Chapter 4

7/17/2019 Chapter 4

http://slidepdf.com/reader/full/chapter-4-568ebe62bc42a 1/73

Computer GraphicsComputer Graphics

Chapter 3Chapter 3Graphics Output PrimitivesGraphics Output Primitives

Andreas SavvaAndreas Savva

7/17/2019 Chapter 4


2

Basic Raster AlgorithmsBasic Raster Algorithms

for 2D Primitivesfor 2D Primitives

Description of picturesSpecified b a set of intensities for the pi!el positions"

Describe it as a set of comple! ob#ects$ such as trees$furniture and %alls$ positioned at specific coordinate

locations %ithin the scene"Graphics programming pac&ages provide functions todescribe a scene in terms of basic geometric structuresreferred to as output primitives"

' Points

' Straight lines

' Circles

' Splines curves and surfaces

' Polgon color areas

' Character strings' (tc"

7/17/2019 Chapter 4


3

)mplementing Application programs)mplementing Application programs

Description of ob#ects in terms of primitives and

attributes and converts them to the pi!els on the

screen"Primitives * %hat%hat is to be generated

Attributes * ho%ho% primitives are to be generated

7/17/2019 Chapter 4


+

PointsPoints

,he electron beam is turned on to illuminate the phosphor

at the selected location -!$ . %here

/ 0 x 0 ma!!

/ 0 y 0 ma!

setpi!el-!$ $ intensit. * loads an intensit value into theframe1buffer at -!$ ."

getpi!el-!$ . * retrieves the current frame1buffer

intensit setting at position -!$ ."

-/$/.

-ma!!$ma!.

CR,

7/17/2019 Chapter 4


inesines

Analog devises$ such as a random1scan displa or avector plotter$ displa a straight line smoothl from one

endpoint to another" inearl varing hori4ontal and

vertical deflection voltages are generated that are

proportional to the re5uired changes in the ! and directions to produce the smooth line"

7/17/2019 Chapter 4


6

Digital devices displa a straight line b plotting discrete

coordinate points along the line path %hich are calculated

from the e5uation of the line"

Screen locations are referenced %ith integer values$ so plotted

positions ma onl appro!imate actual line positions bet%een t%o

specific endpoints"

A computed line position of -7/"+8$ 2/"7. %ill be converted to

pi!el position -7/$ 27." ,his rounding of coordinate values to

integers causes lines to be displaed %ith a stairstep appearance

-the 9 #aggies:."

Particularl noticeable on sstems %ith lo% resolution"

,o smooth raster lines$ pi!el intensities along the line paths must

be ad#usted"

7/17/2019 Chapter 4


;

ine Dra%ing Algorithmsine Dra%ing Algorithms

Cartesian e5uation<

y = mx + c

%herem * slope

c * 1intercept

x

y

x x

y ym

∆

∆=

−

−=

72

72

x7

y7

x2

y2

7/17/2019 Chapter 4


8

SlopeSlope

if =m= > 7

θ > +?

+?

+?

+ve -ve

θ?θ?

θ?

θ?

if =m= < 7

1+? @ θ @ +?

if =m= > 7

+? @ θ @ /? or

1/? @ θ @ 1+?

7/17/2019 Chapter 4


8

;

6

+

3

2 7

/

/ 7 2 3 + 6 ; 8

y = xm > 7c > / y

x

!

/ /

7 7

2 2

3 3

+ +

6 6

; ;

8 8

==mm= > 7= > 7

7/17/2019 Chapter 4


7/

8

;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8

y = x 7m >

c > 7 y

x

! round-.

/ 7 7

7 7" 2

2 2 2

3 2" 3

+ 3 3

3" +

6 + +

; +"

8

==mm== << 77

7/17/2019 Chapter 4


77

==mm== >> 77

8

;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8

y = 3x 1 2m > 3

c > 12 y

x

! round-.

/ 12 12

7 7 7

2 + +

3 ; ;

+ 7/ 7/

73 73

6 76 76

; 7 7

8 22 22outside

7/17/2019 Chapter 4


72

,he Digital Differential Anal4er,he Digital Differential Anal4er

-DDA. Algorithm-DDA. Algorithm

m x

y=

∆∆

!

