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GEOMETRY AND LINE GENERATION[Introduction, Lines, Line segments, Perpendicular Lines, Distance between a point and aLine, Vectors, Pixels and Frame Buffers.]

!e computer is information processing mac!ine, a tool for storing, manipulating, and

correlating data. Data tables can easil" be generated. !e time and effort re#uired tounderstand increases wit! t!e numerous data. $% directl" sol&es t!is problem. It is astud" of tec!ni#ues to impro&e communication between !uman and mac!ine. ' grap!ical

presentation of t!e data eases out t!e problem. !is is a passi&e form of $%.$ommunication can also be a two wa" process. ' dialogue can be establis!ed t!roug! t!egrap!ics medium for interaction. !is is termed interacti&e computer grap!ics.

'pplications of $%()anagement information ma" be displa"ed as bars and c!arts.*cientific t!eories and models ma" be described in pictorial form.It can be used in $'D to displa" mac!ine components, la"outs, blueprints, P$Bs etc.

)aps can be created for all +inds of geograp!ic information.'s a tool for animation and &ideo games

*tandardi ation$- / s"stem%0* 1%rap!ics 0ernel *"stem2P3I%* 1Programmer4s 3ierarc!ical Interacti&e %rap!ics *tandard2$%) 1$omputer %rap!ics )etafile2 is a file format for picture information t!at allowsde&ice independent capture, storage, and transfer.$%I 1$omputer %rap!ics Interface2 is a companion standard to pro&ide a pictorialinterface for t!e $%) primiti&es.

Lines(' point 1a position in a plane2 can be specified wit! an ordered pair of numbers 1x, "2,w!ere x is t!e !ori ontal distance and " is t!e &ertical distance from t!e origin. If two

points 1x5, "52 and 1x6, "62 are specified, t!en an e#uation for t!e line is gi&en b"(

(y y1) / (x x1) = (y2 y1) / (x2 x1) 152

)ultipl"ing b" t!e denominators gi&es t!e form(1" 7 "52 1x6 7 x52 8 1"6 7 "52 1x 7 x52 162

*ol&ing for " gi&es(" 8 [1"6 7 "52 9 1x6 7 x52] 1x 7 x52 : "5 1;2

y = mx + b 1

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!is is called slope- !"e#$ep" %o#m of t!e line. !e slope m is t!e c!ange in !eig!tdi&ided b" t!e c!ange in widt! for two points on t!e line 1t!e rise o&er t!e run2. !eintercept b is t!e !eig!t at w!ic! t!e line crosses t!e ">axis.)ultipl"ing out t!e factors in e#uation 5 and collecting t!em on one side of t!e e#ualsign "ields anot!er different form of line e#uation, called t!e &e!e#'l %o#m .

1"6 7 "52 x 7 1x6 7 x52 " : x6 "5 7 x5 "6 8 ? 1@2or(

# x + sy + " = 1A2=!ere( r 8 1"6 7 "52

s 8 7 1x6 7 x52t 8 x6 "5 7 x5 "6

3ere t!e &alues of r, s, and t are t!e possible &alues and multipl"ing t!em b" an"common factor will produce a new set of &alues, w!ic! satisf" t!e e#uation A anddescribe t!e same line. !e &alues of r, s, and t are sometimes c!osen so t!at

#2 + s 2 = 1 . 1 2$omparing e#uations < and A we see t!at(

m = # / s C b = " /s 1 2

Point of intersection( If two lines cross eac! ot!er t!e po !" o% !"e#se$" o! 1P52 can bee&aluated. !is point P5 lies on bot! t!e lines. *ol&ing t!e e#uations of lines for t!ecommon point "ields t!e &alues of P5.Proof( $onsider t!e slope>intercept forms of t!e two lines as follows(

Line 5( 1" 8 m5 x : b52 and Line 6( 1" 8 m6x : b62 1E2Let t!e common point be P5 1x5, "52 t!en,

"5 8 m5 x5 : b5 and "5 8 m6 x5 : b6 15?2/#uating o&er "5 gi&es( m5 x5 : b5 8 m6 x5 : b6 1552

*ol&ing for x5 "ields( x5 8 1b6 7 b52 9 1m5 7 m62 1562*ubstituting x5 in line 5 or line 6 gi&es(

"5 8 1b6 m5 7 b5 m62 9 1m5 7 m62 15;2!e point of intersection P5 is gi&en b"(

[1b6 7 b52 9 1m5 7 m62, 1b6 m5 7 b5 m62 9 1m5 7 m62] 15

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5. "; 8 mx; : b6. min 1x5, x62 x; max 1x5, x62;. min 1"5, "62 "; max 1"5, "62

!e x coordinate increases uniforml" from x5 to x6. *imilarl" t!e " coordinate increases

uniforml" from "5 to "6. !is can be expressed b" t!e following e#uations.x 8 x5 : 1x6 7 x52 u 15A2" 8 "5 : 1"6 7 "52 u 15 2

!e two e#uations toget!er describe a straig!t line. !is is +nown as parametric form because t!e x and " &alues on t!e line are gi&en in terms of a parameter u.

Le!&"* of t!e segment can be calculated using P"t!agorean t!eorem.L6 8 1x6 7 x52 6 : 1"6 7 "52 6 15 2L = (x2 x1) 2 + (y2 y1) 2, 15E2

!e m . po !" t!at lies !alf wa" between t!e x and " coordinates can be e&aluated easil".1xm, " m2 8 [1x5 : x6296, 1"5 : "6296] 16?2

e#pe!. $0l'# l !es ( Perpendicularit" of t!e gi&en lines can be e&aluated b" examiningt!eir slopes. If two lines are pe#pe!. $0l'# , t!e slope of one will be t!e negati&ereciprocal of t!e ot!er. 1 m1 = 1/m2 2.Proof($onsider two lines

Line 5( " 8 m5 x : b5 165.52Line 6( " 8 m6 x : b6. 165.62

If one line is perpendicular to t!e second, t!en a line parallel to t!e first will also be

perpendicular to t!e second. !e wor+ can be simplified b" ta+ing t!eir parallel lines,w!ic! pass t!roug! t!e origin.Parallel to Line 5( " 8 m5 x 165.;2Parallel to Line 6( " 8 m6 x 165.

