Affine Transforms

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Java Two Dimensional Graphics

with Afne Transorms

Sasank Upadrasta



Contents

• Introduction to 2D usin Java

• Drawin Geometric !rimitives

• "hat are Afne Transorms

• Java #rame "orks $ afne

• %eerences



Introduction to 2D usin Java

• Java2D A!I includes these packaesjava.awt

java.awt.image

java.awt.color

java.awt.font

java.awt.geom

java.awt.print

java.awt.renderable



Shapes and Areas

• Shapes & create standard shape o'(ects anddraw them in one method call instead odrawin each sement with multiple methodcalls) The shapes include rectanles andellipses and also ar'itrar* shapes such as the!ol*on class) The shape classes implementthe Shape interace and provide a !athIteratorthat steps throuh the sements o the ivenshape)

• Areas & com'ine shapes in di+erent wa*s tocreate new areas)



• Stroke & var* the thickness o a line, producear'itrar* dot&dash se-uences, select howcorners and line ends are shaped, etc)

• Paint & .ll the interiors o shapes, as well asoutlines, not (ust with solid colors 'ut with– gradients o one color to another, and

– textured patterns o tiled imaes)

Strokes and !aint



Afne Transormations

• Transformations & afne transormations /i)e)those eometrical transormations that keepparallel lines parallel0 can take a shape and

translate it, rotate it, shear it, compress ore1pand it) ultiple transormations can 'ecompounded toether)



Compositin and Clippin

• Compositing & overla* shapes and imaesaccordin to several optional rules that takeadvantae o the alpha transparenc*)

• Clipping & to speed up perormance, onl*redraw the portion o the displa* that lieswithin a iven outline or clip) "ith Java2d onecan use ar'itrar* clip outlines rather than (ust

rectanles)



Te1t, Color and Imaes

• Text & vast arra* o te1t renderin capa'ilitiesthat rane rom the 'asic drawStrin/0 methodto creatin *our own editor rom scratch)

• Color & lots o tools to help render colors in aconsistent manner reardless o the device onwhich the* are displa*ed)

• Images & use man* o the a'ove tools and

techni-ues that are used with shapes & e))transormations, compositin, clippin etc) & onimaes)



Imae #ilters and !rintin

• Image processing & appl* imae .lters& ede enhancement, 'lurrin,

inversion, etc) are availa'le)• Printing & print the raphics displa* to

the local printer as shown on themonitor)





The Graphics class

• The vast ma(orit* o uses or the Java2D A!I utili3e a small su'set o its

capa'ilities encapsulated in thejava.awt.Graphics class) ost methods othe Graphics class can 'e divided into

two 'asic roups4



5) Draw and .ll methods, ena'lin *ou torender 'asic shapes, te1t, and imaes

2) Attri'utes settin methods, which a+ect

how that drawin and .llin appears

• ethods such as setFont and setColor de.ne how draw and .ll methods

render)



Drawin method e1amples

• drawStrin $ #or drawin te1tg.drawString("Hello", 10, 10);

• drawImae $ #or drawin imaes

g.drawImage(img, 0, 0, width, height, 0, 0,imageWidth, imageHeight, null);

• draw6ine, drawArc, draw%ect, draw7val,drawPolgon ! For drawing geometri sha#es

g$.draw(new %ine$&.&ou'le(0, 0, 0, 0));



Imae scalin

• 8ou can choose one o several methodsin the Graphics class 'ased on whether

*ou want to render the imae at thespeci.ed location in its oriinal si3e orscale it to .t inside the iven rectanle



Transparenc* and #ill

• 8ou can .ll the transparent areas o theimae with color or keep them

transparent• #ill methods appl* to eometric shapesand include *ill+r, *illet, *ill-al,*illPolgon)



2D Coordinates

• "hether *ou draw a line o te1t or an imae,remem'er that in 2D raphics ever* point is

determined '* its 1 and * coordinates) All o

thedraw and .ll methods need this inormationwhich determines where the te1t or imaeshould 'e rendered)



Usin 1,* Coordinates

• #or e1ample, to draw a line, an applicationcalls the ollowin4

/aa.awt.ra#his.draw%ine(int 1, int 1, int

$, int $)

In this code (x1, y1) is the start point o theline, and (x2, y2) is the end point o the line)

• So the code to draw a hori3ontal line is as

ollows4ra#his.draw%ine($0, 100, 1$0, 100);





"eather e1ample /continued0

The setFont and drawString methods render thetemperature and the weather condition)

...

