Computer Graphics Output Primitives Notes

1. OUTPUT PRIMITIVES
Mrs.G.ALAMELU,MCA.,M.PHIL.,
EMG YADAVA WOMEN’S COLLEGE,

2. POINTS AND LINES:
A line connects two points. It is a basic element in
graphics.
To draw a line, you need two points between
which youcandraw aline.
In the following three algorithms, we refer the
one point of line as X0,Y0 and the second point of
lineasX1,Y1.

3. Apointisadimensionlessshape,sinceitrepresentsadotonly,
whereasalineisaone-dimensionalshape.

4. Pointsisthefundamentalelementof picturerepresentation.
Two points represent line or edge and 3 or more points a
polygon.
Curvedlinesarerepresentedbytheshort straightlines.
It is the position in the plan defined as either pair or triplets of
numberdependinguponthedimension.

5. LINE-DRAWING ALGORITHMS:
The process of ”turing on” the pixels for a line segment is
called vector generation or line generation,and the
algorithms for them are known as vector generation
algorithmsorlinedrawingalgorithms.
Before discussing specific line drawing algorithms it is
useful to note the general requirements for such
algorithms.

6. The cartesian slope-intercept equation for a straight line
is y=m.x+b
Slopeofthelinem= y2-y1/x2-x1
Interceptbwith b=y1-m.x1

7.  The line should appears as a straight line and it
shouldstartandendaccurately.
 The line should be displayed with constant
brightness alongit
 lengthandorientation.
 Thelineshouldbedrawnrapidly
Theserequirementspecifythedesiredcharacteristicsofline:

9.  Input: At least one or more inputs must be accept a good
algorithm.
 Output:Atleastoneoutputmustproducedanalgorithm.
 An algorithm should be precise: The algorithm’s each step
mustwell-define.
 Finiteness: Finiteness is require in an algorithm. It signifies
that the algorithm will come to a halt once all of the steps
havebeencomplete.
PropertiesofaLineDrawing Algorithm:

10.  Correctness: An algorithm must implemented
correctly.
 Uniqueness: The result of an algorithm should be
basedonthe given input, andall stepsof the algorithm
shouldbeclearlyanduniquelydefined.
 Effectiveness: An algorithm’s steps must be correct
andefficient.
 Easy to understand: Learners must be able to
understand the solution in a more natural way thanks
toanalgorithm.

11. DDA ALGORITHM:
The digital differential analyser(DDA) is a scan-
conversion line algorithm based on calculating
either ∆yor ∆x.
A linear DDA starts by calculating the smaller of dy
or dxforaunitincrementof theother.
It is a simple and efficient algorithm that works by
using the incremental difference between the x-
coordinates and y-coordinates of the two endpoints
toplot theline.

12.  Thetwo endpointsof thelinesegment, (x1,y1) and(x2,y2).
 Calculate thedifference between thex-coordinates andy-
coordinates oftheendpoints asdxand dyrespectively.
 Calculate theslope ofthe lineasm= dy/dx.
 Set theinitialpointofthelineas(x1,y1).
 Loopthroughthe x-coordinates of theline,incrementingbyone
eachtime, andcalculatethecorresponding y-coordinateusing
theequationy=y1+m(x –x1).
 Plot thepixelatthecalculated(x,y) coordinate.
 Repeat steps 5and6untiltheendpoint(x2,y2) isreached.
Asteps involved inDDAlinegenerationalgorithmare:

14.  Sample at unit x intervals(∆x = 1) and compute each successive y
valueas yk+1 =yk +m.
 Sample at unit y intervals(∆y = 1) and compute each succeeding x
valueas xk+1 =xk + 1/m.
 If this processing is reversed,so that the starting ebdpoint is at
right,then either∆x=-1
 If absolute value of the slope is less than 1 and the start endpoint is at
theleft, ∆x=1

15. The digital differential analyser(DDA) is a scan-conversion line algorithm
based oncalculatingeither∆yor ∆x.
A linear DDA starts by calculating the smaller of dy or dx for a unit
incrementof the other.
It is a simple and efficient algorithm that works by using the incremental
difference between the x-coordinates and y-coordinates of the two
endpoints to plotthe line.

16. BRESENHAM’S LINE ALGORITHM:
The main task is to find all the intermediate points required
fordrawinglineABonthecomputerscreen of pixels.
In this algorithm, every pixel has integer coordinates. Apart
from that, the Bresenham algorithm works on addition and
subtractionoperations.
This algorithmis usedforscanconvertingaline.

17. It’s a line drawing algorithm that determines the points of
an n-dimensional raster that should be selected in order
to form a close approximation to a straight line between
two points.
Bresenham's algorithm - a fundamental method in
Computer Graphics - is a clever way of approximating a
continuous straight line with discrete pixels, ensuring
that the line appears straight and smooth on a pixel-
based display.

