
 We put bit value 1 for highlighted line and 0 for other line.
03
24
23
22
21
20
19
18
17
16 15
14
13
01
02
04
05
06
07
08
09
10
11
12

Contd.
 Example letter V
 Code for letter V = 1 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0
 This technique is not used now a days because:
• It requires more memory to store 24 bit code for single character.
• It requires conversion from code to character.
• It doesn’t provide curve shapes.
03
24
23
22
21
20
19
18
17
16 15
14
13
01
02
04
05
06
07
08
09
10
11
12

Line Attributes
 Basic attributes of a straight line segment are:
• Type
• Dimension
• color
• pen or brush option.

Line Type
 Possible selection for the line-type attribute includes solid lines, dashed
lines, and dotted lines etc.
 We modify a line –drawing algorithm to generate such lines by setting
the length and spacing of displayed solid sections along the line path.
 To set line type attributes in a PHIGS application program, a user invokes
the function: setLinetype(It)
 Where parameter lt is assigned a positive integer value of 1, 2, 3, 4… etc.
to generate lines that are, respectively solid, dashed, dotted, or dotdash
etc.
1
2
3
4
Solid
Dashed
Dotted
Dotdash

Line Width
 Implementation of line-width options depends on the capabilities
of the output device.
 A heavy line on a video monitor could be displayed as adjacent
parallel lines, while a pen plotter might require pen changes.
 To set line width attributes in a PHIGS application program, a user
invokes the function: setLinewidthScalFactor (lw)
 Line-width parameter lw is assigned a positive number to indicate
the relative width of the line to be displayed.
 Values greater than 1 produce lines thicker than the standard line
width and values less than the 1 produce line thinner than the
standard line width.

Contd.
 In raster graphics we generate thick line by plotting
• above and below pixel of line path when slope |m|<1. &
• left and right pixel of line path when slope |m|>1.

Line Width at Endpoints and
Join
 As we change width of the line we can also change line end and
join of two lines which are shown below
Butt caps
Projecting square caps
Round caps
Miter join
Round join
Bevel join

Line color
 The name itself suggests that it is defining color of line displayed
on the screen.
 By default system produce line with current color but we can
change this color by following function in PHIGS package as
follows: setPolylinecolorIndex (lc)
 In this lc is constant specifying particular color to be set.

• In some graphics packages line is
displayed with pen and brush
selections.
• Options in this category include
shape, size, and pattern.
• These shapes can be stored in a
pixel mask that identifies the
array of pixel positions that are
to be set along the line path.
• Also, lines can be displayed with
selected patterns by
superimposing the pattern
values onto the pen or brush
mask.
Pen and Brush Options

Color and Grayscale Levels
 Various colors and intensity-level options can be made available to
a user, depending on the capabilities and design objectives of a
particular system.
 General purpose raster-scan systems, for example, usually provide
a wide range of colors, while random-scan monitors typically offer
only a few color choices, if any.
 In a color raster system, the number of color choices available
depends on the amount of storage provided per pixel in the frame
buffer

Contd.
 Also, color-information can be stored in the frame buffer in two
ways:
• We can store color codes directly in the frame buffer OR
• We can put the color codes in a separate table and use pixel values as an
index into this table
 With direct storage scheme we required large memory for frame
buffer when we display more color.
 While in case of table it is reduced and we call it color table or
color lookup table.

Color Lookup Table
 Color values of 24 bit is stored in lookup table and in frame buffer
we store only 8 bit index of required color.
 So that size of frame buffer is reduced and we can display more
color. Color
Lookup
Table
196
255
0
196 2081 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
To Red Gun
To Green Gun
To Blue Gun
Frame Buffer

Greyscale
 With monitors that have no color capability, color function can be
used in an application program to set the shades of grey
(greyscale) for display primitives.
 Numeric values between 0-to-1 can be used to specify greyscale
levels.
 This numeric values is converted in binary code for store in raster
system. Example: frame buffer with 2 bits per pixel.
Displayed Greyscale
Stored Intensity Values In The
Intensity Code
Binary Code
Frame Buffer
Black
00
0
0.0
Dark grey
01
1
0.33
Light grey
10
2
0.67
White
11
3
1.0

Area-Fill Attributes
 For filling any area we have choice between solid colors or pattern
to fill all these are include in area fill attributes. Which are:
• Fill Styles
• Pattern Fill
• Soft Fill

Fill Styles
 Area are generally displayed with three basic style.
1. hollow with color border
2. filled with solid color
3. filled with some design
 In PHIGS package fill style is selected by following function:
setInteriorStyle (fs)
 Value of fs include hollow ,solid, pattern etc.