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simplification gi&es(? 8 7 6 "5 "6 7 6 x5 x6 16;.52

or("5 9 x5 8 7 x6 9 "6 16;.62

we !a&e "5 8 m5 x5 and "6 8 m6 x6( 1 m5 8 "5 9 x5 and m6 8 "6 9x62m5 8 7 59m6 16

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V 8 V5 : u 1V6 7 V52 1;;2=!ere( V 8 1x, "2M V5 8 1x5, "52M and V6 8 1x6, "62.

xels '!. 4#'me 50%%e#s3 Pixel 1picture element2 is t!e smallest addressable screenelement. /ac! pixel !as a name or address. !e names correspond to t!e coordinates,

w!ic! identif" points. !e displa" screen is considered as a grid 1or arra"2 of pixels. /ac!row and column can be numbered con&enientl" to represent t!e screen coordinates. !ecoordinate 1i, K2 gi&es t!e column and row of a pixel. /ac! pixel is centered at itscoordinates. Line segments are drawn b" setting t!e intensities 1brig!tness2 of a string of

pixels between a starting pixel and an ending pixel. !e maximum number of distinguis!able points, w!ic! a line ma" !a&e, depends on t!e resolution of t!e displa"de&ice.Frame buffer is an arra" 1stored in t!e computer4s memor"2, w!ic! contains an internalrepresentation of t!e image to be displa"ed. It collects and stores t!e pixel &alues for use

b" t!e displa" de&ice. !e grap!ics displa" de&ice accesses t!is arra" to determine t!eintensit" at eac! pixel s!ould be illuminated 1displa"ed2.

e$"o# Ge!e#'" o! ( !e process of turning on t!e pixels for a line segment is called&ector generation. !e problem is to select pixels, w!ic! lie near to t!e line segment. It isnot eas" to select all t!e pixels t!at lie on t!e line. 'not!er problem is t!at t!e apparentt!ic+ness of t!e line would c!ange wit! slope and position. 'n alternati&e would be tostep along t!e columns of pixels and #uer" w!ic! row is closest to t!e line. !e pixel int!at row and column could be turned on. !is wor+s well for gentle slope lines 175 N m N52. But for s!arp slope lines 1steep lines2 t!is met!od lea&es gaps. !is can be !andled b"stepping up t!e rows of pixels and sol&ing for t!e columns. 1'lgorit!ms are discussed int!e later sections.2

A!" 'l s !& o% l !es( Lines ma" !a&e a Kagged or stair>step appearance w!en t!e" stepfrom one row and column to t!e next column and next row. !is is one aspect of a p!enomenon called aliasing. 'liasing produces defects, w!ic! occur w!en t!e scene to bedispla"ed c!anges faster 1or more smoot!l" t!an e&er" two pixels2. *etting pixels to gra"le&els between blac+ and w!ite pro&ides a means to reduce t!is effect. !is tec!ni#ue iscalled antialiasing, and it uses t!e gra" le&els to graduall" turn>off t!e pixels in one rowas it graduall" turns on t!e pixels in t!e next. If a line segment lies in between two pixels,normall" t!e closest one is selected. *uppose t!at instead of pic+ing t!e closest, bot! areturned on and t!e intensit" &alues are set suc! t!at t!e pixel closest to t!e line recei&esmost of its intensit". !e sum of t!e intensit" &alues for t!e two pixels s!ould matc! t!etotal intensit" &alue for t!e line. !e function used can be a simple or complex expression

based on t!e intensit" patterns, pixel s!apes, and !ow lines co&er t!em. 'ntialiasing wit!complicated functions can still be done efficientl" b" storing t!e function &alues in aloo+up table.

T* $6 l !e se&me!"s ( !ese are t!e line segments wit! t!ic+ness greater t!an one pixel.wo &ector generation algorit!ms are run in parallel to turn on t!e pixels along t!e edges

1ie, top and bottom boundaries2. 'nd t!e pixels t!at lie in between t!e boundaries are also

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turned on to produce a t!ic+ line segment. For gentle slope line between 1x5, "52 and 1x6,"62 wit! t!ic+ness w, t!e boundar" lines are as follows(

op boundar" line( 1x5, "5 : w"2 and 1x6, "6 :w"2Bottom boundar" line( 1x5, "5 7 w"2 and 1x6, "6 7 w"2=!ere( w" 8 [1w 752 9 6] [ (x2 x1) 2 + (y2 y1) 2, , / 7x2 x1 J]

!e factor containing x and " &alues is needed to find t!e amount to s!ift up and down inorder to ac!ie&e t!e proper widt! w as measured perpendicular to t!e line direction.*!arpl" sloping lines can be !andled similarl" wit! t!e x and " roles re&ersed.

8*'#'$"e# &e!e#'" o!3 *trings of c!aracters are often displa"ed to label and annotatedrawings and to gi&e instructions 9 information to t!e user. $!aracters are alwa"s builtinto t!e grap!ics displa" de&ice 1usuall" as !ardware, but some times as t!roug!software2. !ere are two primar" met!ods for c!aracter generation. -ne is called t!estro+e met!od and t!e ot!er t!e dot matrix or bitmap met!od.

!e stro+e met!od creates c!aracters out of a series of line segments, li+e stro+es of a pen. !is lends itself to c!anges of scaleM t!e c!aracters ma" be made twice as large b"

simpl" doubling t!e lengt! of eac! segment.!e dot>matrix met!od represents c!aracters b" an arra" of dots 1@x arra"s are oftenused, but xE and Ex5; are also found2. !is arra" is li+e a small frame buffer, Kust bigenoug! to !old a c!aracter. !e dots are t!e pixels for t!is small arra". Placing t!ec!aracter on t!e screen becomes a matter of cop"ing pixel &alues from t!e smallc!aracter arra" into some portion of t!e screen4s frame buffer. !e memor" containingt!e c!aracter dot matrix is arra" is often a !ardware de&ice called a $*'#'$"e#-&e!e#'"o#$* p, but ') ma" also be used w!en man" fonts are desired. !is met!od does notlend itself to &ariable>si ed c!aracters as t!e si e of a dot is fixed. 'ntialiasing tec!ni#uescan be applied to c!aracters to impro&e t!e appearance, particularl" for &er" small fontsand c!aracters w!ere t!e finite resolution of t!e displa" interferes wit! t!eir smoot!cur&ed s!apes.

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GRA 9I8 RIMITI EI!"#o.0$" o!: D spl'y .e; $es: # m " ;e ope#'" o!s: T*e D spl'y-4 le I!"e#p#e"e#:

No#m'l consuming 1can ta+e se&eral seconds2 and ma+es it less suitable for usein real>time animation.