/ 7

7 +

2 ;

3 7/+ 73

76

means that for a unit -7. change in x there is

m1change in y.

i"e" y > 3 x 7 m > 3

m y

x 7=

∆

∆means that for a unit -7. change in y there is

7m change in x.

Do not use y > 3 x 7

to calculate y"

Ese m

7/17/2019 Chapter 4


73

,he DDA Fethod,he DDA Fethod

Eses differential e5uation of the line < mm

)f slope =m= ≤ 7 then increment x in steps of 7 pi!el

and find corresponding y1values"

)f slope =m= > 7 then increment y in steps of 7 pi!el

and find corresponding x1values"

step through in x step through in y

7/17/2019 Chapter 4


7+

,he DDA Fethod,he DDA Fethod

Desired line

- xi$round- yi..

- xi$ yi.

- xi7$round- yim..

- xi7$ yim.

7/17/2019 Chapter 4


7

if slopeif slope mm ≥≥ //

if ==mm== ≤≤ 77

xiC7 > xi 7

yiC7 > yi m

if ==mm== >> 77

yiC7 > yi 7

xiC7 > xi 7m

Left

Right

Left

Right

7/17/2019 Chapter 4


76

Proceeding from right1endpoint to left1endpointProceeding from right1endpoint to left1endpoint

if slopeif slope mm ≥≥ //

if ==mm== ≤≤ 77

xiC7 > xi 1 7

yiC7 > yi 1 m

if ==mm== >> 77

yiC7 > yi 1 7

xiC7 > xi 1 7m

Left

Right

Left

Right

7/17/2019 Chapter 4


7;

if slopeif slope mm @ /@ /

if ==mm== ≤≤ 77

xiC7 > xi 7

yiC7 > yi m

if ==mm== >> 77

yiC7 > yi 1 7

xiC7 > xi 1 7m

Left

Right

Left

Right

7/17/2019 Chapter 4


78

Proceeding from right1endpoint to left1endpointProceeding from right1endpoint to left1endpoint

if slopeif slope mm << //

if ==mm== ≤≤ 77

xiC7 > xi 1 7

yiC7 > yi 1 m

if ==mm== >> 77

yiC7 > yi 7

xiC7 > xi 7m

Left

Right

Left

Right

7/17/2019 Chapter 4


7

(!ample -DDA.(!ample -DDA.

+=+=

≤≤

+=+

+

37

7

737 7

7/

7

ii

ii

y y x x

m

x y

! round-.

/ 7 77 +3 7

2 3 2

3 2 2

+ ;3 2

83 3

6 3 3

; 7/3 3

8 773 +

8;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8

y

x

7/17/2019 Chapter 4


2/

(!ample -DDA.(!ample -DDA.

−−=−=

−<

+−=+

+

.-7

7

83

37

7

7

ii

ii

x x y y

m

x y

y x round- x.

8 / /; 73 /

6 23 7

7 7

+ +3 7

3 3 2

2 2 2

7 ;3 2

/ 83 3

8 ;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8

y

x

7/17/2019 Chapter 4


27

void ineDDA-int !/$ int /$ int !7$ int 7. int d! > !7 * !/$ d > 7 * /$ stepsH

if -abs-d!.Iabs-d.. steps > abs-d!.H

else steps > abs-d.H one of these %ill be 7 or 17 double !)ncrement > -double.d! -double .stepsH double )ncrement > -double.d -double .stepsH

double ! > !/H

double > /H setPi!el-round-!.$ round-..H

for -int i>/H i@stepsH i. ! > !)ncrementH

> )ncrementH setPi!el-round-!.$ round-..H JJ

Kote Kote<< ,he DDA algorithm is faster than the direct use of y > mx c"

)t eliminates multiplicationH onl one addition"

7/17/2019 Chapter 4


22

(!ample(!ample

Dra% a line from point -2$7. to -72$6.

Dra% a line from point -7$6. to -77$/.

;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8 7/ 77 72

7/17/2019 Chapter 4


23

Bresenham ine AlgorithmBresenham ine AlgorithmA more efficient approach

Basis of the algorithm<

Lrom start position decide A or B ne!t

AA

BB

Start position

7/17/2019 Chapter 4


2+

Bresenham ine AlgorithmBresenham ine Algorithm

Lor a given value of xone pi!el lies at distance t i above the line$ andone pi!el lies at distance si belo% the line

True line

si

t i

7/17/2019 Chapter 4


2

Bresenham ine AlgorithmBresenham ine Algorithm

Decision parameter

d i = - si - t i.

