line algorthim

8/13/2019 Line Algorthim

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Bresenham Line

Drawing Algorithm,Circle Drawing &

Polygon Filling



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50Contents

In today’s lecture we’ll have a look at: – Bresenham’s 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



3

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50DDA

Digital differential analyser

Y=mx+c

For m<1

∆y=m∆x

For m>1

∆x=∆y/m



4

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50Question

A line has two end points at (10,10) and(20,30). Plot the intermediate points using

DDA algorithm.



5

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50The Bresenham Line Algorithm

The Bresenham algorithm isanother incremental scan

conversion algorithm

The big advantage of thisalgorithm is that it uses only

integer calculations

J a c k B r e s e n h a mworked for 27 years at

IBM before entering

academia. Bresenham

developed his famous

algorithms at IBM in

t h e e a r l y 1 9 6 0 s



6

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50The Big Idea

Move across the x axis in unit intervals andat each step choose between two different y

coordinates

2 3 4 5

2

4

3

5 For example, fromposition (2, 3) we

have to choose

between (3, 3) and

(3, 4)

We would like the

point that is closer to

the original line

(x k

, y k

)

(x k +1, y k )

(x k +1, y k +1)



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The y coordinate on the mathematical line at

xk +1 is:

Deriving The Bresenham Line Algorithm

At sample position xk +1 the vertical

separations from the

mathematical line arelabelled d upper and d lower

b xm y k )1(

y

y k

y k+ 1

x k + 1

d lower

d upper



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So, d upper and d lower are given as follows:

and:

We can use these to make a simple decision

about which pixel is closer to the mathematical

line


(cont…)

k lower y yd

k k yb xm )1(

y yd k upper )1(

b xm y k k )1(1



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

Let’s substitute m with ∆ y/∆ x where ∆ x and

∆ y are the differences between the end-points:


(cont…)

122)1(2 b y xmd d k k upper lower

)122)1(2()(

b y x x

y xd d x k k upper lower

)12(222 b x y y x x y k k

c y x x y k k 22



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So, a decision parameter pk for the k th stepalong a line is given by:

The sign of the decision parameter pk is the

same as that of d lower

– d upper

If pk is negative, then we choose the lower

pixel, otherwise we choose the upper pixel


(cont…)

c y x x y

d d x p

k k

upper lower k

22

)(



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Remember coordinate changes occur alongthe x axis in unit steps so we can do

everything with integer calculations

At step k +1 the decision parameter is givenas:

Subtracting pk from this we get:


(cont…)

c y x x y p k k k 111 22

)(2)(2 111 k k k k k k y y x x x y p p



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

where yk+1 - yk is either 0 or 1 depending onthe sign of pk

The first decision parameter p0 is evaluated

at (x0, y0) is given as:


(cont…)

)(22 11 k k k k y y x y p p

x y p 20



13

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50The Bresenham Line Algorithm

BRESENHAM’S 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 Δ x, Δ y, 2Δ y, and (2Δ y - 2Δ x)

and get the first value for the decision parameter as:

4. At each xk along the line, starting at k = 0, perform the

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

(xk +1, yk ) and:

x y p 20

y p p k k 21



14

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50The Bresenham Line Algorithm (cont…)

ACHTUNG! The algorithm and derivation

above assumes slopes are less than 1. for

other slopes we need to adjust the algorithm

slightly.

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

5. Repeat step 4 (Δ x – 1) times

x y p p k k 221



15

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50 Adjustment

For m>1, we will find whether we willincrement x while incrementing y each time.

After solving, the equation for decisionparameter pk will be very similar, just the x

and y in the equation will get interchanged.