' l'sm' '!el stores t!e image and allows selecti&e erasing. It contains a gas1at low pressure2 sandwic!ed between !ori ontal and &ertical grids of fine wires. ' large&oltage difference w!en applied between t!ese wires will cause t!e gas to glow. oilluminate a pixel t!e &oltage is increased momentaril" on t!e wires t!at intersect at t!edesired point. o extinguis! a pixel t!e &oltage on t!e corresponding wires is reduceduntil t!e glow cannot be maintained. Plasma panels are &er" durable and are often usedfor militar" applications.

In a li#uid cr"stal displa" 1 L8D 2 t!e lig!t is et!er transmitted or bloc+ed

depending upon t!e orientation of molecules in t!e li#uid cr"stal. !e material issandwic!ed between t!e !ori ontal and &ertical grids of electrodes. 'n electrical signalcan be used to c!ange t!e orientation of t!e molecules to turn t!e pixels on or off. !is isa flat panel displa" tec!nolog", w!ic! ma+es it lig!ter in weig!t, and offers low &oltageand power re#uirements.

9'#. $opy .e; $es ( !ese are t!e raster printers and plotters t!at produce t!e image in asingle sweep across t!e page. !ere are number of tec!nologies w!ic! fall into t!is

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categor" 1&i , film printers, laser printers, electrostatic plotters, t!ermal and t!ermal>transfer printers, in+>Ket printers, and impact dot>matrix printers2. !e de&ices range inresolution from about 5?? dpi to o&er 5??? dpi 1dpi 8 dots per inc!2. Do"-m'"# xp# !"e#s !a&e an arra" of wires, w!ic! can be indi&iduall" triggered to press an in+edribbon to ma+e a dot on t!e paper. B" sweeping t!e arra" across t!e paper images can be

formed. I!6->e" p# !"e#s form tin" droplets of in+, w!ic! can be guided to t!e paper toform dots. !e" use no les to prepare t!e Ket. L'se# p# !"e#s are built on top of copier tec!nolog". ' laser is used to suppl" lig!t pattern. ' rotating mirror sweeps t!e laser in araster pattern, and a lig!t &al&e turns t!e beam on or off to form t!e image. 4 lm p# !"e#suse a laser scanning s"stem. !e laser is focused directl" on p!otograp!ic film to formt!e picture. T*e#m'l p# !"e#s !a&e a print !ead, w!ic! can burn tin" dots on a !eat>sensiti&e paper. T*e#m'l "#'!s%e# p# !"e#s !a&e a similar print !ead, but it is used tomelt dots of wax>based pigment on to t!e paper. Ele$"#os"'" $ p# !"e#s use an arra" of wires to w!ic! a &oltage ma" be selecti&el" applied. 's t!e paper passes across t!e wires,it is gi&en a pattern of electrostatic c!arges. !e paper t!en passes t!roug! a li#uid toner,w!ic! is attracted, to t!e pattern on t!e paper to ma+e permanent impression upon dr"ing.

# m " ;e Ope#'" o!s ( )ost grap!ics s"stems offer a set of similar set of grap!icscommands to perform different operations 1li+e drawing a line segment, displa"ing text,line st"le, etc.2. !e first primiti&e command is t!at for drawing a line segment. ' linesegment is specified b" its two endpoints, it is often t!e case t!at t!ese will be connectedend to end. !e final point of t!e last segment becomes t!e first point of t!e nextsegment. o a&oid specif"ing t!is point twice, t!e s"stem can +eep trac+ of t!e current

pen or electron beam position. !e line segments are drawn using absolute line commandLIG/>'B*>6 1Q, R2, w!ere Q and R are t!e coordinate &alues of t!e final positionH. -n

t!e ot!er !and t!e" can also be drawn using relati&e line command LIG/> /L>6 1DQ,DR2, w!ere DQ and DR are t!e distances to mo&e along x>axis and ">axis from t!ecurrent position.H.

)ost often line segments are disconnected from one anot!er. !is needs t!e pento be mo&ed from current position to anot!er position wit!out drawing a line segment. Inot!er words t!e interconnecting segment is not drawn 1or !idden2. 'gain t!ere can be

bot! absolute and relati&e mo&es( )-V/>'B*>6 1Q, R2, w!ere Q and R are t!ecoordinate &alues of t!e final position.H and )-V/> /L>6 1DQ, DR2, w!ere DQ andDR are t!e distances to mo&e along x>axis and ">axis from t!e current position.H.

Ex'mple 1Line Drawing of a 3ouse2( !is can be constructed b" eit!er using absolute or relati&e commands. If absolute commands are used t!en t!e image will alwa"s be locatedat t!e same position on t!e screen. If, !owe&er, onl" relati&e commands are used, t!en t!e

position of t!e image will depend upon t!e current position. %enerall" t!ese series of commands are stored in a sub program to a&oid t!e repetiti&e coding. !is ma" be usedfor t!e construction of pictures made of repeated instances of t!e basic components. !ust!e subprogram for eac! t"pe of component s!ould be written using onl" relati&ecommands. Drawing t!e entire picture is reduced to positioning t!e beam and callingsubprograms.

*ubprogram 3ouseB/%IG

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LIG/> /L>6 1?, ?.62MLIG/> /L>6 1?.5, ?.62MLIG/> /L>6 1?.5, >?.62MLIG/> /L>6 1?, >?.62MLIG/> /L>6 1>?.6, ?2M

/GDM

!e abo&e subprogram uses relati&e commands. It will start at t!e current position, w!ic!will become t!e lower left corner of t!e drawing. It will draw t!e left wall, t!e roof, t!erig!t wall, and, finall" t!e floor. *ince onl" relati&e commands are used, more number of instances can be displa"ed b" simpl" calling t!is subprogram at different starting

positions. !ree calls of subprogram at different initial pen positions is s!own below(

B/%IG)-V/>'B*>6 1?.5, ?.62M3- */M

)-V/>'B*>6 1?.'B*>6 1?. , ?.62M3- */M

/GDM

D spl'y-4 le I!"e#p#e"e#3 It ser&es as an interface between t!e grap!ics program and t!edispla" de&ice. !e displa">file contains t!e information necessar" to construct t!e

picture. !e information will be in t!e form of instructions suc! as Odraw a line or Omo&e t!e pen . *a&ing instructions suc! as t!is ta+es muc! less storage t!an sa&ing t!e

picture itself. !e displa">file instructions ma" be sa&ed in a file eit!er for displa" at alater time or for transfers to anot!er mac!ine. *uc! files containing imaging instructionsare called me"'% les. !e displa">file interpreter con&erts t!ese instructions into actualimages. *ome grap!ic s"stems are e#uipped wit! special !ardware 1grap!ic processor2.In ot!er s"stems, t!e be!a&ior of a displa" processor is simulated.