)f d i < /$ then closest pi!el is belo% true line - si smaller.

)f d i ≥ /$ then closest pi!el is above true line -ti smaller.

Me must calculate the ne% values for d i as %e move along

the line"

7/17/2019 Chapter 4


26

3d2d

d

(!ample<(!ample<

.2or"/-i"e" /"-slope.gradientet dxdydx

dy<<<

T r u e l i n e

Start pi!el at - x/$ y7.

+d

AtAt x x 11 < s7 > dy t 7 > dx 1 dy

d 7 = - si - t i. > dy 1 -dx 1 dy. > 2dy 1 dx

but 2dy < dx d i < / y stas the same

hence ne!t pi!el is at - x7$ y7.

AtAt x x 22 < s2 > 2dy t 2 > dx 1 2dy

d 2 = - s2 – t 2. > 2dy 1 -dx 1 2dy. > +dy 1 dx

Suppose d2 ≥ / y is incremented


AtAt x x 33 < s3 > 3dy - dx t 2 > 2dx 1 3dy

d 3 = - s2 – t 3. > 6dy 1 3dx < /

so y stas the same


x7 x2 x3 x+ x

x/ y/

y7

y2

y3

y

7/17/2019 Chapter 4


2;

)n General)n General Lor a line %ith gradient 0 7d / > 2dy * dx

if d i < / then yi7 > yi

d i7 > d i 2dyif d i N / then yi7 > yi 7

d i7 > d i 2-dy * dx.

xi7 > xi 7

Lor a line %ith gradient > 7d / > 2dx * dy

if d i < / then xiC7 > xi

d iC7 > d i 2dx

if d i N / then xiC7 > xi 7d iC7 > d i 2-dx * dy.

yiC7 > yi 7

Kote< Lor =m= 0 7 the constants 2dy and 2-dy-dx. can be calculated once$

so the arithmetic %ill involve onl integer addition and subtraction"

7/17/2019 Chapter 4


28

(!ample(!ample * Dra% a line from -2/$7/. to -3/$78.

7

78

7;

76

7

7+

73

72

77

7/

2/ 27 22 23 2+ 2 26 2; 28 2 3/ 37 32

-2/$7/.

-3/$78.dx > 7/

dy > 8

initial decision d / > 2dy * dx > 6

Also 2dy > 76$ 2-dy * dx. > 1+

i d i ( x i +1, y i +1)

0 6 (21,11)

1 2 (22,12)

2 -2 (23,12)

3 14 (24,13)

4 10 (25,14)

5 6 (26,15)

6 2 (27,16)7 -2 (28,16)

8 14 (29,17)

9 10 (30,18)

7/17/2019 Chapter 4


2

void ineBres-int !/$ int /$ int !7$ int 7. // line for |m| < 1 int d! > abs-!7 * !/.$ d > abs-7 * /.H int d > 2 d * d!$ t%oD > 2 d$ t%oDFinusD! > 2 -d * d!.H int !$ H

if -!/ I !7. determines %hich point to use as start$ %hich as end ! > !7H > 7H !7 > !/H J else

! > !/H > /H J setPi!el-!$.H

%hile -! @ !7. !H if -d @ /. d > t%oDH else H d > t%oDFinusD!H J setPi!el-!$ .H

JJ

7/17/2019 Chapter 4


3/

Special casesSpecial cases

Special cases can be handled separatel

* ori4ontal lines -∆ > /.

* Qertical lines -∆! > /.

* Diagonal lines -=∆!= > =∆=.

directl into the frame1buffer %ithout

processing them through the line1plottingalgorithms"

7/17/2019 Chapter 4


37

Parallel ine AlgorithmsParallel ine Algorithms

,a&e advantage of multiple processors"

Given n p processors$ subdivide the line path into n p Bresenhamsegments"

Lor a line %ith slope / < m < 7 and leftpoint - x/$ y/. the distance to

the right endpoint -left endpoint for ne!t segment. is

%here

∆ x > %idth of the line

∆ x p is computed using integer division

Kumbering the segments$ and the processors$ as /$ 7$ 2$ $ n p17$

starting !1coordinate for the & th partition is

xk > x/ k ∆ x p

p

p p

nn x x 7−+∆=∆

7/17/2019 Chapter 4


32

Parallel ine Algorithms -continue.Parallel ine Algorithms -continue.