77 Free4ing, Cold, Cool, Warm, Hot,

77 8lue, reen, 9ellow, -range, edFont *ont 2 new Font("Seri*", Font.P%+I:, );g.setFont(*ont);String tem#String 2 *eels < " " < tem#erature<"F";FontenderContet *r 2((ra#his$&)g).getFontenderContet();...



"eather e1ample /continued0

The setFont and drawString methods render thetemperature and the weather condition)

...

g.setColor(tetColor);

int =et=em# 2 r>?(int)'ounds=em#.get>(); int =et=em# 2 r9?(int)'ounds=em#.get9();g.drawString(tem#String, =et=em#, =et=em#);int =etCond 2 r>?(int)'oundsCond.get>();int =etCond 2 r9?(int)'oundsCond.get9() <(int)'ounds=em#.getHeight(); g.drawString(ondStr,

=etCond, =etCond);



Introduction to 2D usin Java/contd 90

• The *illet method allows *ou to draw a rectanle .lled withthe speci.ed color)...

etangle$& 'ounds=em# 2

*ont.getString8ounds(tem#String, *r);etangle$& 'oundsCond 2 *ont.getString8ounds(ondStr,*r);int w=et 2 @ath.ma((int)'ounds=em#.getWidth(),(int)'oundsCond.getWidth());int h=et 2 (int)'ounds=em#.getHeight() <(int)'oundsCond.getHeight();

int r> 2 (si4e.width?w=et)7$;int r9 2 (si4e.height?h=et)7$;g.setColor(Color.%IH=A+9);g$.*illet(r>, r9, w=et, h=et);...



Geometric !rimitives

• The Java 2D: A!I provides several classes thatde.ne common eometric o'(ects such aspoints, lines, curves, and rectanles) These

eometr* classes are part o the /aa.awt.geom packae)

• The PathIterator interace de.nes methods orretrievin elements rom a path)



Shape Interace

• The Sha#e interace provides a set o methodsor descri'in and inspectin eometric patho'(ects) This interace is implemented '* the

eneralPath class and other eometr* classes)



Geometric !rimitives

• All e1amples represented in this section createeometries '* usin /aa.awt.geom and thenrender them '* usin the ra#his$& class) To

'ein *ou o'tain a ra#his$& o'(ect, ore1ample '* castin the ra#his parameter othe #aint() method)

#u'li oid #aint (ra#his g) 5

ra#his$& g$ 2 (ra#his$&) g;

...

6





!oint 6ocations

• To create a point with the coordinates <,< *ouuse the deaultconstructor, Point$&.&ou'le() )

8ou can use the set%oation method to set the

position o the point as ollows4

set%oation(dou'le , dou'le ) $ To set thelocation o the point& de.nin coordinates asdou'le values



Set 6ocation

• set%oation(Point$& #) $ To set the location othe point usin the coordinates o anotherpoint)

Also, the Point$& class has methods tocalculate the distance 'etween the currentpoint and a point with iven coordinates, orthe distance 'etween two points)



6ine

• The %ine$& class represents a line sement in/1, *0 coordinate space) The %ine$&.Float and%ine$&.&ou'le su'classes speci* lines in ;oat

and dou'le precision) #or e1ample4

77 draw %ine$&.&ou'le

g$.draw(new %ine$&.&ou'le(1, 1, $, $));



Set 6ine methods

• This class includes several set%ine() methodsto de.ne the endpoints o the line)Alternativel*, the endpoints o the line could

'e speci.ed '* usin the constructor or the%ine$&.Float class as ollows4– %ine$&.Float(*loat >1, *loat 91, *loat >$,*loat 9$)

– %ine$&.Float(Point$& #1, Point$& #$)

• Use the StroBe o'(ect in the Graphics2D classto de.ne the stroke or the line path)



=llipse

• The =llipse2d class represents an ellipsede.ned '* a 'oundin rectanle) Thelli#se$&.Float and lli#se$&.&ou'le su'classes

speci* an ellipse in ;oat and dou'le precision)lli#se is ull* de.ned '* a location, a widthand a heiht) #or e1ample4

77 draw lli#se$&.&ou'le

g$.draw(new lli#se$&.&ou'le(, , retwidth,retheight));



%ectanle

• Classes that speci* primitives represented inthe ollowin e1ample e1tend theetangularSha#e class, which implements the

Sha#e interace and adds a ew methods o itsown)

These methods ena'les *ou to et inormationa'out a shape>s location and si3e, to e1aminethe center point o a rectanle, and to set the

'ounds o the shape)



