Bai Giang Ky Thuat Do Hoa

HC VIN CNG NGH BU CHNH VIN THNG ---------------------------------------

NGUYN THU THO

TI LUN VN I MI C CH K HOCH SN XUT KINH DOANH TI BU IN TRUNG NG TRONG GIAI ON 2014 2020

Chuyn ngnh: Qun tr kinh doanh M s: 60.34.01.02

TM TT LUN VN THC S

H NI NM 2014

PTIT

tttvTypewriter

tttvTypewriter

tttvTypewriter

tttvTypewriter

Lun vn c hon thnh ti: HC VIN CNG NGH BU CHNH VIN THNG

Ngi hng dn khoa hc: TS. Cao Th Thin Thu

Phn bin 1: TS. Nguyn Thanh Tng

Phn bin 2: TS. Nguyn Vn Tn

Lun vn s c bo v trc Hi ng chm lun vn thc s ti Hc vin Cng ngh Bu chnh Vin thng Vo lc: 13 gi 30 ngy 15 thng 2 nm 2014

C th tm hiu lun vn ti: - Th vin ca Hc vin Cng ngh Bu chnh Vin thng

PTIT

tttvTypewriter

tttvTypewriter

LI NI U

ho my tnh (Computer Graphics) l mt lnh vc l th v c nhiu ng dng

trong thc t, n gp phn lm cho giao tip gia con ngi v my tnh tr nn thn

thin hn. Giao din kiu vn bn (text) c thay th hon ton bng giao din

ho. Tuy nhin, vic dy v hc k thut ha th khng n gin do ch ny c

nhiu phc tp. K thut ha lin quan n tin hc v ton hc bi v hu ht cc gii

thut v, t cng cc php bin hnh u c xy dng da trn nn tng ca hnh hc

khng gian hai chiu v ba chiu. Hin nay, K thut ha l mt mn hc c ging

dy cho sinh vin chuyn ngnh Cng Ngh Thng Tin.

Trong cun gio trnh ny, ti mun mang li cho bn c cc c s l thuyt v

ho my tnh t n gin nht nh cc thut ton v ng thng, ng trn, a gic, k

t... Tip n cc k thut xn ta, cc php bin i ho trong khng gian 2D v

3D...Chng ta ln lt lm quen vi th gii mu sc thng qua cc h mu: RGB,

CMYK, HSV.... Phc tp hn na l cc php chiu, cc phng php xy dng ng

cong v mt cong cho i tng. Cui chng ta tm hiu v nh sng v hnh hc fractal.

Gio trnh gm chn chng, trong chng mt gip bn c c ci nhn tng

quan v k thut ho t trc n gi cng nh hng tng lai cho lnh vc ny. Cc

chng tip theo, mi chng s l mt vn t n gin n phc tp v c s nn tng

cho ngnh k thut ho. Cui mi chng u c phn bi tp kim tra li kin thc

va c c. Bi tp gm hai dng: dng tnh ton v dng lp trnh, i vi dng lp

trnh bn c th vit bng C/C++ hay BC thm ch bng VB u c. Cui cng l phn

ph lc gm cc hng dn lm bi tp lp trnh, ngn ng hay dng y l C/C++ hay

BC.

B cc r rng, hnh nh phong ph, a dng. Ti hy vng rng gio trnh l mt b

tham kho y cc thng tin hu ch v c tnh thc tin cao cho mn k thut ho.

Trong qu trnh bin son mc d c gng ht sc nhng vn khng trnh khi

nhng sai st, rt mong nhn c s ng gp chn thnh t qu bn c.

Xin chn thnh cm n.

Tc gi

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MC LC

LI NI U ........................................................................................................................ 1

MC LC .............................................................................................................................. 2

CHNG 1: TNG QUAN V K THUT HO ...................................................... 7

1.1. CC KHI NIM TNG QUAN CA K THUT HO MY TNH

(COMPUTER GRAPHICS) ......................................................................................................... 7

1.1.1. L ch s pht trin ................................................................................................. 7

1.1.2. K thut ha vi tnh. ...................................................................................... 8

1.2. CC K THUT HO ....................................................................................... 8

1.2.1. K thut ho im (Sample based-Graphics) ................................................. 8

1.2.2. K thut ho vector .......................................................................................... 9

1.2.3. Phn loi ca ho my tnh ........................................................................... 10

1.2.4. Cc ng dng tiu biu ca k thut ha ....................................................... 11

1.2.5. Cc chun giao din ca h ho ..................................................................... 13

1.3. PHN CNG HO (GRAPHICS HARDWARE) ............................................ 13

1.3.1. Cc thnh phn phn cng ca h ho tng tc........................................... 13

1.3.2. My in ................................................................................................................. 14

1.3.3. Mn hnh CRT .................................................................................................... 14

1.3.4. Mn hnh tinh th lng (Liquid Crystal Display LCD) ................................... 16

Tm tt chng: ............................................................................................................... 17

Bi tp: .............................................................................................................................. 18

CHNG 2: CC GII THUT SINH THC TH C S ............................................ 19

2.1. CC H THNG TO TRONG HO....................................................... 19

2.1.1. H to thc (WCS World Coordinate System) ........................................... 19

2.1.2. H to thit b (DCS Device Coordinate System) ...................................... 19

2.1.3. to thit b chun (NDCS Normalized Device Coordinate System) ........... 20

2.2. IM V ON THNG ....................................................................................... 20

2.2.1. im ................................................................................................................... 20

2.2.2. on thng .......................................................................................................... 20

2.3. CC GII THUT XY DNG THC TH C S ........................................... 21

2.3.1. Gii thut v on thng thng thng .............................................................. 21

2.3.2. Gii thut Bresenham ......................................................................................... 22

2.3.3. Gii thut trung im-Midpoint .......................................................................... 23

2.3.3. Gii thut sinh ng trn (Scan Converting Circles)(Bresenham) ................... 25

2.3.5. Gii thut sinh ng trn Midpoint .................................................................. 28

2.3.6. Gii thut sinh ng ellipse ............................................................................. 30

2.3.7. Gii thut sinh k t .......................................................................................... 33

2.3.8. Gii thut sinh a gic (Polygon) ....................................................................... 34

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Tm tt chng: ............................................................................................................... 39

Bi tp: .............................................................................................................................. 39

CHNG 3: CC PHP BIN I HO .................................................................. 41

3.1. CC PHP BIN I HNH HC HAI CHIU .................................................... 41

3.1.1. Php bin i Affine (Affine Transformations) ................................................. 41

3.1.2. Cc php bin i i tng ............................................................................... 41

3.2. TA NG NHT V CC PHP BIN I ............................................. 45

3.2.1. To ng nht ................................................................................................ 45

3.2.2. Php bin i vi to ng nht .................................................................... 46

3.2.3. Ci t c/c++ cho php quay tam gic quanh 1 im (xq,yq): ........................... 47

3.3. CC PHP BIN I HNH HC BA CHIU ...................................................... 48

3.3.1.Biu din im trong khng gian 3 chiu ............................................................ 48

3.3.2. Php tnh tin ...................................................................................................... 48

3.3.3. Php t l ............................................................................................................. 48

3.3.4. Php bin dng.................................................................................................... 49

3.3.5. Php ly i xng ............................................................................................... 49

3.3.6. Php quay 3 chiu ............................................................................................... 49

3.3.7. Ci t bng c/c++ nh sau: ............................................................................... 53

Tm tt:............................................................................................................................. 54

Bi tp: .............................................................................................................................. 54

CHNG 4: CC GII THUT HO C S........................................................... 57

4.1. M HNH CHUYN I GIA BA H THNG TO ................................. 57

4.1.1. M hnh chuyn i ............................................................................................ 57

4.1.2. Php nh x t ca s vo khung nhn ............................................................... 57

4.2. CC GII THUT XN TI (CLIPPING) ............................................................ 59

4.2.1. Khi nim ........................................................................................................... 59

4.2.2. Clipping im ..................................................................................................... 59

4.2.3. Xn ta on thng .............................................................................................. 59

4.2.4. Gii thut xn ta a gic (Sutherland Hodgman) .............................................. 66

Tm tt chng: ............................................................................................................... 70

Bi tp: .............................................................................................................................. 70

CHNG 5: PHP CHIU PROJECTION ...................................................................... 71

5.1. KHI NIM CHUNG ............................................................................................... 71

5.1.1.Nguyn l v 3D (three-Dimension) ................................................................... 71

5.1.2. c im ca k thut ho 3D ....................................................................... 71

5.1.3.Cc phng php hin th 3D .............................................................................. 71

5.2.PHP CHIU ............................................................................................................. 72

5.3. PHP CHIU SONG SONG (Parallel Projections ) ................................................. 74

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5.3.1.Php chiu trc giao (Orthographic projection) .................................................. 74

5.3.2. Php chiu trc lung (Axonometric) ................................................................ 75

5.3.3. Php chiu xin - Oblique .................................................................................. 78

5.4. PHP CHIU PHI CNH (Perspective Projection) .............................................. 79

5.4.1. Php chiu phi cnh mt tm chiu .................................................................. 80

5.4.2. Php chiu phi cnh hai tm chiu ................................................................... 81

5.4.3. Php chiu phi cnh ba tm chiu .................................................................... 83

Tm tt chng: ............................................................................................................... 83

Bi tp: .............................................................................................................................. 83

CHNG 6: MU SC TRONG HO ...................................................................... 85

6.1. NH SNG V MU SC (light and color) .......................................................... 85

6.1.1. Quan nim v nh sng ....................................................................................... 85

6.1.2. Yu t vt l ....................................................................................................... 85

6.1.3. Cm nhn mu sc ca con ngi (Physiology - Sinh l - Human Vision) ....... 87

6.1.4. Cc c trng c bn ca nh sng ..................................................................... 89

6.2. NH SNG N SC ............................................................................................ 89

6.2.1. Cng sng v cch tnh ................................................................................ 90

6.2.2. Php hiu chnh gama ......................................................................................... 90

6.2.3. Xp x bn tng - halftone .................................................................................. 91

6.2.4. Ma trn Dither v php ly xp x bn tng ....................................................... 93

6.3. CC H MU TRONG MN HNH HA ...................................................... 93

6.3.1. M hnh mu RGB (Red, Green, Blue - , lc, lam) ........................................ 94

6.3.2. M hnh mu CMY (Cyan, Magenta, Yellow - xanh tm, ti, vng) ......... 94

6.3.3. M hnh mu YIQ .............................................................................................. 95

3.4. M hnh mu HSV (Hue, Saturation,Value) - M thut........................................ 96

6.3.5. Biu mu CIE (1931 Commission Internationale de lEclairage) .............. 97

6.4. CHUYN I GIA CC H MU ................................................................... 100

6.4.1. Chuyn i HSV - RGB ................................................................................... 100

6.4.2. Chuyn i RGB sang XYZ ............................................................................. 101

Tm tt:........................................................................................................................... 102

Bi tp: ............................................................................................................................ 102

CHNG 7: NG CONG V MT CONG TRONG 3D ......................................... 104

7.1. NG CONG - CURVE ..................................................................................... 104

7.1.1. im biu din ng cong (curve represents points ) .................................. 104

7.1.2. ng cong a thc bc ba tham bin ............................................................. 104

7.1.3. ng cong Hermite ........................................................................................ 105

7.1.4. ng cong Bezier ........................................................................................... 106