No#m'l * 'Rs 8 3/I%3 S Rn : 3/I%3 >* '

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=!ere( Qs, Rs are screen coordinates and Qn, Rn are normali ed coordinates.!e displa" de&ices are often not s#uare. !e ratio of t!e !eig!t to widt! is called

t!e displa"4s 'spe$" #'" o . If t!e full dimension are used t!e image will be stretc!ed or s#uas!ed. If a s#uare area of t!e displa" is used t!e image is correctl" proportioned butsome of t!e displa" area is wasted. If a larger s#uare area t!an t!e displa" is used t!e

image ma" not entirel" fit on t!e displa".

D spl'y-4 le "#0$"0#e ( !e displa">file contains series of instructions 1commands2 toconstruct t!e image. /ac! command contains two parts, an operation code 1opcode2 ,w!ic! indicates w!at +ind of command it is 1eg( LIG/ or )-V/2, and operands, w!ic!are coordinates of a point 1x, "2. -ne possible met!od of storing t!ese instructions is touse t!ree separate arra"s( one for t!e operation code 1DF>-P2, one for t!e x coordinate1DF>Q2, and one for t!e " coordinate 1DF>R2. !us t!e se&ent! instruction can be DF>-P[ ], DF>Q [ ], and DF>R [ ] of t!e t!ree arra"s. !e displa" file must be large enoug! to!old all t!e commands needed to create t!e image.

't t!is point t!ere are onl" two possible instructions to consider, )-V/ and

LIG/. 'lso relati&e commands can be con&erted to absolute commands before t!e" areentered into displa">file. If an opcode of 5 means a )-V/ command and an opcode of 6means a LIG/ command, ' command to mo&e to position x 8 ?.; and " 8 ?. would loo+ li+e 5, ?.;, ?. .

D4-O ?, 1@D4- ?, ?@D4-Y ?, B@

!e abo&e statements would store t!is instruction in t!e t!ird position of t!edispla">file. If DF>-P [Q [R [file algorit!ms are used to enter instructions into t!e displa">file. !einterpreter will read instructions from a portion of t!e displa">file and carr" out t!eappropriate commands. o displa" t!e picture described in t!e displa">file, t!ree steps areto be carried out.

5. !e current displa" ma" !a&e to be cleared.6. !e displa">file must be interpreted.;. 'n explicit action 1in some cases2 to s!ow t!e contents of t!e frame

buffer.

Tex" ( 'not!er primiti&e operation is t!at of text 1a c!aracter or a string of c!aracters2

output. !e c!aracters ma" be drawn b" stro+e or dot matrix met!od. !eir patterns areoften copied from memor" into t!e frame buffer or created b" special c!aracter generation !ardware. !e ad&antage is speed and a sa&ing of displa">file memor". Insop!isticated displa"s si e, spacing, orientation, font can be selected. !e interpreter can

be extended to include t!e output of text. !is is done b" extending t!e number of operation codes to include one code for eac! c!aracter. !e opcodes range between 7;6and 756A inclusi&e. !e operands decide t!e position and text can be displa"ed &erticall"or diagonall" also. %rap!ics s"stems 1$- /, %0*2 !a&e ric! selections of text

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formatting operationsM t!e" allow c!anging t!e si e, orientation, spacing, and font of t!ec!aracter and t!e direction of t!e line of text.

L !e "yle ( Lines ma" be continuous, das!ed, or dotted. $olor, intensit", or t!ic+ness of t!e lines can also be selected. *pecial opcodes are used to c!ange line st"les

1or color or intensit"2, but suc! a command would not re#uire an" operands. !e opcodesrange between ? and 75@ inclusi&e, for c!ange of line>st"le commands. !e default linest"le is ? for normal straig!t line. !is is t!e default line>st"le w!en s"stem is initiali ed.-t!er line>st"les correspond to codes 75, 76, and so on. For a line printer, c!anging t!eline>st"le is a matter of c!anging t!e c!aracter t!at is placed in t!e frame buffer.

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OINT LOTTING TE89NIC E[$oordinate *"stem, Incremental )et!ods, Line Drawing 'lgorit!ms, $ircle %enerators.]

Point plotting tec!ni#ues !a&e become essential in programming frame buffer displa"s, w!ere t!e intensit" of eac! dot must be separatel" computed. !ese are based

on t!e $artesian $oo#. !'"e sys"em . Points are addressed b" t!eir x and " coordinates.!e &alue of x increases from left to rig!t and t!at of " from bottom to top.Points are plotted in response to t!e digital signals from t!e computer. !is

depends on coordinate precision and resolution of t!e displa" screen. =it! 5? bits of xand " coordinate precision, 1 2 x 1 2 arra" of positions, are possible w!ere a dot can

be displa"ed 15? bit binar" number 8 6 5? 8 5?6< distinct &alues2. esolution is t!enumber of &isible dots t!at can be displa"ed in a gi&en area of t!e screen. It is usuall"expressed in terms of dots per inc! 1dpi2. Displa" screens generall" measure about 5?inc!es s#uare wit! 5?? dpi. !e &alue of 5?6< is popular as it ma+es full use of 5? bitinteger coordinates. Displa"s of different t"pes

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!e s"mmetrical DD' wor+s on t!e principle t!at t!e x and " &alues aresimultaneousl" incremented in small steps proportional to t!e first deri&ati&es of x and ".for a straig!t line t!e first deri&ati&es are constant and proportional to x and ". t!us int!e case of an infinite>precision displa", a straig!t line can be generated b" incrementingt!e x and " &alues b" e x and e ", w!ere e is some small #uantit".

In t!e real world of limited>precision displa"s, addressable points are needed. !iscan be done b" rounding off t!e &alues after eac! incremental step. 'fter rounding a dotcan be displa"ed at t!e resulting x and " coordinate &alues.

