linedrawingBresenhamCirclesAndPolygons by kumbhkar

7/31/2019 linedrawingBresenhamCirclesAndPolygons by kumbhkar

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Contents

In todays lecture well have a look at: Bresenhams line drawing algorithm

Line drawing algorithm comparisons

Circle drawing algorithms A simple technique

The mid-point circle algorithm

Polygon fill algorithms

Summary of raster drawing algorithms


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The Bresenham Line Algorithm

The Bresenham algorithm isanother incremental scanconversion algorithm

The big advantage of thisalgorithm is that it uses onlyinteger calculations

J a c k B r e s e n h a mworked for 27 years atIBM before enteringacademia. Bresenhamdeveloped his famousalgorithms at IBM int h e e a r l y 1 9 6 0 s


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The Big Idea

Move across thex axis in unit intervals andat each step choose between two differentycoordinates

2 3 4 5

2

4

3

5 For example, fromposition (2, 3) wehave to choosebetween (3, 3) and(3, 4)

We would like thepoint that is closer to

the original line

(xk

,yk

)

(xk+1,yk)

(xk+1,yk+1)


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They coordinate on the mathematical line at

xk+1 is:

Deriving The Bresenham Line Algorithm

At sample positionxk+1 the verticalseparations from the

mathematical line arelabelled dupperand dlower

bxmy k )1(

y

yk

yk+1

xk+1

dlower

dupper


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So, dupperand dlowerare given as follows:

and:

We can use these to make a simple decisionabout which pixel is closer to the mathematical

line

Deriving The Bresenham Line Algorithm(cont)

klower yyd

kk ybxm

)1(

yyd kupper )1(

bxmy kk )1(1


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This simple decision is based on the differencebetween the two pixel positions:

Lets substitute m with y/xwhere x and

y are the differences between the end-points:


122)1(2 byxmdd kkupperlower

)122)1(2()(

byxx

yxddx kkupperlower

)12(222 bxyyxxy kk

cyxxy kk 22


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So, a decision parameterpk for the kth stepalong a line is given by:

The sign of the decision parameterpk is the

same as that of dlower

dupper

Ifpk is negative, then we choose the lowerpixel, otherwise we choose the upper pixel


cyxxy

ddxp

kk

upperlowerk

22

)(


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Remember coordinate changes occur alongthex axis in unit steps so we can doeverything with integer calculations

At step k+1 the decision parameter is givenas:

Subtractingpk from this we get:


cyxxyp kkk 111 22

)(2)(2 111 kkkkkk yyxxxypp


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But,xk+1 is the same asxk+1 so:

whereyk+1 - yk is either 0 or 1 depending onthe sign ofpkThe first decision parameter p0 is evaluated

at (x0, y0) is given as:


)(22 11 kkkk yyxypp

xyp 20


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The Bresenham Line Algorithm

BRESENHAMS LINE DRAWING ALGORITHM(for |m| < 1.0)

1. Input the two line end-points, storing the left end-point

in (x0, y0)

2. Plot the point (x0, y0)

3. Calculate the constants x, y, 2y, and (2y - 2x)and get the first value for the decision parameter as:

4. At eachxkalong the line, starting at k = 0, perform the

following test. Ifpk< 0, the next point to plot is

(xk+1, yk) and:

xyp 20

ypp kk 21


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The Bresenham Line Algorithm (cont)

ACHTUNG! The algorithm and derivationabove assumes slopes are less than 1. forother slopes we need to adjust the algorithmslightly

Otherwise, the next point to plot is (xk+1, yk+1) and:

5. Repeat step 4 (x 1) times

xypp kk 221


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Bresenham Example

Lets have a go at thisLets plot the line from (20, 10) to (30, 18)

First off calculate all of the constants:

x: 10

y: 8

2y: 16

2y - 2x: -4

Calculate the initial decision parameterp0:

p0= 2yx = 6


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Bresenham Example (cont)

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292726252423222120 28 30

k pk (xk+1,yk+1)

0

1

2

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4

56

7

8

9


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Bresenham Exercise

Go through the steps of the Bresenham linedrawing algorithm for a line going from(21,12) to (29,16)


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Bresenham Exercise (cont)

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292726252423222120 28 30

k pk (xk+1,yk+1)

0

1

2

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56

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8


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Bresenham Line Algorithm Summary

The Bresenham line algorithm has thefollowing advantages:

An fast incremental algorithm

Uses only integer calculationsComparing this to the DDA algorithm, DDAhas the following problems:

Accumulation of round-off errors can makethe pixelated line drift away from what wasintended

The rounding operations and floating point

arithmetic involved are time consuming


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A Simple Circle Drawing Algorithm

The equation for a circle is:

wherer

is the radius of the circle

So, we can write a simple circle drawing

algorithm by solving the equation fory at

unitx intervals using:

222ryx

22 xry


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A Simple Circle Drawing Algorithm(cont)

20020 220 y

20120 221 y

20220 222 y

61920 2219 y

02020 2220 y


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A Simple Circle Drawing Algorithm(cont)

However, unsurprisingly this is not a brilliantsolution!

Firstly, the resulting circle has large gapswhere the slope approaches the vertical

Secondly, the calculations are not veryefficient

The square (multiply) operations

The square root operation try really hard toavoid these!