7.1.5. ng cong B-spline ........................................................................................ 108

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7.2. M HNH B MT (Surface) V CC PHNG PHP XY DNG ............ 114

7.2.1. Cc khi nim c bn ....................................................................................... 114

7.2.2. Biu din mnh t gic ..................................................................................... 115

7.2.3. M hnh ho cc mt cong (Surface Patches) ................................................... 117

7. ................................................................................................................................. 117

7.2.4. Mt t cc ng cong ..................................................................................... 120

Tm tt:........................................................................................................................... 125

Bi tp: ............................................................................................................................ 125

CHNG 8: NH SNG ................................................................................................. 127

8.1. GII THIU ............................................................................................................ 127

8.1.1. Mc tiu chnh trong ha my tnh .............................................................. 127

8.1.2. Cc gii php trong ha my tnh ................................................................ 127

8.2. CC K THUT CHIU SNG TRONG HA MY TNH ....................... 129

8.2.1. nh gi v cng nh sng ........................................................................ 129

8.2.2. Cng nh sng ........................................................................................... 130

8.2.3. Nhng thuc tnh bao quanh ca vt cht ........................................................ 131

8.2.4. Thuc tnh khuch tn ca vt cht .................................................................. 132

8.2.5. S tng tc b mt/nh sng ........................................................................... 133

8.2.6. S khc x v s truyn sng ........................................................................... 134

8.3. CC CNG NGH ................................................................................................. 134

8.3.1. Raytracing ......................................................................................................... 134

8.3.2. Radiosity ........................................................................................................... 138

8.3.3. Photon Mapping ............................................................................................... 143

8.4. S SO SNH GIA CC K THUT (COMPARISON OF TECHNIQUES) .. 147

8.4.1. Raytracing ......................................................................................................... 148

8.4.2. Radiosity ........................................................................................................... 148

8.4.3. Photon mapping ................................................................................................ 148

Tm tt:........................................................................................................................... 149

CHNG 9: HNH HC FRACTAL ............................................................................... 150

9.1. S RA I V CC NG DNG CA HNH HC PHN HNH .................. 150

9.1.1. S ra i ca l thuyt hnh hc phn hnh ...................................................... 150

9.1.2. Cc ng dng tng qut ca hnh hc phn hnh .............................................. 151

9.2. MT S K THUT CI T HNH HC PHN HNH ................................. 151

9.2.1 H ng VONKOCK ...................................................................................... 151

9.2.2. ng SIERPINSKI ........................................................................................ 154

9.3. CY FRACTAL...................................................................................................... 155

9.3.1. CC CY THC T: ..................................................................................... 155

9.3.2. BIN DIN TON HC CA CY: ............................................................. 156

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9.4. TP MANDELBROT ............................................................................................. 159

9.4.1. t vn ......................................................................................................... 159

9.4.2. CNG THC TON HC ............................................................................. 159

9.4.3. THUT TON TH HIN TP MANDELBROT ........................................ 160

9.5. TP JULIA.............................................................................................................. 161

9.5.1. t vn : ........................................................................................................ 161

9.5.2. Cng thc ton hc: .......................................................................................... 161

9.5.3. Thut ton th hin tp Julia ............................................................................. 161

9.6. H CC NG CONG PHOENIX .................................................................... 163

Bi tp ............................................................................................................................. 165

Chng 10: OpenGL .......................................................................................................... 166

10.1. Gii thiu v OpenGL ........................................................................................... 166

10.1.1. Khi nim ....................................................................................................... 166

10.1.2. Ci t OpenGL trong Visual C++ ................................................................. 166

10.2. Cc thnh phn c bn ca OpenGL ..................................................................... 166

10.2.1. Chng trnh u tin ..................................................................................... 166

10.2.2. V hnh trong Window ................................................................................... 167

10.2.3. Khung nhn ..................................................................................................... 168

10.3. V cc i tng hnh hc c bn trong OpenGL ................................................. 169

10.3.1. V im, ng v a gic (point, line and polygon) .................................... 169

10.4. Php bin i im nhn v bin i m hnh (Viewing and Modeling

transformations) ........................................................................................................................ 173

10.4.1. Php bin i im nhn ................................................................................. 173

10.4.2. Php bin i m hnh .................................................................................... 173

10.4.3. kt hp cc php bin i ............................................................................... 174

10.5. Php chiu phi cnh v php chiu trc giao (Perspective and Orthographic

Projection) ................................................................................................................................ 174

10.5.1. Php chiu phi cnh ...................................................................................... 174

10.5.2. Php chiu trc giao ....................................................................................... 175

PH LC ........................................................................................................................... 177

1. Yu cu ....................................................................................................................... 177

2. Khi to v ng ch ho ................................................................................. 177

3. Cc hm c bn .......................................................................................................... 178

3.1. Bng mu ca mn hnh ho. .......................................................................... 178

3.2. im .................................................................................................................... 179

3.3. ng .................................................................................................................. 179

3.4. Hnh ch nht ...................................................................................................... 179

3.5. Hnh trn .............................................................................................................. 179

3.6. a gic ................................................................................................................. 180

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3.7. Vn bn ................................................................................................................ 180

3.8. Ca s (viewport) ................................................................................................ 181

3.9. To hnh nh chuyn ng ................................................................................... 181

Cc code chng trnh v d cho bi tp lp trnh .......................................................... 183

Bi 1: quay i tng ................................................................................................. 183

Bi 2: xn ta ............................................................................................................... 190

Bi 3: Php chiu ........................................................................................................ 191

TI LIU THAM KHO .................................................................................................. 185

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Chng 1: Tng quan v k thut ho

7

CHNG 1: TNG QUAN V K THUT HO

1.1. CC KHI NIM TNG QUAN CA K THUT HO MY

TNH (COMPUTER GRAPHICS)

1.1.1. L ch s pht trin

Lch s ca ha my tnh l vo thp nin 1960 c nh du bi d n SketchPad

c pht trin ti Hc vin Cng ngh Massachusetts (MIT) bi Ivan Sutherland. Cc

thnh tu thu c c bo co ti hi ngh Fall Joint Computer v y cng chnh l

s kin ln u tin ngi ta c th to mi, hin th v thay i c d liu hnh nh

trc tip trn mn hnh my tnh trong thi gian thc. H thng Sketchpad ny c dng

thit k h thng mch in v bao gm nhng thnh phn sau:

CRT mn hnh

Bt sng v mt bn phm bao gm cc phm chc nng

My tnh cha chng trnh x l cc thng tin

Vi h thng ny, ngi s dng c th v trc tip cc s mch in ln mn

hnh thng qua bt sng, chng trnh s phn tch v tnh ton cc thng s cn thit ca

mch in do ngi dng v nn.

Cng trong nm 1960 ny, William Fetter nh khoa hc ngi M. ng ang

nghin cu xy dng m hnh bung li my bay cho hng Boeing ca M. ng da trn

hnh nh ba chiu ca m hnh ngi phi cng trong bung li ca my bay xy dng

nn mt m hnh ti u cho bung li my bay. Phng php ny cho php cc nh thit

k quan st mt cch trc quan v tr ca ngi li trong khoang. ng t tn cho phng

php ny l ho my tnh (Computer Graphics) .

Mn hnh l thit b thng dng nht trong h ho, cc thao tc ca hu ht cc

mn hnh u da trn thit k ng tia m cc CRT (Cathode ray tube).

K thut ha c lin tc hon thin vo thp nin 1970 vi s xut hin ca

cc chun ha lm tng cng kh nng giao tip v ti s dng ca phn mm cng

nh cc th vin ha.

S pht trin vt bc ca cng ngh vi in t v phn cng my tnh vo thp

nin 1980 lm xut hin hng lot cc v mch h tr cho vic truy xut ha i cng

vi s gim gi ng k ca my tnh c nhn lm ha ngy cng i su vo cuc sng

thc t.

Nhng nm 1980 c raster graphics ( ho im). Bt u chun ho v d nh:

GKS(Graphics Kernel System): European effort (kt qu ca chu u), Becomes ISO 2D

standard.

Thp nin 90 pht trin c bit v phn cng, thit b hnh hc ho Silicon. Xut

hin cc chun cng nghip: PHIGS (Programmers Hierarchical Interactive Graphics

Standard) xc nh cc phng php chun cho cc m hnh thi gian thc v lp trnh

hng i tng. Giao din ngi my Human-Computer Interface (HCI).

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Ngy nay xut hin nh hin thc, cc ho cho my tnh (Graphics cards for

PCs), game boxes v game players. Cng nghip phim nh nh vo ho my tnh

(Computer graphics becoming routine in movie industry), Maya (th gii vt cht tri gic

c).

1.1.2. K thut ha vi tnh.

ha my tnh l mt lnh vc ca khoa hc my tnh nghin cu v c s ton hc, cc

thut ton cng nh cc k thut cho php to, hin th v iu khin hnh nh trn mn

hnh my tnh. ha my tnh c lin quan t nhiu n mt s lnh vc nh i s, hnh

hc gii tch, hnh hc ha hnh, quang hc,... v k thut my tnh, c bit l ch to

phn cng (cc loi mn hnh, cc thit b xut, nhp, cc v mch ha...).

Theo ngha rng hn, ha my tnh l phng php v cng ngh dng trong vic

chuyn i qua li gia d liu v hnh nh trn mn hnh bng my tnh. ha my

tnh hay k thut ha my tnh cn c hiu di dng phng php v k thut to

hnh nh t cc m hnh ton hc m t cc i tng hay d liu ly c t cc i

tng trong thc t.

1.2. CC K THUT HO

1.2.1. K thut ho im (Sample based-Graphics)

Cc m hnh, hnh nh ca cc i tng c hin th thng qua tng pixel (tng

mu ri rc)

c im:C th thay i thuc tnh ca tng im nh ri rc

o Xo i tng pixel ca m hnh v hnh nh cc i tng.

o Cc m hnh hnh nh c hin th nh mt li im (grid) cc pixel ri

rc,

o Tng pixel u c v tr xc nh, c hin th vi mt gi tr ri rc (s

nguyn) cc thng s hin th (mu sc hoc sng)

Tp hp tt c cc pixel ca grid cho chng ta m hnh, hnh nh i tng m

chng ta mun hin th.

Hnh 1.1 nh ho im

Phng php to ra cc pixel

Phng php dng phn mm v trc tip tng pixel mt.

Da trn cc l thuyt m phng (l thuyt Fractal, v.v) xy dng nn hnh nh

m phng ca s vt.

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Phng php ri rc ho (s ho) hnh nh thc ca i tng.

C th sa i (image editing) hoc x l (image processing) mng cc pixel thu

c theo nhng phng php khc nhau thu c hnh nh c trng ca i

tng.

1.2.2. K thut ho vector

Hnh 1.2 M hnh ho vector

M hnh hnh hc (geometrical model) ca i tng

Xc nh cc thuc tnh ca m hnh hnh hc ny,

Qu trnh t trt (rendering) hin th tng im ca m hnh, hnh nh thc ca

i tng

V d v hnh nh ho Vector

Hnh 1.3 V d v ho vector

C th nh ngha ho vector: ho vector = geometrical model + rendering

Cc tham

s t trt

T trt

Thit b ra

M hnh

ha

Muscle Model Wireframe Model Skeletal Model

Skin Model Hair Model Render and Touch up

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So snh gia Raster v Vector Graphics

ho im(Raster Graphics)

- Hnh nh v m hnh ca cc vt th c

biu din bi tp hp cc im ca li (grid)

- Thay i thuc tnh ca cc pixel => thay

i tng phn v tng vng ca hnh nh.