i"e" ∆ x > 7 $ n p > + processors

Starting x1values at x/$ x/ +$ x/ 8$ x/ 72

∆ y p > m∆ x p

At the k th segment$ the starting 1coordinates is

yk > y/ round-k ∆ y p.Also$ the initial decision parameter for Bresenhams

algorithm at the start of the & th subinterval is<

pk > -& ∆ x p.-2∆ y. * round-k ∆ y p.-2∆ x. 2∆ y * ∆ x

++

78

+

7+7==

−+=∆ p x

A i li i f i h li

7/17/2019 Chapter 4


33

Anti1aliasing for straight linesAnti1aliasing for straight lines

ines generated can have #agged or stair1step appearance$

one aspect of phenomenon called aliasing$ caused b factthat pi!els are integer coordinate points"

Ese anti1aliasing routines to smooth out displa of a line

b ad#usting pi!els intensities along the line path

7/17/2019 Chapter 4


3+

Another raster effectAnother raster effect

ine B

ine A

Both lines plotted %ith the same number of pi!els$ but

the diagonal line is longer than the hori4ontal line

Qisual effect is diagonal line appears less thic&"

)f the intensit of each pi!el is I $ then the intensit per unit

length of line A is I $ %hereas for line B it is onl I √2H this

discrepanc is easil detected b the vie%er"

7/17/2019 Chapter 4


3

Circle Generating AlgorithmsCircle Generating Algorithms

Circles and ellipses are common components in

man pictures"

Circle generation routines are often included in

pac&ages"

7/17/2019 Chapter 4


36

Circle (5uationsCircle (5uations

Polar form x > r Cosθ

y > r Sinθ -r > radius of circle.

P >-r Cosθ $ r Sinθ .

r Sin )

r Cos ) x

y

r

7/17/2019 Chapter 4


3;

Dra%ing a circleDra%ing a circle

DisadvantagesDisadvantages

,o find a complete circle θ varies from /? to

36/?

,he calculation of trigonometric functions

is ver slo%"

θ > /?%hile -θ @ 36/?.

x > r Cosθ

y = r Sinθ

setPixel(x,y)

θ = θ 7?end %hile

7/17/2019 Chapter 4


38

Cartesian formCartesian form Ese Pthagoras theorem

x2 + y2 = r 2

x

r y

y

x x

y

r

( )2 2$ P x r x= −

Ci l l i hCi l l ith

7/17/2019 Chapter 4


3

Circle algorithmsCircle algorithms Step through x1a!is to determine y-values

Disadvantages< * Kot all pi!el filled in * S5uare root function is ver slo%

7/17/2019 Chapter 4


+/

Circle AlgorithmsCircle Algorithms

Ese 81fold smmetr and onl compute pi!el positions for the +? sector"

45°

- x$ y.

- y$ x.

-1 x$ y.

- y$ -x.

- x$ 1 y.-1 x$ 1 y.

-1 y$ x.

--y$ -x.

7/17/2019 Chapter 4


+7

Bresenhams Circle AlgorithmBresenhams Circle Algorithm

eneral !rin"i#leeneral !rin"i#le

,he circle function<

and

Consider onl

+? 0 θ 0 /?

2 2 2- $ .circle f x y x y r = + −

if -!$. is inside the circle boundar

if -!$. is on the circle boundar

if -!$. is outside the circle boundar

/- $ . /

/

circle f x y<= =

>

7/17/2019 Chapter 4


+2


p7 p3

p2

D(si ) D(t i )

After point p7$ do %e choose p2 or p3T

yi

yi - 1

x i x i + 1

r

h i l l i h

7/17/2019 Chapter 4


+3


Define< D- si. > distance of p3 from circle D-t i. > distance of p2 from circle

i"e" D- si. > - xi 7.2

yi2

* r 2

Ual%as veV D-t i. > - xi 7.2 - yi * 7 )2 * r 2 Ual%as 1veV

Decision Parameter pi $ D%si ) + D%t i )so if pi @ / then the circle is closer to p3 -point above.

if pi N / then the circle is closer to p2 -point belo%.