We need a more efficient, more accurate

solution


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Eight-Way Symmetry

The first thing we can notice to make our circledrawing algorithm more efficient is that circlescentred at (0, 0) have eight-way symmetry

(x, y)

(y, x)

(y, -x)

(x, -y)(-x, -y)

(-y, -x)

(-y, x)

(-x, y)

2

R


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Mid-Point Circle Algorithm

Similarly to the case with lines,there is an incrementalalgorithm for drawing circles

the mid-point circle algorithmIn the mid-point circle algorithmwe use eight-way symmetry so

only ever calculate the pointsfor the top right eighth of acircle, and then use symmetryto get the rest of the points

The mid-point circle

a l g o r i t h m w a sdeveloped by JackBresenham, who weheard about earlier.Bresenhams patent

for the algorithm canb e v i e w e d h e r e .
http://patft.uspto.gov/netacgi/nph-Parser?u=%2Fnetahtml%2Fsrchnum.htm&Sect1=PTO1&Sect2=HITOFF&p=1&r=1&l=50&f=G&d=PALL&s1=4371933.PN.&OS=PN/4371933&RS=PN/4371933http://patft.uspto.gov/netacgi/nph-Parser?u=%2Fnetahtml%2Fsrchnum.htm&Sect1=PTO1&Sect2=HITOFF&p=1&r=1&l=50&f=G&d=PALL&s1=4371933.PN.&OS=PN/4371933&RS=PN/4371933


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Mid-Point Circle Algorithm (cont)

(xk+1, yk)

(xk+1, yk-1)

(xk, yk)Assume that we havejust plotted point (xk, yk)

The next point is a

choice between (xk+1, yk)and (xk+1, yk-1)

We would like to choose

the point that is nearest tothe actual circle

So how do we make this choice?


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Lets re-jig the equation of the circle slightlyto give us:

The equation evaluates as follows:

By evaluating this function at the midpointbetween the candidate pixels we can make

our decision

222),( ryxyxfcirc

,0

,0

,0

),( yxfcirc

boundarycircletheinsideis),(if yx

boundarycircleon theis),(if yx

boundarycircletheoutsideis),(if yx


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Assuming we have just plotted the pixel at(xk,yk) so we need to choose between(xk+1,yk) and (xk+1,yk-1)

Our decision variable can be defined as:

Ifpk< 0 the midpoint is inside the circle andand the pixel atyk is closer to the circle

Otherwise the midpoint is outside andyk-1 is

closer

222 )2

1()1(

)2

1,1(

ryx

yxfp

kk

kkcirck


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To ensure things are as efficient as possiblewe can do all of our calculationsincrementally

First consider:

or:

whereyk+1 is eitherykoryk-1 depending on

the sign ofpk

2212111

21]1)1[(

21,1

ryx

yxfp

kk

kkcirck

1)()()1(2 122

11 kkkkkkk yyyyxpp


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The first decision variable is given as:

Then ifpk< 0 then the next decision variable

is given as:

Ifpk> 0 then the decision variable is:

r

rr

rfp circ

45

)

2

1(1

)2

1,1(

22

0

12 11 kkk xpp

1212 11 kkkk yxpp


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The Mid-Point Circle Algorithm

MID-POINT CIRCLE ALGORITHM Input radius rand circle centre (xc, yc), then set the

coordinates for the first point on the circumference of acircle centred on the origin as:

Calculate the initial value of the decision parameter as:

Starting with k = 0 at each positionxk, perform thefollowing test. Ifpk< 0, the next point along the circlecentred on (0, 0) is (xk+1, yk) and:

),0(),( 00 ryx

rp 4

50

1211

kkk

xpp


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The Mid-Point Circle Algorithm (cont)

Otherwise the next point along the circle is (xk+1, yk-1)and:

4. Determine symmetry points in the other seven octants5. Move each calculated pixel position (x, y) onto the

circular path centred at (xc, yc) to plot the coordinate

values:

6. Repeat steps 3 to 5 untilx >= y

111 212 kkkk yxpp

cxxx cyyy


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Mid-Point Circle Algorithm Example

To see the mid-point circle algorithm inaction lets use it to draw a circle centred at(0,0) with radius 10

Mid P i t Ci l Al ith E l


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Mid-Point Circle Algorithm Example(cont)

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1

0

8

976543210 8 10

10 k pk (xk+1,yk+1) 2xk+1 2yk+1

0

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Mid-Point Circle Algorithm Exercise

Use the mid-point circle algorithm to drawthe circle centred at (0,0) with radius 15

Mid P i t Ci l Al ith E l


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Mid-Point Circle Algorithm Example(cont)

k pk (xk+1,yk+1) 2xk+1 2yk+1

0

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976543210 8 10

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15 16


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Mid-Point Circle Algorithm Summary

The key insights in the mid-point circlealgorithm are:

Eight-way symmetry can hugely reduce the

work in drawing a circle Moving in unit steps along the x axis at each

point along the circles edge we need to

choose between two possible y coordinates

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Filling Polygons

So we can figure out how to draw lines andcircles

How do we go about drawing polygons?

We use an incremental algorithm known asthe scan-line algorithm

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Scan-Line Polygon Fill Algorithm

2

4

6

8

10 Scan Line

02 4 6 8 10 12 14 16

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Scan-Line Polygon Fill Algorithm

The basic scan-line algorithm is as follows: Find the intersections of the scan line with all

edges of the polygon

Sort the intersections by increasing xcoordinate

Fill in all pixels between pairs of intersectionsthat lie interior to the polygon

37 Scan Line Polygon Fill Algorithm


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Scan-Line Polygon Fill Algorithm(cont)

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Line Drawing Summary

Over the last couple of lectures we havelooked at the idea of scan converting lines

The key thing to remember is this has to be

FASTFor lines we have either DDA or Bresenham

For circles the mid-point algorithm

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Anti-Aliasing

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Summary Of Drawing Algorithms

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linedrawingBresenhamCirclesAndPolygons by kumbhkar

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