- Copy c cc pixel t mt hnh nh ny

sang hnh nh khc.

ho vector(Vector Graphics)

- Khng thay i thuc tnh ca tng im

trc tip

- X l vi tng thnh phn hnh hc c s

ca n v thc hin qu trnh t trt v hin th

li.

- Quan st hnh nh v m hnh ca hnh nh

v s vt nhiu gc khc nhau bng cch

thay i im nhn v gc nhn.

1.2.3. Phn loi ca ho my tnh

Phn loi theo cc lnh vc ca ho my tnh

Phn loi theo h to

K thut ho hai chiu: l k thut ho my tnh s dng h to hai chiu

(h to phng), s dng rt nhiu trong k thut x l bn , th.

K thut ho ba chiu: l k thut ho my tnh s dng h to ba chiu,

i hi rt nhiu tnh ton v phc tp hn nhiu so vi k thut ho hai chiu.

Cc lnh vc ca ho my tnh:

K thut x l nh (Computer Imaging): sau qu trnh x l nh cho ta nh s ca

i tng. Trong qu trnh x l nh s dng rt nhiu cc k thut phc tp: k thut

khi phc nh, k thut lm ni nh, k thut xc nh bin nh.

K thut nhn dng (Pattern Recognition): t nhng nh mu c sn ta phn loi

theo cu trc, hoc theo cc tiu tr c xc nh t trc v bng cc thut ton chn lc

c th phn tch hay tng hp nh cho thnh mt tp hp cc nh gc, cc nh gc

K thut phn tch v to nh

ho hot hnh v ngh thut

K thut nhn dng

X l nh

ho minh ho

CAD/CAM System

K thut ho

Kin to

ho

X l ho

K thut ho

K thut ho 2 chiu

K thut ho 3 chiu PTIT


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ny c lu trong mt th vin v cn c vo th vin ny ta xy dng c cc thut

gii phn tch v t hp nh.

K thut tng hp nh (Image Synthesis): l lnh vc xy dng m hnh v hnh nh

ca cc vt th da trn cc i tng v mi quan h gia chng.

Cc h CAD/CAM (Computer Aided Design/Computer Aided Manufacture

System): k thut ho tp hp cc cng c, cc k thut tr gip cho thit k cc chi

tit v cc h thng khc nhau: h thng c, h thng in, h thng in t.

ho trnh by (Presentation Graphics): gm cc cng c gip hin th cc s liu

th nghim mt cch trc quan, da trn cc mu th hoc cc thut ton c sn.

ho hot hnh v ngh thut: bao gm cc cng c gip cho cc ho s, cc nh

thit k phim hot hnh chuyn nghip lm cc k xo hot hnh, v tranh... V d: phn

mm 3D Studio, 3D Animation, 3D Studio Max.

1.2.4. Cc ng dng tiu biu ca k thut ha

ho my tnh l mt trong nhng lnh vc l th nht v pht trin nhanh nht ca tin

hc. Ngay t khi xut hin n c sc li cun mnh lit, cun ht rt nhiu ngi

nhiu lnh vc khc nhau nh khoa hc, ngh thut, kinh doanh, qun l...Tnh hp dn

ca n c th c minh ho rt trc quan thng qua cc ng dng ca n.

Xy dng giao din ngi dng (User Interface)

Giao din ho thc s l cuc cch mng mang li s thun tin v thoi mi cho

ngi dng ng dng. Giao din WYSIWYG v WIMP ang c a s ngi dng u

thch nh tnh thn thin, d s dng ca n.

To cc biu trong thng mi, khoa hc, k thut

Cc ng dng ny thng c dng tm lc cc d liu v ti chnh, thng k,

kinh t, khoa hc, ton hc... gip cho nghin cu, qun l... mt cch c hiu qu.

T ng ho vn phng v ch bn in t

Thit k vi s tr gip ca my tnh (CAD_CAM)

Lnh vc gii tr, ngh thut v m phng

iu khin cc qu trnh sn xut (Process Control)

Lnh vc bn (Cartography)

Gio dc v o to

PTIT


12

Mt s v d ca ng dng k thut ho:

Hnh 1.4 Cc ng dng ca k thut ho

Hnh 1.5 H ng dng CAD - CAM

PTIT


13

1.2.5. Cc chun giao din ca h ho

Mc tiu cn bn ca cc chun cho phn mm ho l m bo tnh tng thch. Khi

cc cng c c thit k vi hm ho chun, phn mm c th c di chuyn mt

cch d dng t h phn cng ny sang h phn cng khc v c dng trong nhiu ci

t v ng dng khc nhau.

GKS (Graphics Kernel System): chun xc nh cc hm ho chun, c thit

k nh mt tp hp cc cng c ho hai chiu v ba chiu.

GKS Functional Description, ANSI X3.124 - 1985.GKS - 3D Functional

Description, ISO Doc #8805:1988.

CGI (Computer Graphics Interface System): h chun cho cc phng php giao

tip vi cc thit b ngoi vi.

CGM (Computer Graphics Metafile): xc nh cc chun cho vic lu tr v chuyn

i hnh nh.

VRML (Virtual Reality Modeling Language): ngn ng thc ti o, mt hng pht

trin trong cng ngh hin th c xut bi hng Silicon Graphics, sau c

chun ha nh mt chun cng nghip.

PHIGS (Programmers Hierarchical Interactive Graphics Standard): xc nh cc

phng php chun cho cc m hnh thi gian thc v lp trnh hng i tng.

PHIGS Functional Description, ANSI X3.144 - 1985.+ Functional Description,

1988, 1992.

OPENGL th vin ha ca hng Silicon Graphics, c xy dng theo ng

chun ca mt h ha nm 1993.

DIRECTX th vin ho ca hng Microsoft, Direct X/Direct3D 1997

1.3. PHN CNG HO (GRAPHICS HARDWARE)

1.3.1. Cc thnh phn phn cng ca h ho tng tc

CPU:thc hin cc chng trnh ng dng.

B x l hin th (Display Processor): thc hin cng vic hin th d liu ho.

B nh h thng (System Memory): cha cc chng trnh v d liu ang thc

hin.

Gi phn mm ho (Graphics Package): cung cp cc hm ho cho chng

trnh ng dng

Phn mm ng dng (Application Program): phn mm ho ng dng.

B m ( Frame buffer): c nhim v cha cc hnh nh hin th.

B iu khin mn hnh (Video Controller): iu khin mn hnh, chuyn d liu

dng s frame buffer thnh cc im sng trn mn hnh.

PTIT


14

Hnh 1.6 Cc thnh phn cng ca h ho tng tc

1.3.2. My in

Dot size: ng knh ca mt im in b nht m my in c th in c

Addressability: kh nng a ch ho cc im in c th c trn mt n v di (dot per

inch)

Dot size Point per inch

8 - 20/ 100 inch 200, 600

5/1000 inch 1500

My v 6,15/1000 inch 1000, 2000

1.3.3. Mn hnh CRT

Mt chm cc tia in t (tia m cc) pht ra t mt sng in t, vt qua cun li tia dn n v

tr xc nh trn mn hnh c ph mt lp phosphor. Ti mi v tr tng tc vi tia in t ht

phosphor s pht ln mt chm sng nh. Nhng chm sng s m dn rt nhanh nn cn c cch

no n duy tr nh trn mn hnh. Mt trong cc cch l: lp i lp li nhiu ln vic v li nh

tht nhanh bng cch hng cc tia in t tr li v tr c. Gi l lm ti (refresh CRT).

S lng ti a cc im c th hin th trn mt CRT c gi l phn gii

(Resolution). Hay phn gii l s lng cc im c th c v theo chiu ngang v

chiu dc (c xem nh tng s im theo mi hng) ca mn hnh.

Kch thc vt l ca mn hnh ho c tnh t di ca ng cho mn

hnh. Thng dao ng t 12-27 inch, hoc ln hn.

Thuc tnh khc ca mn hnh l t s phng (aspect ratio). N l t l ca cc

im dc v cc im ngang cn pht sinh cc on thng c di n v theo c hai

hng trn mn hnh. Mn hnh c t s phng khc mt, th hnh vung hin th trn

thnh hnh ch nht cn hnh trn thnh hnh ellipse.

PTIT


15

Hnh 1.7 Cng ngh mn hnh CRT

Mn hnh dng im (Raster Display): thng gp nht trong s cc dng mn hnh

s dng CRT trn cng ngh truyn hnh. Mi im trn mn hnh c gi l pixel. Cc

thng tin v nh hin th trn mn hnh c lu tr trong mt vng b nh gi l vng

m lm ti (Refresh buffer) hay l vng m khung (Frame Buffer). Vng lu tr tp

cc gi tr cng sng ca ton b cc im trn mn hnh v lun tn ti mt cch

song nh gia mi im trn mn hnh v mi phn t trong vng ny.

to ra hnh nh en trng, n gin ch cn lu thng tin ca mi Pixel l mt bt

(0,1) (xem hnh 1.8). Trong trng hp nh nhiu mu th cn nhiu bt hn, nu thng tin

mi pixel c lu bng b bt th ta c th c 2b gi tr mu phn bit cho pixel .

SONY Trinitron

CRT

NEC Hybrid

Mask

Hitachi EDP Standard Dot-trio

PTIT


16

V d m hnh ho im ngi nh v ngi sao.

Hnh 1.8 Song nh gia vng m khung v mn hnh

Trong cc mn hnh mu, ngi ta nh ngha tp cc mu lm vic trong mt bng

tra (LookUp Table - LUT). Mi phn t ca LUT c nh ngha mt b ba gi tr

(RGB) m t mt mu no . Khi cn s dng mt mu, ta ch cn ch nh s th t

(index) tng ng ca mu trong LUT, s phn t trong bng LUT chnh l s mu c

th c hin th cng mt lc trn mn hnh.

X: 0 Xmax2 mu/ 1 bit

Y: 0 Ymax16 mu/ 4 bit ;256 mu/ 8bit

216

mu/ 16 bit ; 224

mu/ 24 bit

640 x 480 x 16 Video RAM = 2MB

1024 x 1024 x 24 Video RAM = 24MB

Vic lm ti trn mn hnh dng ny c thc hin tc 60 - 80 khung/giy.

i khi tc lm ti cn c biu din bng n v Hertz (Hz - s chu k trn/giy),

trong mt chu k tng ng vi mt khung (frame). Vy tc lm ti 60

khung/giy n gin l 60 Hz. Khi t n cui mi dng qut, tia in t quay tr li

bn tri ca mn hnh bt u dng qut k tip. Vic quay tr v bn tri mn hnh sau

khi lm ti mi dng qut c gi l tia hi ngang (Horizontal retrace). V ti cui mi

frame, tia in t (tia hi dc - Vertical retrace) quay tr li gc bn tri ca mn hnh

chun b bt u frame k tip.

Hnh 1.9 Qut mnh v qut dng ca mn hnh CRT

1.3.4. Mn hnh tinh th lng (Liquid Crystal Display LCD)

Da vo cng ngh truyn nh sng qua in cc m t gia l cun dy xon. Khi cha

c t trng (cha c dng in) cun dy th nh sng truyn thng, khi c t trng

th nh sng truyn i chiu.