,h Al ith,h Al ith

7/17/2019 Chapter 4


++

,he Algorithm,he Algorithm x & $ &

y& $ r

p& $ '12 + r 2 ( r 2 + '12 + %r -1)2 ( r 2 $ 3 ( 2r

if pi < 0 then

yi +1 = yi

pi+1 = pi + 4x i + *

else if pi ≥ 0 then

yi+1 = yi ( 1 pi +1 = pi + 4% x i – yi ) + 1&

Sto# ,en x i yi and determine s.mmetr.

#oints in t,e ot,er o"tants

x i +1 $ x i + 1

(!ample(!ample

7/17/2019 Chapter 4


+

(!ample(!ample

7/

8

;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8 7/

i pi xi$ i

/ 17; -/$ 7/.7 177 -7$ 7/.

2 17 -2$ 7/.

3 73 -3$ 7/.

+ 1 -+$ . 7 -$ .

6 -6$ 8.

; -;$;.

r > 7/

p/ > 3 * 2r > 17;

)nitial point - x/$ y/. > -/$ 7/.

7/17/2019 Chapter 4


+6

(!ercises(!ercises

Dra% the circle %ith r > 72 using the

Bresenham algorithm"

Dra% the circle %ith r > 7+ and center at

-7$ 7/."

7/17/2019 Chapter 4


+;

Decision ParametersDecision Parameters

Prove that if pi @ / and yiC7 > yi then

piC7 > pi + xi 6

Prove that if pi N / and yiC7 > yi * 7 then

piC7 > pi +- xi – yi. 7/

7/17/2019 Chapter 4


+8

Advantages of Bresenham circleAdvantages of Bresenham circle

Onl involves integer addition$ subtraction

and multiplication

,here is no need for s5uares$ s5uare rootsand trigonometric functions

id i Ci l Al i h

7/17/2019 Chapter 4


+

Fidpoint Circle AlgorithmFidpoint Circle Algorithm

yi yi17

xi xi7 xi2

Fidpoint

x2 y2 * r 2 > /

Assuming that %e have #ust plotted the pi!els at % x i yi )"Mhich is ne!tT % x i +1 .i) OR % x i +1 yi ( 1)0

- ,e one t,at is "loser to t,e "ir"le0

Fid i Ci l Al i h

7/17/2019 Chapter 4


/

Fidpoint Circle AlgorithmFidpoint Circle Algorithm

,he decision parameter is the circle at the midpoint

bet%een the pi!els yi and yi * 7"

)f pi @ /$ the midpoint is inside the circle and the pi!el

yi is closer to the circle boundar" )f pi N /$ the midpoint is outside the circle and the pi!el

yi 1 7 is closer to the circle boundar"

72

2 2 272

- 7$ .

- 7. - .

i circle i i

i i

p f x y

x y r

= + −

= + + − −

D i i PD i i P t

7/17/2019 Chapter 4


7

Decision ParametersDecision Parameters

Decision Parameters are obtained using

incremental calculations

OR

%here yi7 is either yi or yi17 depending on the sign of pi

77 7 7 2

2 2 27

7 2

- 7$ .

- 2. - .

i circle i i

i i

p f x y

x y r

+ + +

+

= + −

= + + − −

2 2 2

7 7 72- 7. - . - . 7i i i i i i i p p x y y y y+ + += + + + − − − +

Kote<

xi7 > xi 7

,h Al ith,h Al ith7 )nitial values< point-/ r.

7/17/2019 Chapter 4


2

,he Algorithm,he Algorithm7" )nitial values<1 point-/$r .

x/ > /

y/ > r

2" )nitial decision parameter

3" At each xi position$ starting at i > /$ perform the

follo%ing test< if pi @ /$ the ne!t point is - xi 7$ yi. and

pi7 > pi 2 xi7 7

)f pi N /$ the ne!t point is - xi7$ yi17. and

pi7 > pi 2 xi7 7 * 2 yi7

%here 2 xi7 > 2 xi 2 and 2 yi7 > 2 yi * 2+" Determine smmetr points in the other octants

" Fove pi!el positions - x,y. onto the circular path centered

on - xc , yc. and plot the coordinates< x > x xc$ y > y yc

6" Repeat 3 * until x N y

move circle origin at -/$/. b x > x * xc and y > y * yc

2 2 7 7/ 2 2 +

-7$ . 7 - .circle p f r r r r = − = + − − = −

(!ample(!ample

7/17/2019 Chapter 4


3

(!ample(!ample

7/

8

;

6

+

3

2

7

/

/ 7 2 3 + 6 ; 8 7/

i pi xi7$ i7 2 xi7 2 yi7

/ 1 -7$ 7/. 2 2/

7 16 -2$ 7/. + 2/

2 17 -3$ 7/. 6 2/

3 6 -+$ . 8 78

+ 13 -$ . 7/ 78

8 -6$ 8. 72 76

6 -;$ ;.

r > 7/

p/ > 7 * r > 1 -if r is integer round p/ > + * r to integer.