'n alternati&e to rounding is t!e use of arit!metic o&erflow. !e x and " &aluesare +ept in registers t!at !a&e two parts 1&i ., integer and fractional2. !e incrementing&alues, w!ic! are bot!, less t!an unit", are repeatedl" added to t!e fractional parts.=!ene&er t!e result o&erflows t!e corresponding integer part is incremented. !e integer

parts of t!e x and " registers are used in plotting t!e line. !is would normall" !a&e t!eeffect of truncating rat!er t!an rounding. o ac!ie&e true rounding t!e DD' is initiali edwit! t!e &alue ?.@ in eac! of t!e fractional parts.

-ne ad&antage of t!is arrangement is t!at it allows detecting t!e c!anges in x and

" and a&oids plotting t!e same point twice. !e o&erflow indicator allows t!is detectionand produces signals to reposition t!e point t!at traces out t!e line. !e precision of t!eincrementing &alues and of fractional parts of t!e registers s!ould be greater t!an t!ecoordinate precision of t!e displa", ot!erwise t!e accurac" will be lost on long lines.

!e appearance of lines depends on t!e &alue c!osen for t!e increment e. !e&alue of e is c!osen suitabl" to a&oid t!e spacing of consecuti&e points from exceedingone screen unit. 1e 8 6 7n, w!ere 6 n 7 5 max1J xJ,J "J2 6 n2. In fact, e is t!e reciprocal of t!e line lengt! estimateM in t!is case t!e line lengt! is 6 n.

It generates accurate lines, since t!e displacement of a displa"ed point from t!etrue line is less t!an one !alf of a screen unit. Logicall" t!e s"mmetrical DD' is simple.

!e use of a negati&e power of 6 for e means t!at t!e implementing &alues can be

determined b" s!ifting t!e dx and d" registers rat!er t!an b" a di&ision. /ac! step iscomputed wit! Kust two additions.

mple DDA3 !e simple DD' is as accurate as its s"mmetrical counterpart but generates a

different se#uence of dots because of its different met!od of estimating line lengt!. Linelengt! estimate is c!osen to be e#ual to t!e largest of t!e magnitudes of dx and d", so t!atedx or ed" is of unit magnitude. !is allows replacing one of t!e DD'4s addresses wit! asimple counter. !erefore it generates unit steps in t!e direction of greatest motion. !esimple DD' is an ideal basis for t!e software line generator, but t!e need for di&isionlogic ma+es it less suitable to !ardware implementation.

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5#ese!*'mFs 'l&o# "*m3 *imilar to simple DD' it is designed so t!at eac! successi&e iteration c!anges one

of t!e coordinate &alues b" 5. !e ot!er coordinate ma" or ma" not c!ange, depending ont!e &alue of an error term maintained b" t!e algorit!m. !is error term records t!edistance, measured perpendicular to t!e axis of greatest mo&ement, between t!e exact

pat! of t!e line and t!e actual dots generated. During t!e iterations, t!e slope of line isadded to t!e error term. Before t!is is done, t!e sign of e is used to determine w!et!er toincrement t!e " coordinate of t!e current point. ' positi&e e &alue indicates t!at t!e exact

pat! of t!e line lies abo&e t!e current point and t!erefore " coordinate is incremented and5 is subtracted from e. If e is negati&e t!e " coordinate &alue is left unc!anged.Bresen!am4s algorit!m a&oids generating duplicate points. Because it also a&oidsmultiplications and di&isions, it is well suited to !ardware implementation.

8 #$le &e!e#'"o#s3 $ircles and circular arcs are fre#uentl" displa"ed 1eg. )ec!anical engineering

drawings2. ' number of incremental met!ods are a&ailable to plot circles and arcs.

$ircle generators are capable of generating closel" spaced dots, suitable for point plottingde&ices.!e principle of DD' can be extended to cur&es. -ne suc! cur&e is t!e circular

arc. !e differential e#uation of a circle wit! center at t!e origin is d"9dx 8 7 x9". !ecircle plotting DD' can be implemented b" using 7 ex and e" as incrementing &alues.

x n:5 8 x n : e " n " n:5 8 " n 7 e x n

!e &alue of e is c!osen suitabl" to a&oid t!e spacing of consecuti&e points fromexceeding one screen unit. 1e 8 6 7n, w!ere 6 n 7 5 r 6 n2.

nfortunatel" t!e abo&e met!od plots a spiral but not a circular arc. /ac! step is

made in a direction perpendicular to a radius of t!e circle. /ac! successi&e point isslig!tl" fart!er from t!e center. !is problem is sol&ed b" using x n:5 rat!er t!an x n for computing " n:5 .

x n:5 8 x n : e " n " n:5 8 " n 7 e x n:5

's a matter of fact, t!e abo&e e#uations generate points on an ellipse instead of acircle. !e eccentricit" of t!e resulting cur&es ma" be #uite noticeable w!en e isrelati&el" large. !is is made negligible if e is +ept small, but increases t!e computationconsiderabl". !e circles 1ellipsesT2 drawn b" t!e DD' need not be centered on t!eorigin. !e displacements in x and " to reposition t!e center can be adKusted for. !is

algorit!m is well suited to !ardware implementation.Finall", it is possible to construct a DD' t!at draws an exact circle using t!etrigonometric e#uations( [x n:5 8 x n cos : " n sin M "n:5 8 " n cos 7 x n sin ]. !etrigonometric &alues are eas" to compute and are constant for an" circle. !is pair of e#uations can be used to ad&antage if multiplications are inexpensi&e.


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LINE DRA ING DI LAYT*e 8RT: I!*e#e!"-Memo#y De; $es: T*e "o#'&e-T0be D spl'y: T*e Re%#es* L !e-

D#' !& D spl'y ,

$omputer>generated figures ma" be di&ided into two classes, line drawings and

continuous tone images. Line drawings are easier to create because t!e algorit!ms for t!eir creation are simple. sing incremental met!ods, line segments can be generated.!e amount of information re#uired to present t!em is less. !e" can be displa"ed on

e#uipment t!at is readil" a&ailable.

!e purpose of a . spl'y .e; $e is to con&ert electrical signals into &isibleimages. !e . spl'y $o!"#olle# sits in between t!e computer and t!e displa" de&ice. Itrecei&es information from t!e computer and con&erts it into signals acceptable to t!ede&ice. !e functions of displa" controller are( &oltage le&el con&ersion between t!ecomputer and t!e displa" de&ice, buffering to compensate for differences in speed of operation, and generation of line segments and text c!aracters. )an" displa" controllers

are furnis!ed wit! additional !ardware to perform functions suc! as scaling and rotationto impro&e t!e speed of response.