Interface to

host computer

Display

processo

r

(Display

commands)

(interaction data)

Keyboard

Data input 00000000000000

00000000000100

0000

00000000000000

00000000000100

0000

00000000000000

00000000011111

0000

00000000011000

00000111111111

1111

00000000111100

00000000011111

0000

00000011111111

00000000000100

0000

00001111111111

11000000000100

0000

00111111111111

11110000000000

0000

00011111111111

11100000000000

0000

00011111111111

11100000000000

0000

00011111111111

11100000000000

0000

00000000000000

00000000000000

0000

Bitmap refresh buffer

(the 1s are accentuated

for contrast)

CRT

PTIT


17

Hnh 1.10 Cng ngh truyn nh sng trong mn hnh tinh th lng

CRT Displays (mn hnh CRT)

Advantages (u im)

p ng nhanh (c phn gii cao)

Mu sc a dng (C su v rng)

Mu sc bo ho v t nhin

Cng ngh khng qu t v hon thin

Gc nhn rng, tng phn v sng cao

Disadvantages (nhc im)

Ln v nng (typ. 70x70 cm, 15 kg)

Tiu tn ngun in cao (typ. 140W)

C hi cho sc kho v trng in t v t tnh

Mn hnh nhp nhy (at 50-80 Hz)

Hnh hay b mo ti 4 gc

LCD Displays (mn hnh tinh th lng)

Advantages (u im)

Hnh dng nh, trng lng nh (approx 1/6 of

CRT, typ. 1/5 of CRT)

Tiu tn ngun thp (typ. 1/4 of CRT)

Mn hnh phng tuyt i nn khng mo ti

cc gc

Mu sc u, nh sinh ng

Khng b hiu ng in t trng

C th mn hnh va ln va rng (>20 inch)

Disadvantages (nhc im)

Gi thnh cao (presently 3x CRT)

Gc nhn hp hn (typ. +/- 50 degrees)

tng phn thp (typ. 1:100)

chi ( ngi) thp hn (typ. 200 cd/m2)

Tm tt chng:

S ra i ca ho my tnh thc s l cuc cch mng trong giao tip gia ngi dng

v my tnh. Vi lng thng tin trc quan, a dng v phong ph c truyn ti qua

hnh nh. Cc ng dng ho my tnh li cun nhiu ngi nh tnh thn thin, d

dng, kch thch kh nng sng to v tng ng k hiu sut lm vic.

PTIT


18

ho my tnh ngy nay c c ng dng rt rng ri trong nhiu lnh vc

khoa hc, k thut, ngh thut, kinh doanh, qun lCc ng dng ho rt a dng,

phong ph v pht trin lin tc khng ngng. Ngy nay, hu nh khng c chng trnh

ng dng no m khng s dng k thut ho lm tng tnh hp dn cho mnh.

Mt h thng ho bao gi cng gm hai phn chnh l phn cng v phn

mm. Phn cng bao gm cc thit b hin th (thit b xut) v cc thit b nhp. Tiu

biu nht l mn hnh, c hai loi mn hnh thng dng l CRT v LCD.

Bi tp:

1. Cu to v nguyn l hot ng ca mn hnh dng im. Nu cc khi nim

vng m khung, phn gii, t s phng.... ca mn hnh loi ny?

2. ngha v hot ng ca bng tra LUT?

3. Tnh Video Ram ca cc mn hnh ln lt c phn gii l 640x480,

1024x768, 1280x1024 m c mi pixel c m t ln lt l 8bt, 12 bit, 24

bit.

4. Nu chng ta dng cc gi tr 12bit cho mi pixel trong mt bng tham chiu

lookup table, c bao nhiu hng mc m lookup table c c?

5. Ti sao phi chun ho cc phn mm ho? Lit k cc chun ha .

PTIT

Chng 2: Cc gii thut sinh thc th c s

19

CHNG 2: CC GII THUT SINH THC TH C S

2.1. CC H THNG TO TRONG HO

Trong lnh vc k thut ha, chng ta phi hiu c rng thc cht ca ha l lm

th no c th m t v bin i c cc i tng trong th gii thc trn my tnh.

Bi v, cc i tng trong th gii thc c m t bng ta thc. Trong khi , h

ta thit b li s dng h ta nguyn hin th cc hnh nh. y chnh l vn

c bn cn gii quyt. Ngoi ra, cn c mt kh khn khc na l vi cc thit b khc

nhau th c cc nh ngha khc nhau. Do , cn c mt phng php chuyn i tng

ng gia cc h ta v i tng phi c nh ngha bi cc thnh phn n gin

nh th no c th m t gn ng vi hnh nh thc bn ngoi.

Hai m hnh c bn ca ng dng ha l da trn mu s ha v da trn c

trng hnh hc. Trong ng dng ha da trn mu s ha th cc i tng ha

c to ra bi li cc pixel ri rc. Cc pixel ny c th uc to ra bng cc chng

trnh v, my qut, ... Cc pixel ny m t ta xc nh v tr v gi tr mu. Thun li

ca ng dng ny l d dng thay i nh bng cch thay i mu sc hay v tr ca cc

pixel, hoc di chuyn vng nh t ni ny sang ni khc. Tuy nhin, iu bt li l khng

th xem xt i tng t cc gc nhn khc nhau. ng dng ha da trn c trng

hnh hc bao gm cc i tng ha c s nh on thng, a gic,.... Chng c lu

tr bng cc m hnh v cc thuc tnh. V d : on thng c m hnh bng hai im

u v cui, c thuc tnh nh mu sc, dy. Ngi s dng khng thao tc trc tip

trn cc pixel m thao tc trn cc thnh phn hnh hc ca i tng.

2.1.1. H to thc (WCS World Coordinate System)

Mt trong nhng h ta thc thng c dng m t cc i tng trong th gii

thc l h ta Descartes. Vi h ta ny, mi im P c biu din bng mt cp

ta P(xp,yp,zp) vi xp, yp,zp R

Hnh 2.1 H ta thc.

Ox,Oy, Oz l trc to

xp ,yp,zp : to ca P

2.1.2. H to thit b (DCS Device Coordinate System)

H ta thit b (device coordinates) c dng cho mt thit b xut c th no , v

d nh my in, mn hnh,.. Trong h ta thit b th cc im cng c m t bi cp

PTIT


20

ta (x,y). Tuy nhin, khc vi h ta thc l x, y N. iu ny c ngha l cc

im trong h ta thc c nh ngha lin tc, cn cc im trong h ta thit b

l ri rc. Ngoi ra, cc ta x, y ca h ta thit b ch biu din c trong mt gii

hn no ca N. V d : phn gii ca mn hnh trong ch ha l 640x480. Khi

, x(0,639) v y(0,479) (xem hnh 2.2).

Hnh 2.2 H ta trn mn hnh

2.1.3. to thit b chun (NDCS Normalized Device Coordinate System)

Do cch nh ngha cc h ta thit b khc nhau nn mt hnh nh hin th c trn

thit b ny l chnh xc th cha chc hin th chnh xc trn thit b khc. Ngi ta xy

dng mt h ta thit b chun i din chung cho tt c cc thit b c th m t cc

hnh nh m khng ph thuc vo bt k thit b no.

Trong h ta chun, cc ta x, y s c gn cc gi tr trong on t [0,1].

Nh vy, vng khng gian ca h ta chun chnh l hnh vung n v c gc tri

di (0, 0) v gc phi trn l (1, 1).

Qu trnh m t cc i tng thc nh sau (xem hnh 2.3):

Hnh 2.3 H ta trn mn hnh.

2.2. IM V ON THNG

2.2.1. im

Trong h to hai chiu (mt phng) th im c biu din P(x,y), ngoi ra n cn c tnh

cht mu sc. V d v mt im ta c hm putpixel(x,y,color).

2.2.2. on thng

Biu din tng minh: y = f(x)

Mt on thng c xc nh nu bit 2 im thuc n. Phng trnh on thng i

qua 2 im P (x1,y1) v Q(x2,y2) nh sau:

PTIT


21

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

(y-y1)(x2-x1)=(x-x1)(y2-y1)

(x2-x1)y=(y2-y1)x + y1(x2-x1) - x1(y2-y1)

y = ((y2-y1)/(x2-x1))x + y1 - ((y2-y1)/(x2-x1))x1

y = kx + b

k = (y2-y1)/(x2-x1) dc hay h s gc ca ng

b = y1- kx1 on chn trn trc y

y = kx (tc l khi x thay i th y thay i theo)

Hnh 2.4 V on thng PQ

Biu din khng tng minh: ax+by+c=0

Ta c : (y2-y1)x - (x2-x1)y + (x2-x1)y1 - (y2-y1)x1 = 0

(y2-y1)x - (x2-x1)y + x2y1 - x1y2 = 0

hay ax + by + c = 0

Trong a = (y2-y1), b = -(x2-x1 ) v c = x2y1 - x1y2

Biu din thng qua tham s:

P(u) = P1 + u(P2 - P1) c u [0,1]

x(u) = x1 + u( x2 - x1 )

y (u)= y1 + u( y2 - y1 )

2.3. CC GII THUT XY DNG THC TH C S

2.3.1. Gii thut v on thng thng thng

Nguyn l chung: cho mt thnh phn to x hay y bin i theo tng n v v tnh

nguyn cn li sao cho gn vi to thc nht.

Ta c 1112

12 yxxxx

yyy

Cho x thay i tm y, trong bi ny cho x1 thay i tin ti x2 ta chn n v nh

nht ca mn hnh x=1.

Gii thut thng thng:

void dline(int x1,int y1, int x2,int y2, int color) {

float y; int x;

for (x=x1; x


22

2.3.2. Gii thut Bresenham

1960 Bresenham thuc IBM tm ra cc im gn vi ng thng da trn phn gii

hu hn. Gii thut ny loi b c cc php ton chia v php ton lm trn nh ta

thy trong gii thut trn.

Xt on thng vi 0 < k < 1

Hnh 2.5 M t gii thut Bresenham (0 yi+1 = yi +1

t D = d1 - d2= 2k(xi + 1) - 2yi + 2b - 1

C k=y/x v t Pi = xD = x (d1 - d2)

Pi = x(2y/x(xi +1)- 2yi +2b-1) = 2yxi +2y -2xyi + 2bx -x

Ta tnh bc tip:

Pi+1 = 2yxi+1 +2y -2xyi+1 + 2bx -x

Pi+1 - Pi = -2x(yi+1 -yi) + 2y(xi+1 -xi)

C xi+1 =xi+1 nn:

Pi+1 - Pi = - 2x(yi+1 -yi) + 2y = 2y - 2x(yi+1 -yi)

Nu Pi 0 th yi+1 = yi +1 Pi+1 = Pi + 2y - 2x

Tnh gi tr u: P1?