)nitial point - x/$ y/. > -/$ 7/.

7/17/2019 Chapter 4


+

(!ercises(!ercises

Dra% the circle %ith r > 72 using the

Fidpoint1circle algorithm

Dra% the circle %ith r > 7+ and center at

-7$ 7/."

7/17/2019 Chapter 4


(!ercises(!ercises

Prove that if pi @ / and yiC7 > yi then

piC7 > pi 2 xiC7 7

Prove that if pi N / and yiC7 > yi17 then

piC7 > pi 2 xiC7 7 * 2 yiC7

Fidpoint functionFidpoint function

7/17/2019 Chapter 4


6

Fidpoint functionFidpoint functionvoid plotpoints(int x, int y){

setpixel(xcenter+x, ycenter+y);

setpixel(xcenter-x, ycenter+y);setpixel(xcenter+x, ycenter-y);setpixel(xcenter-x, ycenter-y);setpixel(xcenter+y, ycenter+x);setpixel(xcenter-y, ycenter+x);setpixel(xcenter+y, ycenter-x);setpixel(xcenter-y, ycenter-x);

}

void circle(int r){

int x = 0, y = r; plotpoints(x,y);int p = 1 – r;

while (x<y) {

x++;if (p<0) p += !x + 1;else {y--;

p += !(x-y) + 1;}

plotpoints(x,y);

}}

(llipse Generating Algorithms(llipse Generating Algorithms

7/17/2019 Chapter 4


;

(llipse1Generating Algorithms(llipse1Generating Algorithms

lli#selli#se * A modified circle %hose radius varies from a

ma!imum value in one direction -ma#or a!is. to a minimumvalue in the perpendicular direction -minor a!is."

P=(x,y) F 7

F 2

d 7

d 2

,he sum of the t%o distances d 7 and d 2$ bet%een the fi!ed positions F 7 and F 2 -called the foci of the ellipse. to an point P on the ellipse$ is the same value$ i"e"

d 1 + d 2 $ "onstant

(llipse Properties(llipse Properties

7/17/2019 Chapter 4


8

(llipse Properties(llipse Properties

(!pressing distances d 7 and d 2 in terms of the focal

coordinates F 7 > - x7$ x2. and F 2 > - x2$ y2.$ %e have<

Cartesian coordinates<

Polar coordinates<

2 2 2 2

7 7 2 2- . - . - . - . constant x x y y x x y y− + − + − + − =

r yr x

22

7c c

x y

x x y yr r

− −+ = ÷ ÷ ÷

cos

sin

c x

c y

x x r

y y r

θ

θ

= +

= +

7/17/2019 Chapter 4


(llipse Algorithms(llipse Algorithms

Smmetr bet%een 5uadrants Kot smmetric bet%een the t%o octants of a 5uadrant

,hus$ %e must calculate pi!el positions along the

elliptical arc through one 5uadrant and then %e obtain

positions in the remaining 3 5uadrants b smmetr

- x$ y.-1 x$ y.

- x$ 1 y.-1 x$ 1 y.

r x

r y


7/17/2019 Chapter 4


6/


Decision parameter<

2 2 2 2 2 2- $ .ellipse y x x y

f x y r x r y r r = + −

7

Slope > 17

r x

r y 2

/ if - $ . is inside the ellipse

- $ . / if - $ . is on the ellipse/ if - $ . is outside the ellipse

ellipse

x y

f x y x y x y

<

= =>

2

2

2

2

y

x

r xdylope

dx r y= = −

(lli Al ith(lli Al ith Slope 7

7/17/2019 Chapter 4


67


Starting at -/$ r y. %e ta&e unit steps in the x direction until

%e reach the boundar bet%een region 7 and region 2"