Contd.
 Another values for fill style is hatch, which is patterns of line like
parallel line or crossed line.
 For setting interior color in PHIGS package we use:
setInteriorColorIndex (fc)
 Where fc specify the fill color.
Hollow Solid Pattern Diagonal
Hatch Fill
Diagonal Cross-
Hatch Fill

• We select the pattern with
setInteriorStyleIndex (pi)
• Where pattern index parameter pi
specifies position in pattern table
entry.
• For example:
– SetInteriorStyle( pattern ) ;
– setInteriorStyleIndex ( 2 ) ;
– fillArea (n, points);
Pattern Fill
Pattern(cp)
Index(pi)
1
2
Pattern Table

Contd.
 We can also maintain separate table for hatch pattern.
 We can also generate our own table with required pattern.
 Other function used for setting other style as follows:
setpatternsize (dx, dy)
 setPaternReferencePoint (position)
 We can create our own pattern by setting and resetting group of
pixel and then map it into the color matrix.

Soft Fill
 Soft fill is filling layer of color on back ground color so that we can
obtain the combination of both color.
 It is used to recolor or repaint so that we can obtain layer of
multiple color and get new color combination.
 One use of this algorithm is soften the fill at boundary so that
blurred effect will reduce the aliasing effect.
 For example if we fill t amount of foreground color then pixel color
is obtain as:

 Where F is foreground color and B is background color

Contd.
 If we see this color in RGB component then:



 Then we can calculate t as follows:

 If we use more then two color say three at that time equation
becomes as follow:

 Where the sum of coefficient , , and is 1.

Character Attributes
 The appearance of displayed characters is controlled by attributes
such as:
• Font
• Size
• Color
• Orientation.
 Attributes can be set for entire string or may be individually.

Text Attributes
 In text we are having so many style and design like italic fonts,
bold fonts etc.
 For setting the font style in PHIGS package we have function:
setTextFont (tf)
 Where tf is used to specify text font.
 For setting color of character in PHIGS we have function:
setTextColorIndex (tc)
 Where text color parameter tc specifies an allowable color code.
 For setting the size of the text we use function:
setCharacterheight (ch)
 Where ch is used to specify character height.

Contd.
 For scaling the character we use function:
setCharacterExpansionFacter (cw)
 Where character width parameter cw is set to a positive real
number that scale the character body width.
 Spacing between character is controlled by function:
 setCharacterSpacing (cs)
 Where character spacing parameter cs can be assigned any real
value.

Contd.
 The orientation for a displayed character string is set according to
the direction of the character up vector:
setCharacterUpVector (upvect)
 Parameter upvect in this function is assigned two values that
specify the x and y vector components.
 Text is then displayed so that the orientation of characters from
baseline to cap line is in the direction of the up vector.
 For setting the path of the character we use function:
setTextPath (tp)
 Where the text path parameter tp can be assigned the value:
right, left, up, or down.

Contd.
 For setting the alignment we use function:
setTextAlignment (h, v)
 Where parameter h and v control horizontal and vertical
alignment respectively.
 For specifying precision for text display is given with function:
setTextPrecision (tpr)
 Where text precision parameter tpr is assigned one of the values:
string, char, or stroke.
 The highest-quality text is produced when the parameter is set to
the value stroke.

Marker Attributes
 A marker symbol display single character in different color and in
different sizes.
 For marker attributes implementation by procedure that load the
chosen character into the raster at defined position with the
specified color and size.
 We select marker type using function: setMarkerType (mt)
 Where marker type parameter mt is set to an integer code.

Contd.
 Typical codes for marker type are the integers 1 through 5,
specifying, respectively:
1. a dot (.)
2. a vertical cross (+)
3. an asterisk (*)
4. a circle (o)
5. a diagonal cross (x).
 Displayed marker types are centred on the marker coordinates.

Contd.
 We set the marker size with function:
SetMarkerSizeScaleFactor (ms)
 Where parameter marker size ms assigned a positive number
according to need for scaling.
 For setting marker color we use function:
setPolymarkerColorIndex (mc)
 Where parameter mc specify the color of the marker symbol.

Aliasing
 Primitives generated in raster graphics by various algorithms have
stair step shape or jagged appearance.
 Jagged appearance is due to integer calculation by rounding actual
values.
 This distortion of actual information due to low frequency
sampling is called aliasing.

 Minimise effect of aliasing by some way is known as antialiasing.
 In periodic shape distortion may be occurs due to under sampling.
 To avoid losing information from such periodic objects, we need to
set the sampling frequency to at least twice that of the highest
frequency occurring in the object.
 This is referred to as the Nyquist sampling frequency (or Nyquist
sampling rate):
Antialiasing

Contd.
 In other words sampling interval should be no larger than one-half the
cycle. Which is called nyquist sampling interval.
 One way to solve this problem is to display image on higher resolution.
 But it has also limit that how much large frame buffer we can maintain
along with maintaining refresh rate 30 to 60 frame per second.
 And also on higher resolution aliasing will remains up to some extents.
 With raster systems that are capable of displaying more than two
intensity levels (color or grayscale), we can apply antialiasing methods to
modify pixel intensities.
 By appropriately varying the intensities of pixels along the boundaries of
primitives, we can smooth the edges to lessen the aliasing effect.