P1 = x(d1 - d2) = x(2y/x(x1 +1)- 2y1 +2b-1)

= 2yx1 +2y -2xy1 + 2bx -x

C y1=kx1 + b = y/x x1 +b

P1 = 2yx1 +2y -2x((y/x)x1 +b) + 2bx -x

= 2yx1 +2y -2yx1 - 2bx + 2bx -x

P1 = 2y - x

PTIT


23

Hnh 2.6 S khi thut ton

Bresenham cho ng thng

void Bre_line(int x1, int y1,

int x2, int y2, int c){

int x, y, dx, dy,p;

y = y1;

dx = x2 - x1;

dy = y2 - y1;

p = 2*dy - dx;

for (x=x1; x


24

Hnh 2.7 M t gii thut Midpoint

So snh hay kim tra M s c thay bng vic xt gi tr d.

d > 0 im B c chn khi yi+1 = yi

d < 0 im A c chn khi yi+1 = yi + 1

Trng hp d = 0 chng ta c th chn im bt k hoc A, hoc B.

S dng phng php biu din khng tng minh

f(x,y)= ax +by +c =0 (1) dx =x2-x1 dy =y2-y1

Biu din tng minh:

y= (dy/dx)x +B hay f(x,y)=0= xdy - ydx +Bdx (2)

So snh (1) v (2) ta c a=dy, b=-dx v c= Bdx

C f(x,y)=0 vi mi (x,y) thuc ng thng

t di=f(xi+1,yi+1/2) = a(xi+1) +b(yi +1/2) +c

Nu chn A (d0) th M s tng theo x

di+1=f(xi+2,yi+1/2) = a(xi+2) +b(yi +1/2) +c

di+1 - di = a Hay di+1 = di + dy

Tnh d1 ? d1 = f(x1+1,y1+1/2) = a(x1+1) +b(y1 +1/2) +c

= ax1 +by1 +c +a +1/2 b = f(x1,y1) +a +b/2

C (x1,y1) l im bt u, nm trn on thng nn f(x1,y1) = 0

Vy d1 = a+ b/2 = dy - dx/2

A

B

d0

A

PTIT


25

Hnh 2.8 S khi gii thut Midpiont

cho on thng

/* Thuat toan Midpoint de ve

doan thang (0


26

Hnh 2.10 M t gii thut Bresenham

Gi s bt u xi vy xi+1 = xi +1

y2 = r2 - (xi +1)2

d1 = yi2 - y2 = yi

2 - r2 - (xi +1)2

d2 = y2 - (yi - 1)

2 = r2 - (xi +1)2 - (yi - 1)

2

pi = d1 - d2 = 2(xi +1 )2 + yi

2 + (yi - 1)2 -2r2

Xt: pi =d2) chn im nm trong ng trn yi+1 = yi +1

pi = 2(xi +1 )2 + 2yi

2 - 2yi 1 - 2r2

pi+1 = 2(xi +2 )2 + 2yi+1

2 - 2yi+1 + 1 - 2r2

pi+1 = pi + 4xi +6 + 2yi+12 - 2yi

2- 2yi+1 + 2yi

pi+1 = pi + 4xi +6 + 2(yi+12 - yi

2 )- 2(yi+1 - yi )

Nu pi =0 hay yi+1 = yi -1

pi+1 = pi + 4xi +6 - 4yi + 2 + 2

pi+1 = pi + 4(xi - yi ) + 10

Tnh P1 ? khi ng vi x1=0 v y1 =r

p1 = 2(x1 +1)2 + y1

2 + (y1 - 1)2 -2r2

= 2 + r2 + (r-1)2 - 2r2= 3 - 2r

Gii thut l:

PTIT


27

Hnh 2.11 S khi gii thut Bresemham

cho ng trn

void Bre_circle(int xc, int yc,

int Radius, int color)

{

int x, y, p;

x = 0;

y = Radius;

p = 3 - 2 * Radius;

while (x


28

y--;

}

x++;

pc(xc,yc, x,y);

pc(xc,yc, y,x);

}

pc(xc,yc, y,y); // ve 4 diem phan giac x=y

}

void main(){

int gr_drive = DETECT, gr_mode;

initgraph(&gr_drive, &gr_mode, "");

Bresenham_Circle(getmaxx() / 2, getmaxy() / 2, 150, 4);

getch();

closegraph();

}

2.3.5. Gii thut sinh ng trn Midpoint

Phng trnh ng trn khng tng minh:

f(x,y) = x2+y2-R2 =0

Hnh 2.12 M t gii thut Midpoint

Nu f(x,y) = 0 th nm trn ng trn

f(x,y) > 0 th nm bn ngoi ng trn

f(x,y) < 0th nm bn trong ng trn

Thc hin gii thut trn 1/8 ng trn v ly i xng cho cc gc cn li.

Vi M l im gia ca AB

Vi di l gi tr ca ng trn ti mt im bt k

Ta c: di = f(xi+1,yi - 1/2) = (xi +1 )2 + (yi - 1/2)

2 - r2

di < 0 chn A th im k cn s dch chuyn theo x mt n v

di+1 = f(xi +2, yi -1/2)

= (xi +2)2 + (yi - 1/2)

2 - r2

di+1 - di = (xi +2)2 - (xi +1 )

2 = 2xi +3

di+1 = di + 2xi+3

di >= 0 chn B th im k cn s dch chuyn theo x 1 n v, theo y -1

n v

di+1 = f(xi +2, yi -3/2)

PTIT


29

= (xi +2)2 + (yi - 3/2)

2 - r2

di+1 - di = 2xi - 2yi +5

di+1 = di + 2xi - 2yi +5

Tnh d1? ti im (0, r)

d1 =f(1,r-1/2)= 12 + (r-1/2)2 - r2

d1 = 5/4 -r

Thut ton nh sau:

Hnh 2.13 S khi gii thut Midpiont v

ng trn

void Mid_circle(int xc, int yc,

int Radius, int color)

{

int x, y, d;

x = 0;

y = Radius;

d = 1- Radius;

while (x


30

d += 2 * x + 3;

else {

d += 2 * (x-y) + 5;

y--;

}

x++; }}

2.3.6. Gii thut sinh ng ellipse

Tnh i xng c thc hin trn 4 cch

Hnh 2.14 M t gii thut sinh ng ellipse

Vector vi tip tuyn gradient =1

Ta c tip tuyn vi cung trn ( dc) = -1= dy/dx = - fx/fy

Trong fx=2b2x o hm ring phn ca f(x,y) vi x

V fy=2a2y o hm ring phn ca f(x,y) vi y

Gi s ta ch xt trn gc phn t th nht: gi s ta chia cung t (0,b) n (a,0) ti

Q, c dc -1

Trn phn 1: x thay i th y thay i theo

Trn phn 2: y thay i th x thay i theo

Xt trn phn 1:

Bt u t (0,b), bc th i (xi,yi) chn tip

A(xi+1, yi)

B(xi+1,yi-1)

Tham s quyt nh:

Pi = f(xi+1,yi-1/2) = b2(xi+1)

2 + a2(yi-1/2)2 -a2b2

Pi+1 = f(xi+1+1,yi+1-1/2) = b2(xi+1+1)

2 + a2(yi+1-1/2)2 -a2b2

Pi+1 - Pi = b2((xi+1+1)

2 - (xi+1)2 )+ a2((yi+1-1/2)

2 - (yi-1/2)2 )

Pi+1 = Pi + 2b2xi+1+ b

2 + a2((yi+1-1/2)2 - (yi-1/2)

2 )

Nu Pi


31

Nu Pi >=0 chn B

xi+1=xi+1

yi+1=yi -1

Pi+1 = Pi + 2b2xi(xi+1) +b

2 + a2((yi-1 -1/2)2 - (yi-1/2)

2 )

= Pi + 2b2xi +3b

2 +a2(-3yi +9/4 +yi -1/4)

= Pi + 2b2xi +3b

2 +a2(-2yi +2)

Hay Pi+1 = Pi + b2(2xi +3) + a

2(-2yi +2)

Tnh P1? ti (0,b)

P1 = f(x1+1,y1-1/2) = b2 + a2(b-1/2)2 -a2b2

P1 = b2 - a2b +a2/4

Xt trn phn 2:

Ta ly to d ca Pixel sau cng trong phn 1 ca ng cong tnh gi tr ban u

cho phn 2.

Gi s pixel (xk,yk) va chuyn qut cui cng ca phn 1 nhp vo bc j cho phn

2 (xj,yj).

Pixel k tip c th l:

C(xj,yj-1)

D(xj+1, yj-1)

Tham s quyt nh:

qj = f(xj+1/2,yj-1) = b2(xj+1/2)

2 + a2(yj-1)2 -a2b2

qj+1 = f(xj+1+1/2,yj+1-1) = b2(xj+1+1/2)

2 + a2(yj+1-1)2 -a2b2

qj+1 - qj = b2((xj+1+1/2)

2 - (xj+1/2)2 )+ a2((yj+1-1)

2 - (yj-1)2 )

qj+1 = qj + b2((xj+1+1/2)

2 - (xj+1/2)2 )+ a2- 2a2yj+1

Nu qj =0 chn C

yj+1=yj -1

xj+1= xj

qj+1 = qj + a2- 2a2(yj-1)

Hay qj+1 = qj + a2(3 - 2yj )

Tnh q1?

q1 = f(xk+1/2,yk -1) = b2(xk+1/2)

2 + a2(yk-1)2 -a2b2

PTIT


32

Cu hi: lc ly i xng 4 cch tm 1 Ellipse hon chnh t cc to pixel

c to ra vi cung phn t th 1. C hin tng overstrike xy ra hay khng?

Tr li: hin tng overstrike xy ra ti:

(0,b); (0,-b); (a,0); (-a,0)

Thut ton

#include

#include

#define ROUND(a) ((long)(a+0.5))

void plot(int xc, int yc, int x, int y, int color){

putpixel(xc+x, yc+y, color);

putpixel(xc-x, yc+y, color);

putpixel(xc+x, yc-y, color);

putpixel(xc-x, yc-y, color);

}

void Mid_ellipse(int xc, int yc, int a, int b, int color){

long x, y, fx, fy, a2, b2, p;

x = 0;

y = b;

a2 = a * a; //a2

b2 = b * b; // b2

fx = 0;

fy = 2 * a2 * y; // 2a2y

plot(xc, yc, x,y, color);

p = ROUND(b2-(a2*b)+(0.25*a)); // p=b2 - a2b + a2/4

while (fx < fy){

x++;

fx += 2*b2; //2b2

if (p0){

y--;

fy -= 2*a2; // 2a2

if (p>=0)

p+=a2*(3 - 2*y); //p =p + a2(3-2y)

else{

x++;

fx += 2*b2; // 2b2

p += b2*(2*x+2) + a2*(-2*y +3); //p=p + b2(2x +2) +a2(-2y +3)

}

plot(xc, yc, x, y, color);

}

}

PTIT


33

void main(){

int gr_drive = DETECT, gr_mode;

initgraph(&gr_drive, &gr_mode, "");

Mid_Ellipse(getmaxx() / 2, getmaxy() / 2, 150, 80, 4);

getch();

closegraph();

}

2.3.7. Gii thut sinh k t

Trong mn hnh text, truy xut cc k t trn mn hnh c h tr bi phn cng. Cc k

t c lu tr trong b nh ROM, di dng bitmap hay cc ma trn nh. Phn cng s

a k t ln mn hnh ti v tr xc nh, tnh ton cun trang v xung dng.

Trong ho:

Vector: nh ngha cc k t theo nhng ng cong mm bao ngoi ca chng,

tn km v mt tnh ton.