,hen %e ta&e unit steps in the y direction over the

remainder of the curve in the first 5uadrant"

At the boundar

therefore$ %e move out of region 7 %henever

7

Slope > 17

r x

r y 2

2 2

7 2 2 y x

dy

r x r ydx = − ⇒ =

2 22 2 y xr x r y≥

Fid i t (lli Al ithFid i t (lli Al ith

7/17/2019 Chapter 4


62

Fidpoint (llipse AlgorithmFidpoint (llipse Algorithm

yi

yi17

xi xi7 xi2

Fidpoint

Assuming that %e have #ust plotted the pi!els at % x i yi )"

,he ne!t position is determined b<

72

2 2 2 2 2 272

7 - 7$ .

- 7. - .

i ellipse i i

y i x i x y

p f x y

r x r y r r

= + −= + + − −

)f p7i @ / the midpoint is inside the ellipse ⇒ yi is closer

)f p7i N / the midpoint is outside the ellipse ⇒ yi * 7 is closer

i i - i .D i i P -R i 7.

7/17/2019 Chapter 4


63

Decision Parameter -Region 7.Decision Parameter -Region 7.

At the ne!t position U xi7 7 > xi 2V

%here yi7 > yi

or yi7 > yi * 7

77 7 7 2

2 2 2 2 2 27

7 2

7 - 7$ .

- 2. - .

i ellipse i i

y i x i x y

p f x y

r x r y r r

+ + +

+

= + −

= + + − −

2 2 2 2 2 27 77 7 2 2

7 7 2 - 7. - . - .i i y i y x i i p p r x r r y y+ + = + + + + − − −


7/17/2019 Chapter 4


6+

Decision Parameter -Region 7.Decision Parameter -Region 7.Decision parameters are incremented b<

Ese onl addition and subtraction b obtaining

At initial position %& r y)

2 2

7

2 2 2

7 7

2 if 7 /

2 2 if 7 /

y i y i

y i y x i i

r x r p

increment r x r r y p

+

+ +

+ <= + − ≥

2 2

2 and 2 y xr x r y

2

2 2

2 2 2 2 27 7/ 2 2

2 2 27

+

2 /

2 2

7 -7$ . - .

y

x x y

ellipse y y x y x y

y x y x

r x

r y r r

p f r r r r r r

r r r r

=

=

= − = + − −

= − +

Region 2Region 2

7/17/2019 Chapter 4


6

Region 2Region 2Over region 2$ step in the negative direction and midpoint ista&en bet%een hori4ontal pi!els at each step"

yi

yi17

xi xi7 xi2

Fidpoint

Decision parameter<

72

2 2 2 2 2 272

2 - $ 7.

- . - 7.

i ellipse i i

y i x i x y

p f x y

r x r y r r

= + −= + + − −

)f p2i I / the midpoint is outside the ellipse ⇒ x i is closer

)f p2i 0 / the midpoint is inside the ellipse ⇒ x i 7 is closer

D i i P -R i 2.D i i P t -R i 2.

7/17/2019 Chapter 4


66


At the ne!t position U yi7 * 7 > yi * 2V

%here xi7 > xi

or xi7 > xi 7

77 7 72

2 2 2 2 2 27

7 2

2 - $ 7.

- . - 2.

i ellipse i i

y i x i x y

p f x y

r x r y r r

+ + +

+

= + −

= + + − −

2 2 2 2 27 77 7 2 2

2 2 2 - 7. - . - .i i x i x y i i p p r y r r x x+ + = − − + + + − +


7/17/2019 Chapter 4


6;


Decision parameters are incremented b<

At initial position % x & y&) is ta&en at the last position

selected in region 7

2 2

7

2 2 2

7 7

2 if 2 /

2 2 if 2 /

x i x i

y i x i x i

r y r pincrement

r x r y r p

+

+ +

− + >=

− + ≤

7/ / /2

2 2 2 2 2 27/ /2

2 - $ 7.

- . - 7.

ellipse

y x x y

p f x y

r x r y r r

= + −

= + + − −


7/17/2019 Chapter 4


68

Fidpoint (llipse Algorithmp p g

7" )nput r x$ r y$ and ellipse center - xc$ yc.$ and obtain the first

point on an ellipse centered on the origin as

- x/$ y/. > -/$ r y.