%ectanle

• The etangle$& class represents arectanle de.ned '* a location /1, *0and dimension /w 1 h0) Theetangle$&.Float andetangle$&.&ou'le su'classes speci* arectanle in ;oat and dou'le precision)#or e1ample4

77 draw etangle$&.&ou'leg$.draw(new etangle$&.&ou'le(, , retwidth,retheight));



%ounded %ectanle

• The oundetangle$& class represents arectanle with rounded corners de.ned '* alocation /1, *0, a dimension /w 1 h0, and the

width and heiht o the corner arc) Theoundetangle$&.Float andoundetangle$&.&ou'le su'classes speci* a

round rectanle in ;oat and dou'le precision)

S i i % d d



Speci*in %ounded%ectanle

• The rounded rectanle is speci.ed withollowin parameters4

– 6ocation– "idth

– ?eiht

– "idth o the corner arc

– ?eiht o the corner acr



%ounded %ectanle location

• To set the location, si3e, and arcs o aoundetangle$& o'(ect, use the methodsetoundet(dou'le a, dou'le , dou'le w,

dou'le h, dou'le arWidth, dou'le arHeight))#or e1ample4

77 draw oundetangle$&.&ou'le

g$.draw(new oundetangle$&.&ou'le(, ,

retwidth, retheight, 10, 10));

D i G t i



Drawin Geometric!rimitives /Contd 90

• The e1ample athttp4@@(ava)sun)com@docs@'ooks@tutorial@2d@eometr*@e1amples@ShapesDemo2D)(ava code

contains implementations o+ all the descri'edeometric primitives)

• #or more inormation a'out classes andmethods represented in this section see the

http4@@(ava)sun)com@(avase@@docs@api@(ava@awt@eom@packae&summar*)html speci.cation)



"hat is an Afne TransormB

• The Ane Transform class represents a2D Afne transorm that perorms a linearmappin rom 2D coordinates to other 2D

coordinates that preserves thestraihtness and parallelness o lines)Afne transormations can 'e constructedusin se-uences o translations, scales,

;ips, rotations, and shears)

"h t i f T



"hat is afne Transorm /contd90

• An Ane Transform is a linear transorm, so thetransormation can 'e e1pressed in the matri1 notationo linear ale'ra) An ar'itrar* Ane Transform can'e mathematicall* e1pressed '* si1 num'ers in a

matri1 like this4 sx shx tx

shy sy ty• The symols or factors sx and sy are scale factors that

accomplish scaling! The factors tx and ty are scalefactors that accomplish translation! The factors shx

and shy are scale factors that accomplish shear!

"hat is afne Transorm



"hat is afne Transorm /contd90

• 8our raphics can 'e transormed in theollowin wa*s 'eore 'ein displa*ed

on the output device4Scalin

Translation

Shear

%otation

/rotation is a com'ination o scalin and shearin, sothere are not separate rx and ry num'ers) 0





Translation

Translation moves the oriin o thecoordinate s*stem in device space)

Another use o translation /incom'ination with scalin0 is to ;ip thedeault positive direction o the verticala1is so that increasin positive values o

up instead o down, which is the deault)



Shear

• Jonathan Enudsen descri'es shear in his 'ookentitled Java 2D Graphics as4

FTake a stack o paper /like *ou miht place in

*our laser printer0 and draw a picture on theside o the stack) Then deorm the stack sothat the side o the stack on which *ou drewthe picture takes the orm o a paralleloram)e especiall* careul to keep the oppositesides parallel) Then look at *our picture) Itwill have 'een su'(ected to shear in onedimension)H



Afne Transorm class

• #u'li lass +**ine=rans*ormetends -'/etim#lements Clonea'le, Seriali4a'le

• The +**ine=rans*orm class represents a 2Dafne transorm that perorms a linearmappin rom 2D coordinates to other 2Dcoordinates that preserves the straihtnessand parallelness o lines) Afne

transormations can 'e constructed usinse-uences o translations, scales, ;ips,rotations, and shears)

Afne Transorms as matri1



Afne Transorms as matri1multiplications

• Such a coordinate transormation can 'erepresented '* a row '* column matri1 with animplied last row o < < 5 K) This matri1 transormssource coordinates /1, *0 into destinationcoordinates /1L, *L0 '* considerin them to 'e acolumn vector and multipl*in the coordinatevector '* the matri1 accordin to the ollowinprocess4