In most applications of $% t!e #ualit" of t!e displa"ed image is &er" important.' great deal of effort !as been directed towards t!e de&elopment !ig!>#ualit" computer displa" de&ices. In t!e 5E@?s, t!e $ was t!e onl" a&ailable de&ice. 'long wit! t!econtinuous de&elopment of t!e $ t!ere !as been an intensi&e searc! for alternati&es.

!is !as lead to t!e de&elopment of a number of new tec!ni#ues for con&erting electricalsignals into images. 1eg. DV* , Plasma Panel, Laser *can Displa", L$D, etc.2. Gone of t!em so far could displace t!e $ .

T*e 8RT3

It is an e&acuated and sealed conical glass tube. It is fitted wit! an electron gun att!e narrow end. ' "o+e 1s"stem of electromagnetic coils2 is mounted on t!e outside of tube at t!e base of t!e nec+. !e ot!er end is coated on t!e inside wit! p!osp!or. !eele$"#o! &0! emits a !ig! &elocit", finel" focused beam of electrons. !e beam of electrons 1cat!ode ra"s2 passes t!roug! focusing and deflection s"stems t!at direct t!e

beam toward specified positions on t!e p!osp!or coated screen. !e p!osp!or t!en emitsa small spot of lig!t at eac! position contacted b" t!e beam. !e beam of electrons s!ouldfulfill t!e following re#uirements(

5. It must be '$$0#'"ely %o$0se. so t!at it produces a s!arp spot w!ere it stri+es t!e p!osp!or.6. It must !a&e * &* ;elo$ "y, since t!e brig!tness of t!e image depends on t!e

&elocit" of t!e electron beam.;. )eans must be pro&ided to control t!e flow of electrons so t!at t!e !"e!s "y of

t!e beam can be controlled.

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!e electron gun contains a number of separate parts. ' $'"*o.e !eated b" anelectric filament generates electrons. *urrounding t!e cat!ode is a c"lindrical metal$o!"#ol . , wit! a !ole at one end t!at allows electrons to escape. !e control grid is+ept at a lower potential t!an t!e cat!ode. !is creates an electrostatic field t!at directst!e electrons t!roug! a point sourceM t!is simplifies t!e subse#uent focusing process. B"

altering t!e control>grid potential, t!e rate of flow of electrons 1or beam current2 can be&aried. !us t!e brig!tness of t!e image can be controlled.

Focusing is ac!ie&ed b" a %o$0s !& s"#0$"0#e containing two or more c"lindricalmetal plates at different potentials. !ese set up a toroidal electrostatic field t!ateffecti&el" catc!es stra"ing electrons and deflects t!em bac+ toward t!e axis of t!e beam.

!e result is a beam t!at is extremel" finel" focused and !ig!l" concentrated at t!e precise moment at w!ic! it stri+es t!e p!osp!or.

'n '$$ele#'" !& s"#0$"0#e is generall" combined wit! t!e focusing structure. Itconsists of two metal plates mounted perpendicular to t!e beam axis wit! !oles at t!eir

centers t!roug! w!ic! t!e beam can pass. !e two plates are maintained at a sufficientl"!ig! relati&e potential to accelerate t!e beam to t!e necessar" &elocit". 'ccelerating potentials of se&eral t!ousand &olts are used.

!e resulting electron>gun structure !as t!e ad&antage t!at it can be built as asingle p!"sical unit and mounted inside t!e $ en&elope. -t!er t"pes of gun exist,w!ose focusing is performed b" a coil mounted outside t!e tube. !is is calledelectromagnetic focusing to distinguis! it from t!e more common electrostatic met!oddescribed in t!e preceding paragrap!. !e electromagnetic tec!ni#ue can result in finer focusing, but t!e electrostatic met!od is generall" preferred in grap!ic displa"s because itleads to a c!eaper gun construction.

!e De%le$" o! ys"em ( ' set of coils, or "o+e, mounted at t!e nec+ of t!e tube, forms part of t!e deflection s"stem responsible for addressing in t!e $ . wo pairs of coilsare used, one to control !ori ontal deflection, t!e ot!er &ertical. ' primar" re#uirement of t!e deflection s"stem is t!at it deflects rapidl", since speed of deflection determines !owmuc! information can be displa"ed wit!out flic+er. o ac!ie&e fast deflection, we mustuse large>amplitude currents in t!e "o+e. 'n important part of t!e deflection s"stem ist!erefore t!e set of amplifiers t!at con&ert t!e small &oltages recei&ed from t!e displa"controller into currents of t!e appropriate magnitude.

!e &oltages used for deflection are generated b" t!e displa" controller fromdigital &alues pro&ided b" t!e computer. !ese &alues normall" represent coordinates t!atare con&erted into &oltages b" digital>to>analog 1D9'2 con&ersion. o draw a &ector a

pair of graduall" c!anging &oltages must be generated for t!e !ori ontal and &erticaldeflection coils. *e&eral met!ods !a&e been used, including t!e following(

5. Integrators( 'n integrator is a circuit, w!ic!, if pro&ided wit! a constant&oltage input, generates a linearl" increasing or decreasing &oltage as output.

!us if t!e x and " &alues defining a &ector are con&erted into &oltages and

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used as inputs to a pair of integrators, t!e appropriate deflection signals will be generated.

6. Digital )et!ods( ' fast digital &ector generator, suc! as a DD', can beconstructed from !ardware and used toget!er wit! a pair of D9' con&erters./&er" time a fres! x or " coordinate is generated, t!e coordinate &alue is

con&erted to a deflection &oltage, and a dot is displa"ed.

*osp*o#s3 !e p!osp!ors used in a grap!ic displa" are normall" c!osen for t!eir color

c!aracteristics and persistence. Ideall" t!e persistence, measured as t!e time for t!e brig!tness to drop to one>tent! of its initial &alue, s!ould last about 5?? milliseconds or less, allowing refres! at ;? !ert rates wit!out noticeable smearing as t!e image mo&es.$olor s!ould preferabl" be w!ite, particularl" for applications w!ere dar+ informationappears on a lig!t bac+ground. !e p!osp!or s!ould also posses a number of ot!er attributes( small grain si e for added resolution, !ig! efficienc" in terms of electricenerg" con&erted to lig!t, and resistance to burning under prolonged excitation.