2" Calculate the initial parameter in region 7 as

3" At each xi position$ starting at i > /$ if p7i @ /$ the ne!t

point along the ellipse centered on -/$ /. is - xi 7$ yi. and

other%ise$ the ne!t point is - xi 7$ yi * 7. and

and continue until

2 2 27/ +

7 y x y x p r r r r = − +

2 2

7 77 7 2i i y i y p p r x r + += + +

2 2 2

7 7 77 7 2 2i i y i x i y p p r x r y r + + += + − +

2 2

2 2 y xr x r y≥


7/17/2019 Chapter 4


6

p p gp p g+" - x/$ y/. is the last position calculated in region 7" Calculate the

initial parameter in region 2 as

" At each yi position$ starting at i > /$ if p2i I /$ the ne!t point

along the ellipse centered on -/$ /. is - xi$ yi * 7. and

other%ise$ the ne!t point is - xi 7$ yi * 7. and

Ese the same incremental calculations as in region 7" Continue

until y > /"

6" Lor both regions determine smmetr points in the other three

5uadrants"

;" Fove each calculated pi!el position -!$ . onto the elliptical

path centered on - xc$ yc. and plot the coordinate values

x $ x + x c y $ y + yc

2 2 2 2 2 27

/ / /22 - . - 7. y x x y p r x r y r r = + + − −

2 2

7 72 2 2i i x i x p p r y r + += − +

2 2 2

7 7 72 2 2 2i i y i x i x p p r x r y r + + += + − +

(!ample(!ample

7/17/2019 Chapter 4


;/

pp

i pi xi7$ i7 2r y2 xi7 2r x

2 yi7

/ 1332 -7$ 6. ;2 ;68

7 122+ -2$ 6. 7++ ;68

2 1++ -3$ 6. 276 ;68

3 2/8 -+$ . 288 6+/

+ 17/8 -$ . 36/ 6+/

288 -6$ +. +32 72

6 2++ -;$ 3. /+ 38+

r x $ r y $ *

2r y2 x $ & -%ith increment 2r y

2 > ;2.

2r x

2

y $ 2r x

2

r y -%ith increment 12r x2

> 1728.egion 1egion 1

% x & y&) $ %& *)2 2 27

/ +7 332 y x y x p r r r r = − + = −

Fove out of region 7 since

2r y2

x 6 2r x

2

y

(!ample(!ample

7/17/2019 Chapter 4


;7

pp

6

+ 3

2

7

/

/ 7 2 3 + 6 ; 8

i pi xi7$ i7 2r y2 xi7 2r x

2 yi7

/ 177 -8$ 2. ;6 26

7 233 -8$ 7. ;6 728

2 ;+ -8$ /. 1 1

Region 2Region 2

% x & y&) $ %7 3) -ast position in region 7.

7/ 22 -; $2. 77ellipse p f = + = −

Stop at y > /

7/17/2019 Chapter 4


;2

(!ercises(!ercises

Dra% the ellipse %ith r x > 6$ r y > 8"

Dra% the ellipse %ith r x > 7/$ r y > 7+"

Dra% the ellipse %ith r x > 7+$ r y > 7/ and

center at -7$ 7/."

Fidpoint (llipse LunctionFidpoint (llipse Lunction

7/17/2019 Chapter 4


dpo t pse u ct op pvoid ellipse(int "x, int "y){

int "x = "x ! "x, "y = "y ! "y;int two"x = ! "x, two"y = "y ! "y;int p, x = 0, y = "y;

int px = 0, py = two"x ! y;

ellise#lot#oints(xcenter, ycenter, x, y);$$ "e%ion 1

p = ro&nd("y – ("x ! "y) + (0' ! "x)); while (px < py) {x++;

px += two"y;if (p < 0) p += "y + px;

else {y--; py -= two"x; p += "y + px – py;}ellise#lot#oints(xcenter, ycenter, x, y);}$$ "e%ion

p = ro&nd("y ! (x+0') ! (x+0') + "x ! (y-1)!(y-1) – "x ! "y; while (y 0) {

y--; py -= two"x;if (p 0) p += "x – py;else {x++;

px += two"y; p += "x – py + px;}

Date post:	08-Jan-2016
Category:	Documents
Upload:	abishek-purushothaman
View:	6 times
Download:	0 times

Chapter 4

Documents