D E D m00 m01 m0$ D D m00 < m01 < m0$ D E 2 D m10 m11 m1$ D 2 D m10 < m11 < m1$

D 1 D 0 0 1 D 1 D 1



%otation

• %otation is eas* to visuali3e /'ut thecom'ination o rotation and shear can 'e ver*difcult to visuali3e0) To visuali3e rotation,

draw a picture on a piece o paper) Use asinle tack to attach it to a 'ulletin 'oard)

Then rotate the paper a ew derees aroundthe tack) 8ou will have rotated the picture

around the coordinates o the tack '* thespeci.ed num'er o derees)



%otation in Java 2D

• Java 2D works the same wa*) In the process otransormin a raphic rom user space todevice space, *ou can cause it to 'e rotated '*

a speci.ed anle around a speci.ed coordinateposition)

• ?owever, in Java 2D, the anle must 'especi.ed in radians instead o derees) I *ouare unamiliar with the use o radian measure

to speci* an anle, (ust remem'er theollowin identit*4



%adiand and Derees

• !I radians M 5N< derees

• !I is the numeric value *ou learned a'out ineometr* class)

• !I has a value o )5O5PQ))))))))))) ?owever, *oudon>t have to remem'er the value o !I) It isavaila'le as a static .nal constant in the Math class) 8ou can access the value as Math.PI)



Dividin !I %adians

• I this sounds conusin, *ou miht alsowant to remem'er the ollowin

identities4• !I@2 M Q< derees!I@O M OP derees!I@N M 22)P derees

!I@5 M 55)2P derees, etc)



%otation direction

• !ositive anles o rotation proceed in aclockwise direction) This ma* 'e

'ackwards rom what *ou areaccustomed to in various science andenineerin endeavors where counter&clockwise rotation is oten thouht o as

the positive direction o rotation)



Java #rame "orks & afne

• JavaLs A'stract "indow Toolkit /A"T0 providesa Graphics class with methods or drawinlines, ovals, rectanles, te1t, pol*ons, and

other shapes) A !anel, #rame, or other A"Tcomponent draws itsel usin the Graphicsinstance passed to its paint method) The GUIenvironment automaticall* calls this method

whenever the component needs redrawin)



Java #rame "orks $ afne

• Sun added the Java 2D ramework on top othe oriinal A"T Graphics class, addin man*new eatures to the Java core A!I) 7ne

important Java2D eature is known as theafne transormation) It works '* transorminthe coordinate s*stem 'eore an* drawin isdone on the Graphics o'(ect)



J#rame e1ample

• To illustrate this, weLll skew the coordinateswith a shearin transormation and draw somete1t and raphics to see how it looks with and

without the transormation) #irst, letLs drawsome te1t, usin the deault coordinates, "eLlldo this in the paint method o a GFrame4



=1ample /pae 50

im#ort /aa.awt.Font;im#ort /aa.awt.ra#his;im#ort /aa.awt.ra#his$&;im#ort /aa.swing.GFrame; #u'li lass @=rans*orm etends GFrame 5

#u'li @=rans*orm(String title) 5

su#er(title);6 #u'li oid #aint(ra#his g) 5 ra#his$& g$d 2 (ra#his$&) g;

g$d.setFont(Font.deode("arial?'old?"));g$d.drawString("+**init", 100, 100);

g$d.draw-al(100, 10, , 0); g$d.draw-al(1, 10, , 0);

6



=1ample /pae 20

#u'li stati oid main(StringD args) 5 @=rans*orm a*t 2 new @=rans*orm("+**init");a*t.setSi4e(00,00);

a*t.set&e*aultClose-#eration(GFrame.>I=A-:AC%-S);

a*t.setisi'le(true);6

6



Java afne #rameworks

• In all afne transormations , the chanea+ects the entire coordinate s*stem and not

(ust individual o'(ects) "e call the methods to

appl* the transorms 'eore callin thedrawin methods) 6etLs displa* the a'ove te1twith a shearin transorm, This is a transormwhere lines are shited in a new direction 'utparallel lines still remain parallel) The call to

the shear method happens 'eore the drawinoccurs) ?ere is code to perorm the transorm4



odi.cations to =1ample

#u'li oid #aint(ra#his g) 5

ra#his$& g$d 2 (ra#his$&) g;

g$d.shear(0.,0.);

g$d.setFont(Font.deode("arial?'old?"));g$d.drawString("+**init", 100, 100);g$d.draw-al(100, 10, , 0);

g$d.draw-al(1, 10, , 0);6



%eerences

http4@@(ava)sun)com@

http4@@www)developer)com@

http4@@www)particle)kth)se@htmlnew@

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