Hnh 2.15 K t vector

u nhc im:

- phc tp (tnh ton phng trnh)

- lu tr gn nh

- cc php bin i da vo cng thc bin i

- Kch thc ph thuc vo mi trng (khng c kch

thc c nh)

Bitmap: nh ngha mi k t vi 1 font ch cho trc l 1 nh bitmap hnh ch

nht nh.

Hnh 2.16 K t bitmap

- n gin trong vic sinh k t

(copypixel)

- Lu tr ln

- Cc php bin i(I,B,U, scale) i hi

lu tr thm

- Kch thc khng i

bitmap: s dng hm copypixel (copy im nh) c lu tr trong b nh c nh

- Fontcache, a vo b nh m hin th. Mi 1 k t nh 1 ma trn 2 chiu ca

cc im nh - mt n.

Hm_sinh_ki_tu (mask)

{xmax, ymax, xmin, ymin //cc gii hn ca mt n

xo, yo //im gc trn b m hin th

for (i=ymin;i< ymax ;i++)

for (j=xmin; j< xmax ; j++)

if (mask(i,j) 0)

copypixel ((mask(i,j), pixel(xo+j, yo+i));

}

PTIT


34

K t fontcache bitmap n gin ca SRGP lu tr cc k t theo chui lin tip

nhau trong b nh. Nhng rng cc k t khc nhau, truy nhp cc fontcache thng

qua bn ghi v cu trc cho tng k t.

Cu trc font ch

typedef struct {

int leftx;

int width;

} Charlocation; //V tr ca text

struct {

int CacheId;

int Height; // rng ch

int CharSpace; // Khong cch gia cc k t

Charlocation Table [128]; //bng ch ci

} fontcache;

K t vector

Xy dng theo phng php nh ngha cc k t bi ng cong mm bao ngoi

ca chng d dng thay i kch thc ca k t cng nh ni suy ra cc dng ca k t.

Hon ton c lp vi thit b.

Ti u nht: lu tr font di dng ng bao. Khi cc chng trnh ng dng s

dng l bitmap tng ng vi chng.

2.3.8. Gii thut sinh a gic (Polygon)

a. Thut gii v ng bao a gic

Vic biu din a gic thng qua:

Tp cc on thng

Tp cc im thuc a gic

Cc loi a gic:

Hnh 2.17 Cc loi a gic

a gic li: l a gic c ng thng ni bt k 2 im bn trong no ca a gic

u nm trn trong a gic. a gic khng li l a gic lm.

Cc ng thng bao a gic - cnh ca a gic. Cc im giao ca cnh - nh ca

a gic. Thng tin cn thit xc nh a gic:

S cnh

To cc nh ca a gic

Gii thut:

Polygon (arrayx, arrayy,n)

Tam gic li lm t ct min

PTIT


35

{ if (n


36

Gii thut dng qut (scanline) cho vic t mu vng

Gii thut da trn tng s dng mt ng qut trn trc y ca mn hnh i t

ymax n ymin ca vng cn c t mu.

Vi mi gi tr y = yi ng thng qut ct cc ng bin ca vng cn t to ra

on thng y = yi vi x [xmin, xmax]. Trn on thng chng ta t mu cc im tng

ng i t xmin n xmax c cc im t (xi, yi) y = yi.

n gin nht v d t mu hnh ch nht:

void scanline_rectg(x1,y1,x2,y2,c){ int i,j;

for(i=y1; i>=y2; i--)

for(j=x1; j


37

Nu s giao im tm c gia cc cnh a gic v dng qut l l (iu ny

ch xy ra khi dng qut s i qua cc nh ca a gic) khi ta s tnh s

im l 2 th c th t khng chnh xc. Ngoi ra, vic tm giao im ca

dng qut vi cc cnh nm ngang l trng hp t bit...

gii quyt cc vn trn ta c cc phng php sau:

Danh sch cc cnh kch hot (AET - Active Edge Table)

Mi cnh ca a gic c xy dng t 2 nh k nhau Pi(xi,yi) v Pi+1(xi+1,yi+1) gm

cc thng tin sau:

ymin: gi tr nh nht trong 2 nh ca cnh

xIntersect: honh giao im ca cnh vi dng qut hin ang xt

DxPerScan: gi tr 1/m (m l h s gc ca cnh)

DeltaY: khong cch t dng qut hin hnh ti nh ymax

Danh sch cc cnh kch hot AET: danh sch ny dng lu cc tp cnh ca a

gic c th ct ng vi dng qut hin hnh v tp cc im giao tng ng. N c mt s

c im:

Cc cnh trong danh sch c sp xp theo th t tng dn ca cc honh giao

im c th t mu cc on giao mt cch d dng.

Thay i ng vi mi dng qut ang xt, do danh sch ny s c cp nht lin

tc trong qu trnh thc hin thut ton. u tin ta c danh dch cha ton b cc cnh

ca a gic gi l ET (Edge Table) c sp xp theo th t tng dn ca ymin, ri sau mi

ln dng qut thay i s di chuyn cc cnh trong ET tho iu kin sang AET.

Mt dng qut y=k ch ct 1 cnh ca a gic khi v ch khi k>=ymin v y>0. Chnh

v vy m vi cc t chc ca ET (sp theo th t tng dn ca ymin) iu kin chuyn

cc cnh t ET sang AET s l k>=ymin; v iu kin loi mt cnh ra khi AET l

y


38

Hnh 2.20 Qui tc tnh: mt giao im (A) v hai giao im (B)

Hnh 2.21 lu thut ton scan - line

Gii thut t vng kn theo mu (Pattern filling)

object (nh mu)

A[m,n]

Vn : xc nh im mu v nhiu im tng ng vi chng trn mn hnh.

Phng php 1:

Tm im neo u tri nht hng u tin ca a gic.

Pi Pi

Pi

Pi

Pi-1

Pi-1

Pi-1

Pi-1 Pi+1

Pi+1

Pi+1

Pi+1

A B

PTIT


39

Nhc im: khng c im nh v tr phn bit mt cch r rng cho mu t trong

mt a gic bt k.

Phng php 2: s dng cho SRGP

Ly im neo gc to , gi s ta coi c mn hnh c lt bi mu t cc thc

th l cc ng bin cho cc vng t, vy nu ngoi cc thc th cc mu t khng c

php th hin.

Hnh 2.22 Phng php ly im neo

Tm tt chng:

c th hin th cc i tng ho trn thit b hin th dng im m in hnh l

mn hnh, cn phi c mt qu trnh chuyn cc m t hnh hc ca cc i tng ny

trong h to th gii thc v dy cc pixel tng ng gn vi chng nht trn to

thit b. Qu trnh ny cn c gi l qu trnh chuyn i bng dng qut. Yu cu quan

trng nht i vi qu trnh ny ngoi vic phi cho kt qu xp x tt nht cn phi cho

tc ti u.

Ba cch tip cn v on thng gm thut ton DDA, thut ton Bresenham, thut

ton Midpiont u tp trung vo vic a ra cch chn mt trong hai im nguyn k tip

khi bit im nguyn bc trc. Thut ton DDA n gin ch dng thao tc lm

trn nn phi dng cc php ton trn s thc, trong khi thut ton Bresenham v

Midpiont a ra cch chn phc tp hn nhng cho kt qu tt hn. Tng t dng hai

gii thut Bresenham v Midpiont v ng trn v ellpise v mt s ng cong

khc.

Cc thut ton t mu vng gm thut ton loang ( qui) hay thut ton dng qut.

C th t cng mt mu hay t theo mu.

Bi tp:

1. Ch nh cc v tr mnh no s c chn bi thut ton Bresenham lc chuyn

qut mt ng thng t to pixel (1,1) sang to pixel (8,5).

2. Dng gii thut Bresenham vit hm sinh on thng (xt tt c cc trng hp ca

h s gc).

3. Dng gii thut Midpiont vit hm sinh on thng (xt tt c cc trng hp ca

h s gc).

x

y

neo

neo

x

y

PTIT


40

4. Dng gii thut Midpiont vit hm sinh ng trn (to tm (xc,yc) v bn knh

r).

5. Dng gii thut Midpiont vit hm sinh ng ellipse (to tm (xc,yc) v bn

knh rx v ry ).

6. T hm v ng thng thit k v ci t hm v cc hnh sau: hnh ch nht, a

gic, ngi nh....

7. Vit gii thut tm giao im hai on thng.

8. Vit chng trnh t mu Floodfill (s dng qui vi 4-connected).

9. a ra lu thut ton t mu theo dng qut.Vit thut ton t mu scan-line.

PTIT

Chng 3: Cc php bin i ho

41

CHNG 3: CC PHP BIN I HO

3.1. CC PHP BIN I HNH HC HAI CHIU

3.1.1. Php bin i Affine (Affine Transformations)

Php bin i Affine l php bin i tuyn tnh ta im c trng ca i tng

thnh tp tng ng cc im mi to ra cc hiu ng cho ton i tng.

V d: php bin i ta vi ch 2 im u cui ca on thng to thnh 2 im

mi m khi ni chng vi nhau to thnh on thng mi. Cc im nm trn on thng

s c kt qu l im nm trn on thng mi vi cng php bin i thng qua php ni

suy.

3.1.2. Cc php bin i i tng

Cc i tng phng trong ho 2 chiu m t tp cc im phng. im trong ho 2

chiu biu din thng qua to , vit di dng ma trn gi l vect v tr.

C 2 dng biu din:

Mt hng v 2 ct: yx

Hai hng v 1 ct:

y

x

Trong gio trnh chng ta chn biu din im l ma trn hng (mt hng v 2 ct).

Tp cc im c lu tr trong my tnh s c vit di dng ma trn v tr ca

chng. Chng c th l ng thng, ng cong, nh....tht d dng kim sot cc i

tng thng qua cc php bin i chng, thc cht cc php bin i ho ny c

m t di dng cc ma trn.

3.1.2.1. Php bin i v tr

Gi s ta c im P = [ x y ] trong mt phng vi [x y] l vect v tr ca P, k hiu l

[P]. Gi ma trn T l ma trn bin i s c dng:

Hnh 3.1 Php bin i v tr

Ta c im P sau php bin i thnh P c gi tr [x y].

Thc thi php bin i ng trn 1 im nh s ng vi ton b i tng.

dc

baT

y

x

P P

PTIT


42

Hay ta c: x = ax + cy

y = bx + dy

Xt ma trn bin i T:

Php bt bin:

Khi : a = d =1 v b = c = 0 v ma trn cho php bt bin l:

''10

01** yxyxyxTX

Vy x=x v y = y hay l P = P chng t bt bin qua php bin i.

Php bin i t l (scaling):

Nu d=1 v b = c = 0 th ma trn bin i l:

10

0aT

x = ax

y = y

P dch chuyn theo trc x vi t l a xc nh.