In attempts to impro&e performance in one or anot!er of t!ese respects, man"different p!osp!ors !a&e been produced, using &arious compounds of calcium, cadmium,and inc, toget!er wit! traces of rare>eart! elements. !ese p!osp!ors are identified b" anumbering s"stem, using names li+e P5, Ppersistence blue p!osp!or t!at lea&es a greenafterglow, and P;5. P!osp!ors wit! muc! longer persistence t!an an" of t!ese do exist

but rarel" used because of smearing problems.

!e 5e'm- e!e"#'" o! 8RT3

!e normal $ can generate images of onl" single color, due to t!e limitationsof its p!osp!or. ' color $ de&ice for line>drawing displa"s !as been de&eloped,!owe&er( it uses a multipla"er p!osp!or and ac!ie&es color control b" modulating anormall" constant parameter, namel" t!e beam accelerating potential.

!e arrangement of t!e beam>penetration $ is similar to t!at of normal $ sMt!e onl" unusual component is t!e multipla"er p!osp!or, in w!ic! a la"er of red p!osp!or is deposited be!ind t!e initial la"er of green p!osp!or. If a fairl" low>potential electron

beam stri+es t!e tube face, it excites onl" t!e red p!osp!or and t!erefore produces a redtrace. =!en t!e accelerating potential is increased, t!e &elocit" of t!e beam stri+ing t!e

p!osp!or is greater, and as a result t!e beam penetrates into t!e green p!osp!or,increasing t!e green component of t!e lig!t output. ' limited range of colors, includingred, orange, "ellow and green can be generated in t!is wa".

!e principal problem wit! t!e beam>penetration $ is t!e need to c!ange t!e beam>accelerating potential b" significant amount in order to switc! colors. =!en t!eaccelerating potential c!anges, t!e deflection s"stem must react to compensate. !e!ardware or software must be designed to introduce ade#uate dela"s between c!anges incolor, so t!at t!ere is time for &oltages to settle. In order to pre&ent fre#uent dela"s and


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conse#uent flic+er, it is necessar" to displa" all t!e red elements of t!e pictureconsecuti&el", t!en c!ange t!e accelerating potential and displa" t!e "ellow elements,and so on t!roug! all t!e different colors.

!e *'.o -M's6 8RT (

!e s!adow mas+ color $ can displa" a muc! wider range of colors t!an t!e beam penetration $ and is used in t!e maKorit" of color tele&isions and monitors. Uust be!ind t!e p!osp!or coated face of t!e $ , is a metal plate, s!adow mas+, pierced wit!t!e small round !oles in a triangular pattern. In place of single electron gun, t!e s!adow>mas+ tube uses t!ree guns, grouped in a triangle or delta. !ese t!ree guns areresponsible for t!e red, green and blue components of t!e lig!t output of t!e $ .

!e deflection s"stem of t!e $ operates on all t!ree>electron beams

simultaneousl", bringing all t!ree to t!e same point of focus on t!e s!adow mas+. =!eret!e t!ree beams encounter !oles in t!e mas+, t!e" pass t!roug! and stri+e t!e p!osp!or.

*ince t!e" originate at t!ree different points, !owe&er, t!e" stri+e t!e p!osp!or in t!reeslig!tl" different spots. !e p!osp!or of t!e s!adow>mas+ tube is t!erefore laid down&er" carefull" in groups of t!ree spots 1red, green, and blue2 under eac! !ole in t!e mas+,in suc! a wa" t!at eac! spot is struc+ onl" b" electrons from t!e appropriate gun.

!e effect of t!e mas+ is t!us to Os!adow t!e spots of red p!osp!or from all butt!e red beam, and li+ewise for t!e green and blue p!osp!or spots. )odulation of t!e

beam current is possible. !us control of lig!t output in eac! of t!e t!ree componentcolors is ac!ie&ed.

%reat impro&ements !a&e been made in t!e performance of t!e s!adow>mas+ tube. Ge&er t!e less it !as remained relati&el" expensi&e compared wit! t!e monoc!rome$ , and still !as a relati&el" poor performance in all respects except color range. !es!adow>mas+ $ compares particularl" unfa&orabl" in resolution and in efficienc" of lig!t output. Bot! t!ese effects are caused b" t!e use of t!e s!adow>mas+. !e grain of t!e triangular pattern of !oles sets a limit on t!e attainable resolution, and t!e mas+ tendsto bloc+ a large portion of t!e a&ailable beam energ", reducing t!e total brig!tness. =it!t!e use of &er" !ig! accelerating potentials it is, !owe&er, possible to matc! t!e

brig!tness of monoc!rome images.

'not!er problem is con&ergence. It is extremel" difficult to adKust t!e t!ree gunsand beam deflection s"stem so t!at t!e electron beams are exactl" toget!er, all t!reecon&erging on t!e same !ole in t!e s!adow>mas+. =!en t!e" fail to con&erge, poorl"colored image results. -ften it is possible to ac!ie&e ade#uate con&ergence o&er onl" alimited area of t!e screen.

!e con&ergence problem toget!er wit! t!e relati&el" poor resolution and lig!toutput of s!adow>mas+ $ , !a&e tended to discourage its use in line>drawing displa"s.It is !owe&er, a &er" popular de&ice for use wit! a frame buffer.


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I!*e#e!" memo#y .e; $es

efres! line>drawing displa"s based on t!e $ !a&e t!e disad&antages of !ig!cost and tendenc" to flic+er w!en t!e displa"ed picture is complex. !ese two problems!a&e led to t!e de&elopment of displa" de&ices wit! in!erent image storage capabilit".

!e most widel" used of t!ese de&ices is t!e direct &iew storage tube 1DV* 2M ot!ersinclude t!e plasma panel and t!e laser scan displa".

T*e D #e$" e "o#'&e T0be 1D T 2(

!e DV* is li+e a $ wit! an extremel" long>persistence p!osp!or. ' linedrawn on t!e screen will remain &isible for upto an !our. !ese are useful for displa"ing!ig!l" complex and static images wit!out flic+er. It consists of two electron guns 1&i ,

primar" and flood guns2. !e primar" gun is used to store t!e picture pattern on a finewire grid 1storage mes!2, coated wit! dielectric and mounted Kust be!ind t!e screen. '

pattern of positi&e c!arge is deposited on t!e grid.