18. Points meant by group of pixels.

19. So, d_upperandd_lower aregivenasfollows:
d_lower =y-yk
=m*(xk+1)+ b-yk
d_upper=(yk + 1)-y
=yk + 1-m*(xk +1) -b
 Slope formula:y=m(xk +1)+b

20.  Thedifference between two-pixel places served thebasis forthis
straightforward decision:
d_lower -d_upper=2m(xk +1) - 2yk +2b -1
 Let’ssubstitute mwith ∆y/∆xwhere ∆xand∆yarethedifferences
between theendpoints:
∆x(d_lower -d_upper)=∆x( (2∆y/∆x)(xk +1)-2yk) +(2b -1)
=2∆y.xk -2∆x.yk +2∆y +∆x(2b -1)
=2∆y.xk -2∆x.yk + c
 So, atthekthstep alongaline, adecisionparameterpk isgivenby:
pk=∆x(d_upper-d_lower)
=2∆y.xk -2∆x.yk +c

21.  Thevalues ofsuccessive decisionparameters using
incremental integercalculation.Atstep k+1, the decision
parameterisevaluated from as:
pk+1=2∆y.(xk+1) -2∆x.(yk+1) + c
 From thepreceding equation:
pk+1 -pk =2∆y.(xk+1 - xk) -2∆x (yk+1 -yk)
 But xk+1= xk +1
pk+1 =pk +2∆y -2∆x(yk+1 -yk)

22. Whereyk+1-yk iseither 0or1dependingonthe
signofpk
Theinitialdecisionparameterp0 isevaluated at
(x0,y0)andiswritten asfollows:
p0 =2∆y-∆x
 Bresenham Line drawing algorithm is used for scan converting a line. It is an
efficient method because it involves integer addition, subtraction, and
multiplication operations. Because these actions may be completed quickly,
linescanbegenerated quickly.

23. Inputthetwoendpointsoftheline,savetheleftendpointin(x0,y0)
Plotthepoint(x0 ,y0)
CalculatetheconstantsΔx,Δy,2Δy,and(2Δy -2Δx)andobtainthefirstvalueforthe
decisionparameteras:
p0 =2Δy-Δx
Performthefollowingtestateachxk alongtheline,beginningatk=0.If pk <0,thenthe
nextpointtotheplotis(xk+1 , yk)and:
pk+1 =pk +2Δy
Otherwise,thenextpointtoplotis(xk+1 ,yk+1)and:
pk+1 =pk +2Δy-2Δx
 Repeatstep4,(Δx –1) times
Thestep-wise Bresenham’s linedrawingalgorithmisgivenbelow:

24. PARALLEL LINE ALGORITHM:
Aparallel algorithm,as opposed toatraditional serial algorithm,isan
algorithmwhichcandomultipleoperations inagiventime. Ithasbeena
traditionof computer sciencetodescribeserial algorithms inabstract
machinemodels, often the oneknown asrandom-access machine.
Theprocessing sothatpixel positions canbecalculatedefficiently in
parallel.

26.  Foralinewithslope 0<m<1andleft endpointcoordinate
position (x0,y0),Thedistancebetween beginningxpositions of
adjacentpartitionscanbecalculatedas:
∆xp=(∆xnp-1)/np
 Where ∆xisthewidthofthe line,andthevalue forpartition
width ∆xpiscomputed using integer division.
xk=x0 k∆xp

27.  Thechange∆yp,intheydirectionover eachpartitionis
calculatedfromthelineslope mandpartitionwidth ∆xp:
∆yp=m∆xp
 Atthekthpartition,the startingycoordinateisthen
yk=y0 round(k∆yp)
 Theinitialdecisionparameterfor Bresenham’s algorithmat the
startof thekthsubintervalis obtainedfrom
pk=(k∆xp)(2∆y)-round(k∆yp)(2∆x) 2∆y-∆x

29.  Examples ofParallel Algorithms:
Primes.
Sparse MatrixMultiplication.
PlanarConvex-Hull.
ThreeOther Algorithms.
 Twolines areparallellines ifthey donotintersect. Theslopes ofthe lines
arethesame. f(x)=m1x+b1 andg(x)=m2x+b2 areparallel ifm1=m2f( x)=m1
x+b1andg(x)=m2x+b2 areparallelifm1=m2. I

30.  Analgorithmisasequenceof steps thattakeinputsfromthe user
andaftersomecomputation,producesanoutput.
 Aparallelalgorithmisanalgorithmthatcanexecute several
instructionssimultaneously ondifferent processing devices andthen
combinealltheindividualoutputsto producethefinalresult.
 Calculate pixelpositions alongalinepathsimultaneously by
partitioningthecomputationsamongthevarious processors
available.

31. THANKS YOU!!!