Lines Midpoint

7/25/2019 Lines Midpoint

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ihar Ranjan Roy

Scan Conversion of Lines

MIDPOINT LINE DRAWING ALGORITHM


2/25

Midpoint Algorithm

Given a point just drawn, determine whether we move E orNE on next step

EW

N

Nihar Ranjan Roy,https://sites.google.com/site/niharranjanroy/


3/25

Midpoint Algorithm

Given a point just drawn, determine whether we move E or NEon next step?

Is the line above or below ?

Below: move E

Above: move NE

),1( 21 yx

),( yxNihar Ranjan Roy,https://sites.google.com/site/niharranjanroy/


4/25

Midpoint Algorithm Implicit

Forms

How do we tell if a point is above or below

the line?Bmxy

BxyLH

LH

xx

yy

))(()( LHxxyy

LH xxBxyxxLH

LH

Bxxxyyyxx LHLHLH )()()(



5/25


Forms

How do we tell if a point is above or below the line?

0)()()( Bxxyxxxyy LHLHLH

Bxxxyyyxx LHLHLH )()()(



6/25


Forms

How do we tell if a point is above or below the line?

cbyaxyxfd ),(

yyya LH

xxxb LH )(

)(LH

xxBc

Location of midpoint d=?

f(x,y)=0 (x,y) on the line

f(x,y)0 (x,y) below the line

0)()()( Bxxyxxxyy LHLHLH



7/25


Forms

x

y

)2,0(

)5,6(

263 xy



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Forms

x

y

)2,0(

263 xy

)5,6(1263),( yxyxf



f(x,y)0 (x,y) below the lineNihar Ranjan Roy,

https://sites.google.com/site/niharranjanroy/


9/25


Forms

x

y

)2,0(

263 xy

18)5,0( f

)5,6(1263),( yxyxf



f(x,y)0 (x,y) below the lineNihar Ranjan Roy,

https://sites.google.com/site/niharranjanroy/


10/25


Forms

x

y

)2,0(

263 xy

18)1,4( f

)5,6(




1263),( yxyxf



11/25


Forms

x

y

)2,0(

263 xy

0)4,4( f

)5,6(




1263),( yxyxf



12/25


Forms

x

y

)2,0(

263 xy

)5,6(




1263),( yxyxf



13/25


d=f(x,y)=0 (x,y) on the line

d=f(x,y)0 (x,y) below the line

Midpoint Algorithm

Need value of to determine E or NE

Build incremental algorithm

Assume we have value of

Find value of if E chosen

Find value of if NE chosen

),1( 21 yxf

),1(21

yxf

),2( 21 yxf),2(

23

yxf

E

NE

d?Nihar Ranjan Roy,https://sites.google.com/site/niharranjanroy/


14/25

Midpoint Algorithm d


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Midpoint Algorithm d>0

If NE was chosen, find ),2( 23 yxf

E

NE

dold dnew

cybxayxfdnew )()2(),2(23

23



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If NE was chosen, find

Midpoint Algorithm d>0

),2( 23 yxf

xydd

badd

yxfbayxfd

oldnew

oldnew

new

),1(),2(21

23

3



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Midpoint Algorithm d0 ?

What about starting value?

cybxayxfd LLLL )()1(),1( 21

21

0



18/25

Midpoint Algorithm d0 ?



21

0

bayxfyxfd LLLL 21210 ),(),1(



19/25

Midpoint Algorithm



21

0

bayxfyxf LLLL 2121 ),(),1(

is on the line! =0),( LL yx

bayxfd LL 21

21

0 ),1(



20/25

xyyxfd LL 2),1( 210

Midpoint Algorithm

Multiplying coefficients by 2 doesnt change sign of f

bayxfd LL 2),1( 21

0

bayxfd LL 21

21

0 ),1(

5ydd oldnew 2

)(2 xydd oldnew 6

4

Similarly equation 2 and 3 becomes

E is Chosen

NE is Chosen



21/25

Midpoint Algorithm Summary

int dx, dy, d, incE, incNE, x, y;dx = x2 - x1;

dy = y2 - y1;

d = 2*dy - dx;

incE = 2*dy;

incNE = 2*(dy - dx);

y = y1;

for (x=x1; x0)

{ d = d + incNE;

y = y + 1; }

else

{ d = d + incE; }

}


d=f(x,y)=0 (x,y) on the lined=f(x,y)0 (x,y) below the line

E

NE



22/25

Problem

Plot all the raster points that make a line

from (0,2) to (6,4) using Mid-point line

drawing algorithm.



23/25

int dx, dy, d, incE, incNE, x, y;

dx = x2 - x1;

dy = y2 - y1;

d = 2*dy - dx;

incE = 2*dy;

incNE = 2*(dy - dx);y = y1;

for (x=x1; x0)

{ d = d + incNE;y = y + 1; }

else

{ d = d + incE; }

}

Midpoint Algorithm Example

d E NE (x,y)

-2 (1,2) (1,2)

2 (2,3) (2,3)

-6 (3,3) (3,3)

-2 (4,3) (4,3)

2 (5,4) (5,4)

-6 (6,4) (6,4)

Draw a line from (0,2) to (6,4)dx=6 dy=2 d0=-2

incE=4

incNE=-8



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output

)2,0(

)4,6(d E NE (x,y)

-2 (1,2) (1,2)

2 (2,3) (2,3)

-6 (3,3) (3,3)

-2 (4,3) (4,3)

2 (5,4) (5,4)

-6 (6,4) (6,4)



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Midpoint Algorithm vs DDA

Only integer operations

Faster than DDA

No multiplication no division


Documents

Lines Midpoint