!is pattern is transferred to t!e p!osp!ors b" a continuous flood of electronsissuing from a separate flood gun. !e flood gun is used to maintain t!e picture displa".Uust be!ind t!e storage mes!, anot!er grid 1collector2 is placed to smoot! out t!e flow of flood electrons. !ese electrons pass t!roug! t!e collector at a low &elocit", and areattracted to t!e positi&el" c!arged portions of t!e storage mes!, but repelled b" t!e rest./lectrons not repelled b" t!e storage mes! pass rig!t t!roug! it and stri+e t!e p!osp!or.

In order to increase t!e energ" of t!ese relati&el" slow mo&ing electrons and t!uscreate a brig!t picture, t!e screen is maintained at a !ig! positi&e potential. !is is done

b" appl"ing &oltage to a t!in aluminum coating placed between t!e p!osp!or and t!e tube

face. !e flood electrons mo&e slowl" until t!e" pass t!e storage mes! and do not affectt!e stored c!arge on t!e mes!.

!e image displa"ed cannot be altered. !e entire screen is to be erased anddrawn afres!. !e normal erasing met!od is to appl" a positi&e &oltage to t!e storagemes!. !is remo&es t!e c!arge but also generates an unpleasant flas! o&er t!e entirescreen. !is is still time>consuming 1can ta+e se&eral seconds2 and ma+es it less suitablefor use in real>time animation.

-t!er problems are gradual degradation of t!e picture #ualit" as bac+ground glowaccumulates. !is glow is caused b" t!e small amounts of c!arge deposited on t!e

storage mes! b" t!e repelled flood electrons. !e performance is somew!at inferior to t!erefres! $ . -nl" a single le&el of line intensit" can be displa"ed and onl" green p!osp!or tubes are a&ailable.

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T*e l'sm' '!el3

It stores t!e image and allows selecti&e erasing. It contains a gas 1at low pressure2sandwic!ed between !ori ontal and &ertical grids of fine wires. ' large &oltage differencew!en applied between t!ese wires will cause t!e gas to glow. o illuminate a pixel t!e

&oltage is increased momentaril" on t!e wires t!at intersect at t!e desired point. oextinguis! a pixel t!e &oltage on t!e corresponding wires is reduced until t!e glow cannot be maintained. Plasma panels are &er" durable and are often used for militar"applications.

It consists of two s!eets of glass wit! t!in, closel" spaced gold electrodes attac!edto t!e inner faces and co&ered wit! a dielectric material. !e s!eets are placed a fewt!ousandt!s of an inc! apart. !e inter&ening space is filled wit! a neon>based gas andsealed. B" appl"ing &oltages between t!e electrodes t!e gas wit!in t!e panel is made to

be!a&e as if it were di&ided into tin" cells, eac! one independent of its neig!bors.

B" an ingenious mec!anism, t!e cells can be made to glow, and t!us a picture isgenerated. ' cell is made to glow b" placing a firing &oltage across it b" means of t!eelectrodes. !e gas wit!in t!e cell begins to disc!arge, and t!is de&elops &er" rapidl"into a glow. !e glow can be sustained b" maintaining a !ig!>fre#uenc" alternating&oltage across t!e cellM furt!ermore, if t!e signal amplitude is c!osen correctl", cells t!at!a&e not been fired will not be affected. In ot!er words, eac! cell is bistable( it !as twostable states.

$ells can be switc!ed on b" momentaril" increasing t!e sustaining &oltageM t!iscan be done selecti&el" b" modif"ing t!e signal onl" in t!e two conductors t!at interest att!e desired cell. *imilarl", if t!e sustaining signal is lowered, t!e glow is remo&ed. !us

t!e plasma panel allows bot! selecti&e writing and selecti&e erasure, at speeds of about6? microseconds per cell. !is speed can be increased b" writing or erasing se&eral cellsin parallel.

!e plasma panel produces a &er" stead" image, totall" free of flic+er, and is aless bul+" de&ice t!an a $ of comparable screen si e. Its main disad&antages are itsrelati&el" poor resolution, of about A? dots per inc!, and its complex addressing andwriting re#uirements. Its in!erent memor" is useful but is not as flexible as a frame>

buffer memor". Digital memories are now so inexpensi&e t!at a raster>scan displa" cancost less t!an a plasma panel. 's a result, plasma panels are not used in &er" man" of toda"4s displa"s.


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T*e L'se#- $'! D spl'y3

!e laser>scan displa" is one of t!e few !ig!>resolution, large>screen displa"de&ices. It is capable of displa"ing an image measuring ; b" < feet and still !as arelati&el" small spot si e of about 595?? inc!. It !as been used in displa"ing maps, !ig!

#ualit" text, and elaborate circuit diagrams.

!e principle of t!is displa" is &er" simple. ' laser is deflected b" a pair of mirrors and traces out t!e desired image on a s!eet of p!oto>c!romic film. !is materialis usuall" transparent, but t!e lig!t from t!e laser lea&es a dar+ trace on it. ' Olig!t

proKection s"stem is used to proKect t!e image onto a large screen. o produce afres!image, t!e displa" simpl" winds t!e role of film to bring in blan+ region under t!e laser.

!e deflecting mirrors are extremel" small and are controlled b" electrical signalsrecei&ed from t!e displa" controller. ' complex correction mec!anism compensates for t!e inertia of t!ese mirrors. For interacti&e purposes, a second laser displa"s t!e cursor

nondestructi&el" on t!e screen.


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T*e "o#'&e-T0be D spl'y

!e t"pical storage tube displa" incorporates a b" 5? inc! DV* and a built>inalp!anumeric numeric +e"board. !e screen coordinate s"stem is di&ided into 5?6 plotting commands. =!en t!e displa" is in text mode, t!e instructions are interpreted as'*$II c!aracter codes and t!e corresponding c!aracter is displa"ed on t!e screen at t!ecurrent position. *ome displa"s 1 e+tronix

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Basic Instruction epertoire 1 e+tronix *torage ube Displa"25. !e first &ector command simpl" sets t!e current position.6. !e !ig!er order bits can be omitted for small c!anges.

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/G / % 'P3I$* )-D/ ??555?5L/'V/ % 'P3I$* )-D/ ??55555

e$"o# Rep#ese!"'" o!?5 R 1!ig!2 55 R 1low2 ?5 Q 1!ig!2 5? Q 1low2

/G$-DIG% 1Lea&ing 3ig!er -rder Bits2??555?5?5 5 55 ?5 5 5?55 6 5? 6?5 6 55 5A 5???555?5?5 ? 55 ;? ?5 6 5? 5A?5 5 55 5 ?5 ; 5?

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