Nu b = c =0 th ma trn bin i l:

d

aT

0

0

''0

0** yxdyax

d

ayxTX

Hay tng qut hn gi Sx, Sy ln lt l t l theo trc x v trc y, th ma trn t l

s l:

Sy

SxT

0

0

Khi Sx , Sy >1 gi l php phng to

Khi Sx, Sy


43

i xng qua y i xng qua x i xng qua gc to

Hnh 3.2 Cc php i xng trn 2D

Php bin dng

Khi a = d = 1 th to ca P ph thuc vo thay i ca b v c

Xt c = 0

Hnh 3.3 Php bin dng theo trc oy

C P khng thay i gi tr to x, cn y thay i ph thuc vo c b v x

Xt b = 0

''1

01**X yxycyx

cyxT

Hnh 3.4 Php bin dng theo trc ox

Ch : im gc to P[0 0] bt bin vi mi php bin i

Php quay

C >0 ngc chiu kim ng h

P P

Sx=-1 P

P

Sy=-1

P

P

Sx=Sy=-1

''10

1** yxybxx

byxTX

P

P

y=bx+y

bx

P P

cy x=x+cy

PTIT


44

Hnh 3.5 Php quay trn 2D

Ta c:

P[x y]= [rcos rsin]

P[x y] = [rcos(+) rsin(+)]

P[x y] = [r(coscos - sinsin)r(cossin + sincos)]

= [(xcos - ysin)(xsin + ycos)]

Vy: x = xcos - ysin

y = xsin + ycos

Ta c:

[X]= [X]* [T] = [(xcos - ysin) (xsin + ycos)]

Vy T tng qut khi quay i tng quanh gc to 1 gc bt k l:

cossin

sincosT

3.1.2.2. Php bin i tng hp

Phng php bin i s dng php nhn ma trn vi to im thng qua cc vect v

tr tht s hiu qu v em li cng c mnh v ho cho ngi s dng. Nhng thc t

cc thao tc thng cn khng ch mt m nhiu php bin i khc nhau. Ta c php

hon v khi nhn ma trn l khng thc hin nhng kh nng t hp cc php nhn li cho

php to ra mt ma trn bin i duy nht. Lm gim bt ng k khi lng tnh ton

trong qu trnh bin i, lm tng tc cc chng trnh ng dng v to iu kin cho vic

qun l cc bin i trong ng dng.

Gi s ta c P vi [X] = [x y], c hai php bin i [T1] quay quanh gc to 900:

01

101T

V [T2] ly i xng P qua gc to :

10

012T

Ta c:

PTIT


45

xyyxTXX

01

10*1*'

xyxyTXX

10

01*2*'"

Gi s [T3] l ma trn tng hp [T1] v [T2]

xyyxTXX

01

10*3**

Kt lun: bin i qua nhiu ma trn thnh phn s tng ng vi php bin i

qua ma trn tng hp t cc php bin i .

3.2. TA NG NHT V CC PHP BIN I

3.2.1. To ng nht

Ta xt php tnh tin:

x= x + dx

y = y + dy

Vy P = P + [T]

dy

dxT

Vy php bin i tng hp:

P=P*[T] + [T] = (P + [T]) [T] + [T]

R rng khng th biu din thng qua ma trn tng hp 2 chiu 2x2 c. Php

tnh tin a ra ma trn bin i tng hp l khng th.

Th no l phng php biu din to ng nht ? l phng php biu din m

rng thng qua to ng nht ca cc vect v tr khng ng nht [x y] l ng dng

ca php bin i hnh hc m to im c m t di ma trn [x* y* h] vi

x=x*/h, y=y*/h c h l mt s thc tu . Vy mt vect v tr [xy] bt k trn mt phng

xoy bng tp v s cc im ng nht [hx hyh]. V d: [2 5] s biu din bng [4102],

[6153]..... n gin nht l [251].

Vy to ng nht ca vect v tr [X]= [ x y 1]. Khi ma trn bin i s l

3x3:

1

0

0

dydx

dc

ba

T

1

0

0

*1.'

dydx

dc

ba

yxTPP

x=ax + cy +dx

PTIT


46

y=bx + dy + dy

[X]= [x y1]

Trong dx, dy l khong tnh tin theo trc x v y.

3.2.2. Php bin i vi to ng nht

Ma trn bin i ng nht

Php tnh tin

C a=d=1 v b=c=0

1

010

001

nm

T

Ch : trong mt phng to , mi im k c gc to u c th bin i.

Php bin i tng hp ca hai php tnh tin theo khong [m1 n1] v [m2n2] bng

php bin i duy nht mt khong c gi tr bng tng ca hai php bin i

[m1+m2n1+n2].

Tht vy:

12121

010

001

122

010

001

*

111

010

001

2*1

nnmmnmnm

TT

Php t l

Tng t ma` trn t l:

100

00

00

Sy

Sx

Ts

Ch : Php bin i tng hp ca hai php t l Sx1, Sx2 v Sy1,Sy2 bng php

bin i duy nht c t l l tch hai php bin i trn Sx1*Sx2, Sy1*Sy2.

Ci t c/c++ cho php t l on thng to t (x1,y1) n (x2,y2):

x11=int(x1*1.1); x22=int(x2*0.9);

y11=int(y1*1.1); y22=int(y2*0.9);

Php quay

1

0

0

][

nm

dc

ba

T

]1[

1

010

001

]1[]1''[ nymx

nm

yxyx

]1..[

100

00

00

]1[]1''[ SyySxxSy

Sx

yxyx

PTIT


47

Hnh 3.6 Php quay

Ch : Php bin i tng hp ca hai php quay 1 v 2 l

Ci t c/c++ cho php quay tam gic quanh gc to :

void quay(int &x,int &y,float

goc){

goc=goc*3.14/180;

int t=x;

x=x*cos(goc)-y*sin(goc);

y=t*sin(goc)+y*cos(goc); }

tamgiac(x1,y1,x2,y2,x3,y3,4);

quay(x1,y1,goc);

quay(x2,y2,goc);

quay(x3,y3,goc);

tamgiac(x1,y1,x2,y2,x3,y3,mau);

3.2.3. Ci t c/c++ cho php quay tam gic quanh 1 im (xq,yq):

#define RADS 0.017453293

void Quay(int x1,int y1, int x2, int y2, int x3, int y3,int xq, int yq,float

goc ){

float x11,y11,x22,y22,x33,y33;

float anpha = RADS *goc;

x11=int(x1*cos(anpha)-y1*sin(anpha)+(1-cos(anpha))*xq + sin(anpha)*yq);

y11=int(x1*sin(anpha)+y1*cos(anpha)-sin(anpha)*xq+(1 - cos(anpha))*yq);


y22=int(x2*sin(anpha)+y2*cos(anpha)-sin(anpha)*xq+(1-cos(anpha))*yq);


y33=int(x3*sin(anpha)+y3*cos(anpha)-sin(anpha)*xq+(1 - cos(anpha))*yq);

tamgiac(x11,y11,x22,y22,x33,y33,12);

}

1)cossin()sincos(

100

0cossin

0sincos

]1[]1''[

yxyx

yxyx

y

( x, y )

x

( x, y )

PTIT


48

3.3. CC PHP BIN I HNH HC BA CHIU

Cc php bin i chuyn v - translation, t l-scaling v quay-rotation s dng trong

khng gian 2D u c th m rng trong khng gian 3D.

3.3.1.Biu din im trong khng gian 3 chiu

[x* y* z* h] hay[x*/h y*/hz*/h 1]

Vit gn hn:[x y z 1]

Ma trn bin i tng qut trong khng gian 3D vi ta ng nht (4x4)

Hay

1

0

0

0

dzdydx

hig

fed

cba

T

3.3.2. Php tnh tin

y l php bin i n gin nht, m rng t php bin i trong khng gian 2D ta c:

[X'] = [ X ] . [ T(dx,dy,dz) ]

[ x' y' z' 1 ] = [ x y z 1 ].[ T(dx,dy,dz) ]

= [ x+dx y+dy z+dz 1 ]

Hnh 3.7 Php tnh tin trn 3D

3.3.3. Php t l Tng t trong 2D ta c php t l trong 3D l :

1000

0Sz00

000

000

Sy

Sx

Ts

Ta c Sx,Sy v Sz l cc h s t l trn cc trc to

[X] = [X] . [T(Sx,Sy,Sz) ]

= [x.Sxy.Sy z.Sz 1]

Hnh 3.8 Php t l trn 3D

snml

rjig

qfed

pcba

][T

1

0100

0010

0001

)],,([

dzdydx

dzdydxT

1000

000

000

000

]1[]1'''[Sz

Sy

Sx

zyxzyx

PTIT


49

3.3.4. Php bin dng

Ta c tt c cc phn t nm trn ng cho chnh bng 1

Cc phn t chiu v tnh tin bng 0

[X] = [X] . [Tsh]

Hnh 3.9 Cc php bin dng trn 3D

3.3.5. Php ly i xng

i xng qua trc ox i xng qua mt xoy i xng qua gc O

1000

0100

0010

0001

Mox

1000

0100

0010

0001

Mxoy

1000

0100

0010

0001

Mo

3.3.6. Php quay 3 chiu 3.3.6.1. Quay quanh cc trc to n gin nht l php quay quanh cc trc to ox,oy v oz vi gc dng:

Hnh 3.10 Xc nh gc quay dng trn 3 trc to

Khi ny php quay li a v php quay khng gian 2D quanh gc to

Quay quanh trc oz:

1000

01

01

01

]1[]1'''[ig

fd

cb

zyxzyx

]1[ zfycxizybxgzydx

PTIT


50

Hnh 3.11 Quay quanh trc oz

zz

yxy

yxx

'

'

'

cossin

sincos

Quay quanh trc ox:

Hnh 3.12 quay quanh trc ox

cossin

sincos'

'

'

zyz

zyy

xx

Quay quanh trc oy:

Hnh 3.13 quay quanh trc oy

cossin

sincos

'

'

'

zxz

yy

zxx

Ghi ch: php quay trong khng gian 3D s dng php nhn ma trn bin i khng

c kh nng hon v cc ma trn.

3.3.6.2. Quay quanh mt trc bt k song song vi cc trc ta

u tin chuyn dch i tng cho n khi to a phng ca i tng

trng vi trc to m trc a phng song song.

Quay i tng xung quanh trc ca n (chnh l trc to )

a i tng v to trc khi dch chuyn ta c ma trn tng hp l:

[T//] = [Ttt-].[T].[Ttt+]

1000

0100

00cossin

00sincos

][

Tz

1000

0cossin0

0sincos0

0001

][

Tx

1000

0cos0sin

0010

0sin0cos

][

TyPTIT


51

V d: quay i tng xung quanh mt trc // vi trc z vi khong dch chuyn l

x,y v gc quay l .

Gii:

10)sin)cos1(sin)cos1(

0100

00cossin

00sincos

10

0100

0010

0001

.

10)cossin()sincos(

0000

00cossin

00sincos

10

0100

0010

0001

.

1000

0100

00cossin

00sincos

.

10

0100

0010

0001

..//

xyyx

yxyxyx

yxyx

TTTT ttttz

3.3.6.3. Quay i tng quanh mt trc bt k

Xt bi ton sau, hy tm ma trn bin i a mt trc bt k c hng: V=ax

+ by + cz v trng vi trc oz theo chiu dng.

Hnh 3.14 Quay i tng quanh mt trc

bt k i qua tm

Ta thc hin cc bc nh sau:

Quay V quanh trc y mt gc (-) sao cho

nm trn mt phng (y, z) c V1

Quay V1 quanh trc x mt gc sao cho

trng vi trc z c V2

PTIT


52

Bc 1:

Hnh 3.15 Quay V=ax+by+cz quanh trc oy

Ta tnh ?