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50Bresenham Example

Let’s have a go at this Let’s plot the line from (20, 10) to (30, 18)

First off calculate all of the constants:

– Δ x: 10

– Δ y: 8

– 2Δ y: 16

– 2Δ y - 2Δ x: -4

Calculate the initial decision parameter p0:

– p0 = 2Δ y – Δ x = 6



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50Bresenham Example (cont…)

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292726252423222120 28 30

k pk (xk+1,yk+1)

0

1

2

3

4

56

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50Bresenham Exercise

Go through the steps of the Bresenham linedrawing algorithm for a line going from

(21,12) to (29,16)



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50Bresenham Exercise (cont…)

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292726252423222120 28 30

k pk (xk+1,yk+1)

0

1

2

3

4

56

7

8



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50Bresenham Line Algorithm Summary

The Bresenham line algorithm has thefollowing advantages:

– An fast incremental algorithm

– Uses only integer calculationsComparing this to the DDA algorithm, DDA

has the following problems:

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

intended

– The rounding operations and floating point

arithmetic involved are time consuming



21

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50 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 for y at

unit x intervals using:

222 r y x

22 xr y



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A Simple Circle Drawing Algorithm

(cont…)

20020 22

0 y

20120 22

1 y

20220 22

2 y

61920 2219 y

02020 22

20 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



24

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50Polar coordinates

X=r*cosθ+xc

Y=r*sinθ+yc

0º≤θ≤360º

Or

0 ≤ θ ≤6.28(2*π)

Problem:

– Deciding the increment in θ

– Cos, sin calculations



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50Eight-Way Symmetry

The first thing we can notice to make our circledrawing algorithm more efficient is that circles

centred 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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50Mid-Point Circle Algorithm

Similarly to the case with lines,there is an incremental

algorithm for drawing circles –

the mid-point circle algorithmIn the mid-point circle algorithm

we use eight-way symmetry so

only ever calculate the pointsfor the top right eighth of a

circle, and then use symmetry

to get the rest of the points

The mid-point circle

a l g o r i t h m w a sdeveloped by Jack

Bresenham, who we

heard about earlier.

Bresenham’s 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/4371933

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/4371933



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50Mid-Point Circle Algorithm (cont…)

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2 3 41

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2 3 41

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(x k +1, y k )

(x k +1, y k -1)

(x k , y k ) Assume that we have just 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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Let’s 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),( r y x y x f circ

,0

,0

,0

),( y x f circ

boundarycircletheinsideis),(if y x

boundarycircleon theis),(if y x

boundarycircletheoutsideis),(if y x



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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:

If pk < 0 the midpoint is inside the circle andand the pixel at yk is closer to the circle

Otherwise the midpoint is outside and yk -1 is

closer

222 )2

1()1(

)2

1,1(

r y x

y x f p

k k

k k circk

33



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

First consider:

or:

where yk+1 is either yk or yk -1 depending on

the sign of pk

22

1

2

111

21]1)1[(

21,1

r y x

y x f p

k k

k k circk

1)()()1(2 1

22

11 k k k k k k k y y y y x p p

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The first decision variable is given as:

Then if pk < 0 then the next decision variable

is given as:

If pk > 0 then the decision variable is:

r

r r

r f p circ

45

)

2

1(1

)2

1,1(

22

0

12 11 k k k x p p

1212 11 k k k k y x p p

35



35

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50The Mid-Point Circle Algorithm

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

coordinates for the first point on the circumference of a

circle centred on the origin as:

• Calculate the initial value of the decision parameter as:

• Starting with k = 0 at each position xk , perform the

following test. If pk < 0, the next point along the circle

centred on (0, 0) is (xk +1, yk ) and:

),0(),( 00 r y x

r p 4

50

1211

k k k

x p p

36



36

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50The 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 until x >= y

111 212 k k k k y x p p

c x x x c y y y

37



37

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50Mid-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

38 Mid Point Circle Algorithm Example



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Mid-Point Circle Algorithm Example

(cont…)

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5

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2

1

0

8

976543210 8 10

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

0

1

2

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6

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50Mid-Point Circle Algorithm Exercise

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

40 Mid Point Circle Algorithm Example



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Mid-Point Circle Algorithm Example

(cont…)

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

0

1

2

3

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5

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78

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1

0

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50Mid-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 circle’s edge we need to

choose between two possible y coordinates

42



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50Filling 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

43



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50Scan-Line Polygon Fill Algorithm

2

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10 Scan Line

02 4 6 8 10 12 14 16

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44

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50Scan-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 intersections

that lie interior to the polygon

45 Scan Line Polygon Fill Algorithm



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Scan-Line Polygon Fill Algorithm

(cont…)

46



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50Line 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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line algorthim

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