Antialiasing Methods
1. Supersampling Straight Line Segments
2. Pixel-Weighting Masks
3. Area Sampling Straight Line Segments
4. Filtering Techniques
5. Pixel Phasing
6. Compensating For Line Intensity Differences
7. Antialiasing Area Boundaries

Supersampling Straight Line
Segments
 For the greyscale display of a straight-line segment, we can divide
each pixel into a number of sub pixels and determine the number
of sub pixel along the line path.
 Then we set intensity level of each pixel proportional to number of
sub pixel along the line path.
 E.g. in figure area of each pixel is
divided into nine sub pixel and then we
determine how many number of sub
pixel are along the line ( it can be 3 or 2
or 1 as we divide into 9 sub pixel).
 Based on number 3 or 2 or 1 we assign
intensity value to that pixel.

Contd.
 We can achieve four intensity levels by dividing pixel into 16 sub
pixels and five intensity levels by dividing into 25 sub pixels etc.
 Lower intensity gives blurred effect and hence performs
antialiasing.
 Other way is we considered pixel areas of finite size, but we
treated the line as a mathematical entity with zero width.
 Actually, displayed lines have a width approximately equal to that
of a pixel.
 If we take finite width of the line into account, we can perform
supersampling by setting each pixel intensity proportional to the
number of sub pixels inside the polygon representing the line
area.

Contd.
 A sub pixel can be considered to be
inside the line if its lower left corner is
inside the polygon boundaries.
 Advantage of this is that it having
number of intensity equals to number of
sub pixel.
 Another advantage of this is that it will distribute total intensity
over more pixels.
 E.g. in figure pixel below and left to (10, 20) is also assigned some
intensity levels so that aliasing will reduce.
 For color display we can modify levels of color by mixing
background color and line color.

Pixel-Weighting Masks
 Supersampling method are often implemented by giving more
weight to sub pixel near the center of pixel area.
 As we expect centre sub pixel to be more important in
determining the overall intensity of a pixel.
 For the 3 by 3 pixel subdivisions we have considered so far, a
weighting scheme as in fig. could be used.
 The center sub pixel here is weighted four times that of the corner
sub pixels and twice that of the remaining sub pixels.
 By averaging the weight of sub pixel which are
along the line and assign intensity proportional to
average weight will reduce aliasing effect.

Area Sampling Straight Line
Segments
 In this scheme we treat line as finite width rectangle, and the
section of the line area between two adjacent vertical or two
adjacent horizontal screen grid lines is then a trapezoids.
 Overlap areas for pixels are calculated by determining how much
of the trapezoid overlaps each pixel in that vertical column (or
horizontal row).
 E.g. pixel with screen grid coordinates
(10, 20) is about 90 percent covered by
the line area, so its intensity would be
set to 90 percent of the maximum
intensity.

Contd.
 Similarly, the pixel at (10 21) would be set to an intensity of about
15-percent of maximum.
 With color displays, the areas of pixel overlap with different color
regions is calculated and the final pixel color is taken as the
average color of the various overlap areas.

Filtering Techniques
 It is more accurate method for antialiasing.
 Common example of filter is rectangular, conical and Gaussian
filters.
 Methods for applying the filter function are similar to applying a
weighting mask, but now we integrate over the pixel surface to
obtain the weighted average intensity.
 For reduce computation we often use table look up.

Pixel Phasing
 On raster system if we pass the electron beam from the closer sub
pixel so that overall pixel is shifted by factor ¼, ½, or ¾ to pixel
diameter.
 Where beam strike at that part of pixel get more intensity then
other parts of the pixel and gives antialiasing effect.
 Some systems also allow the size of individual pixels to be
adjusted as an additional means for distributing intensities.

Compensating For Line Intensity
Differences
 Antialiasing a line to soften the aliasing effect also compensates
for another raster effect, illustrated fig.
 As both lines are display with same number of pixel and then also
length of diagonal line is greater than horizontal line by factor .
 So that diagonal line is display with less intensity then horizontal
line.
 For compensating this we display diagonal line with high intensity
and horizontal line with low intensity so that this effect is
minimize.
 In general we set intensity according to slope of
the line.

Antialiasing Area Boundaries
 Methods we discuss for antialiasing line can also be applied for
area boundary.
 If system capabilities permit the repositioning of pixels, area
boundaries can be smoothen by adjusting boundary pixel
positions.
 Other method is to adjust each pixel intensity at boundary
position according to percent of area inside the boundary.

Contd.
 In fig. pixel at (x, y) is assigned half the intensity as its ½ area is
inside the area boundary.
 Similar adjustments based on percent of area of pixel inside are
applied to other pixel.

Graphics 2 Output Primitives Slide.pdf

Graphics 2 Output Primitives Slide.pdf

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