Chiu V trn mt phng (x,z) c V. Ta c:

22

sinca

a

22

cosca

c

1000

00

0010

00

1000

0cos0)sin(

0010

0)sin(0cos

2222

2222

ca

c

ca

a

ca

a

ca

c

T y

Vect: 221 ,,0( cabV

Bc 2:

Hnh 3.16 Quay vector V1 quanh trc ox

222sin

cba

b

222

22

coscba

ca

y

V

V

a

a

b

c x

z

y

V V1

a

a

b

c x

z

V2

PTIT


53

1000

00

00

0001

1000

0cossin0

0sincos0

0001

222

22

222

222222

22

cba

ca

cba

b

cba

b

cba

ca

T x

Vy [Tv] = [T-y]. [Tx]

Nu a ngc li ta c:

yxxyv TTTTT .. 11

Cho trc quay L (vect V = ax + by + cz) v mt im P nm trn trc L. Hy

tm ma trn bin i quay i tng xung quanh trc L mt gc .

Gii:

Ta thc hin nh sau:

1. Tnh tin P v gc ta .

2. Chuyn trc L v trng vi trc OZ

3. Quay i tng xung quanh trc OZ mt gc .

4. Thc hin ngc li 2,1

Vy:

ttvZvttL TTTTTT .... 1,, 3.3.7. Ci t bng c/c++ nh sau:

i 3D sang 2D

#define RADS 0.017453293

struct point{

int x,y,z;

}

point Diem3d(int &x, int &y, int &z) {

point p;

p.x = int(getmaxx()/2+ y - x*cos(RADS*45));

p.y = int(getmaxy()/2 - z + x*cos(RADS*45));

return p;

}

Quay quanh cc trc to

point quay(int x,int y,int z,float goc,int truc){

point p;

if (truc==1){ //Quay quanh OX

p.y=y*cos(RADS*goc)-z*sin(RADS*goc);

p.z=y*sin(RADS*goc)+z*cos(RADS*goc);

p.x=x;

}

if (truc==2){ //Quay quanh OY

p.x=x*cos(RADS*goc)+z*sin(RADS*goc);

p.z=-x*sin(RADS*goc)+z*cos(RADS*goc);

p.y=y;

PTIT


54

}

if (truc==3){

p.x=x*cos(RADS*goc)-y*sin(RADS*goc);

p.y=x*sin(RADS*goc)+y*cos(RADS*goc);

p.z=z;

}

return p;

}

Tm tt:

Cc php bin i hnh hc cho php d dng thao tc trn cc i tng c to ra.

Chng lm thay i m t v to ca cc i tng, t i tng s c thay i

v hng, kch thc v hnh dng. Cc php bin i hnh hc c s bao gm tnh tin,

quay v bin i t l. Ngoi ra mt s php bin i khc cng thng c p dng

l php i xng v bin dng.

Cc php bin i hnh hc 2D u c biu din di dng ma trn ng nht 3x3

tin cho vic thc hin cc thao tc kt hp gia chng. Trong h to ng nht, to

ca mt im c m t bi mt vector dng bao gm ba gi tr, hai gi tr u tng

ng vi to Descartes ca im , v gi tr th ba l 1. Vi cch biu din ny, ma

trn ca php bin i c c t s kt hp ca cc php bin i c s s bng tch ca

cc ma trn ca cc php bin i thnh phn.

Php bin i hnh hc 3D l s m rng ca php bin i 2D. Tng t n cng

c cc php: tnh tin, t l, bin dng v quay. Phc tp nht l php quay, nn chng ta

kho st ln lt t n gin n phc tp: quay i tng quanh cc trc to , quanh 1

trc song song vi trc to , quanh 1 trc bt k....Do kho st cc php bin i affine

vi biu din dng ma trn ng nht (4x4 vi 3D) nn cng vic kh n gin v nht

qun.

Lu php tnh tin v quay c chung thuc tnh l: sau khi bin i hnh dng v

kch thc ca i tng khng thay i m chng ch b thay i v tr v nh hng

trong khng gian.

Bi tp:

1.

a. Hy tm ma trn biu th php quay mt i tng mt gc 600 quanh gc

to .

b. Tm to mi ca P(-3,3) sau khi thc hin php quay trn?

2.

a. Hy vit dng tng qut ca ma trn iu chnh t l tng ng vi mt

im c nh Q(a,b)?

b. Phng ln t gic c cc nh A(0,0), B(1,3), C(4,2) v D(3,1) ln hai ln

kch thc ban u ca n trong khi vn gi im D(3,1)?

3.

PTIT


55

a. M t php bin i nhm quay 1 i tng mt gc xung quanh mt

tm c nh Q(a,b)?

b. Thc hin php quay tam gic ABC c A(0,0), B(1,1) v C(4,2) mt gc

450 xung quanh im (-1, -1)?

4. Cho ABC c cc to nh l A(2,2), B(3,1) v C(4,3). Xc nh ma trn bin

i affine bin i tam gic ny thnh ABC bit nh A(4,3), B(4,5) v C(7,3).

5.

a. hy tm mt ma trn dnh cho php phn chiu i xng gng xung quanh

mt ng thng G vi dc m v tung gc (0,g)?

b. To phn x i xng gng a gic m nh ca n: A(-1,0), B(0,-2),

C(1,0) v D(0,2) xung quanh ng G trong cc trng hp sau:

i. x=2

ii. y=3

iii. y=2x+3

6. Cho hnh chp ABCD c cc to A(0,0,0), B(1,0,0), C(0,1,0) v D(0,0,1). Quay

hnh chp quanh ng L (v=x+y+z) i qua im C mt gc 450. Tm to hnh

chp mi.

7.

a. Hy tm php bin i dnh cho php i xng gng tng ng vi mt

mt phng cho (c vector php tuyn M, trn c mt im P).

b. p dng tm ma trn cho php i xng gng vi mt phng i qua gc

to v vector php tuyn c hng M=x+y+z.

8. Vit chng trnh vi i tng (ng thng, tam gic, t gic...) trong mt

phng 2D

a. Dch chuyn (dng cc phm dch chuyn)

x

y P

P

G

0

g

x

z

y

M

Q

Q

PTIT


56

b. Phng ln, thu nh (dng phm dch chuyn hay g vo t bn phm)

c. Quay vi gc quay c g vo t bn phm (gc g vo c tnh bng

)

9. Vit chng trnh vi i tng (hnh kim cng, hnh lp phng ...) trong 3D

a. Dch chuyn

b. Phng ln, thu nh

c. Quay mt gc

PTIT

Chng 4: Cc gii thut ho c s

57

CHNG 4: CC GII THUT HO C S

4.1. M HNH CHUYN I GIA BA H THNG TO

4.1.1. M hnh chuyn i

H ta Descartes l d thch ng cho cc chng trnh ng dng miu t cc hnh

nh (picture) trn h ta th gii thc (World Coordinate System). Cc hnh nh c

nh ngha trn h ta th gii thc ny sau c h ha v ln cc h ta

thit b (Device Coordinate). in hnh, mt vng ha cho php ngi s dng xc

nh vng no ca hnh nh s c hin th v bn mun t n ni no trn h ta

thit b. Mt vng n l hoc vi vng ca hnh nh c th c chn. Nhng vng ny

c th c t nhng v tr tch bit, hoc mt vng c th c chn vo mt vng

ln hn. Qu trnh bin i ny lin quan n nhng thao tc nh tnh tin, bin i t l

vng c chn v xa b nhng phn bn ngoi vng c chn.

Hnh 4.1 WCS

World Coordinate System

Hnh 4.2 NDCS Normalized

Device Coordinate System

Hnh 4.3 DCS

Device Coordinate System

Qui trnh chuyn i cc i tng trong WCS sang NDCS c gi l

php nh x ca s sang khung nhn hay php bin i chun ho (Window

to Viewport mapping or Normalization Transformation)

Qui trnh c th p cc to thit b hin th chun ho sang cc thit b ri

rc c gi l php bin i trm lm vic (Workstation Transformation)

4.1.2. Php nh x t ca s vo khung nhn

Hnh 4.4 i tng trong ca s Hnh 4.5 i tng ti khung nhn

Mt ca s (window) c ch nh bi bn to thc (WCS): Xwmin,

Xwmax, Ywmin, Ywmax

PTIT


58

Mt khung nhn (viewport) c m t bi bn to thit b chun ho

(NDCS): Xvmin, Xvmax, Yvmin, Yvmax

Mc ch ca php nh x ny l chuyn i cc to thc (Xw,Yw) ca mt im

tu sang thit b chun ho tng ng (Xv,Yv). gi li khong cch ca im trong

khung nhn bng vi khong cch ca im trong ca s, vi yu cu:

minmax

min

minmax

max

ww

ww

vv

vv

xx

xx

xx

xx

minmax

min

minmax

max

ww

ww

vv

vv

yy

yy

yy

yy

Ta c:

maxmin

minmax

minmax )()(

)(vww

ww

vvv xxx

xx

xxx

maxmin

minmax

minmax )()(

)(vww

ww

vvv yyy

yy

yyy

Ta c tm gi tr trn xc nh ca s v khung nhn u l hng s. Vy chng

ta hon ton tnh c (Xv,Yv) t (Xw, Yw) qua php bin i:

[XvYv1] = [XwYw 1]. [T]

[T] = [Ttt-]. [Ts] . [Ttt]

Ma trn tnh tin theo window:

1

010

001

ww

tt

yx

T

Ma trn chuyn i t l window vo viewport l:

100

00

00

minmax

minmax

minmax

minmax

ww

vv

ww

vv

syy

yy

xx

xx

T

Ma trn tnh tin theo to viewport:

1

010

001

vv

tt

yx

T

Vy ma trn bin i t ca s vo khung nhn:

PTIT


59

1..

00

00

minmax

minmaxminmin

minmax

minmaxminmin

minmax

minmax

minmax

minmax

ww

vvwv

ww

vvwv

ww

vv

ww

vv

yy

yyyy

xx

xxxx

yy

yy

xx

xx

T

4.2. CC GII THUT XN TI (CLIPPING)

4.2.1. Khi nim

Xn ta l tin trnh xc nh cc im ca mt i tng nm trong hay ngoi ca s hin

th. Nm trong c hin th, nm ngoi loi b.

Vic loi tng im nh ca i tng thng chm nht l khi i tng m phn

ln nm ngoi ca s hin th.

4.2.2. Clipping im

Gi s (x,y) l to ca mt im, vy im c hin th khi tho mn:

Xmin


60

Hnh 4.6 Mt phng m trong cc trng hp clipping on thng

M vng c xc nh theo 9 vng ca mt phng m cc im cui nm vo .

Mt bt c ci t true (1) hoc false (0).

Bt 1: im cui bn trn ca s = sign(y-ymax)

Bt 2: im cui bn di ca s = sign(ymin-y)

Bt 3: im cui bn phi ca s = sign(x-xmax)

Bt 4: im cui bn tri ca s = sign(xmin-x)

Qui c sign(a) = 1 nu a dng

= 0 nu a m

Bc 2 :Qu trnh kim tra v tr ca on thng so vi ca s. Tt c im u v im

cui ca on thng c m.

Gii thut nh sau :

Bc 1 :Nu m ca P1 hoc P2 u = 0000 th ton b on thng thuc phn hin th.

If (P1.M OR P2.M == 0000) then c on thng thuc ca s hin th

Bc 2 : Nu m ca P1 v P2 c cng mt v tr m P1 v P2 => cng pha

If (P1.M AND P2.M != 0000) then 2 im nm v 1 pha ca ca s

Bc 3: Xt giao im:

Hnh 4

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