wØgvwÎK iæcvšÍiY ( 2D Transformation )

Kw¤úDUvi MÖvwd· Gi g‡a¨ me‡P‡q mvaviY Ges ¸iZ¡c~Y© KvR¸‡jvi GKwU MÖvwdKvj `„k¨ ev K¨v‡giv, †h `„k¨ Ae‡R‡±i ¯’vbv¼(Ae¯’vb, Awf‡hvRb Ges AvKvi) G iæcvšÍwiZ nq|

Avgiv mnR fv‡e ej‡Z cvwi †h, 2D Gi g‡a¨ Translation(¯’vbvšÍiY), Rotation(N~Y©b), Scaling, Mirroring, Shearing I Affine iæcvšÍiY Ges djvdj ¸‡jv‡K ewa©Z (extend) Kiv Ges Ae‡k‡l wKfv‡e GKwaK iæcvšÍ‡ii mnR GKwU †hŠwMK cwieZ©b n‡Z cv‡i Zv †`Lv|

Kw¤úDUvi MÖvwd· G †Kb R¨vwgwZK iæcvšÍiY cÖ‡qvRb nq?

Kvib¸‡jv njt

1| †`Lvi mvnv‡q¨2| g‡Wwjs Uz‡ji mvnv‡h¨3| GKwU B‡gR g¨vwbcy‡jkb Uz‡ji wn‡m‡e

Translation( ¯’vbvšÍiY)

¯’vbvšÍiY ev Abyev` GKwU Acv‡ikb †h GKwU Ae‡R±‡K ¯’vbPz¨Z K‡i GKwU wbw`©ó w`‡K wbw`©ó `~i‡Z¡i gva¨‡g| hw` ¯’vbPz¨wZ †f±i V = txI + tyJ bZzb Ae‡R± c‡q›U P’ (x’, y’) iæcvšÍi cÖ‡qvR‡bi Øviv cvËqv †h‡Z cv‡i Tv ‡_‡K P(x, y), P’

= Tv (P).

¯’vbvšÍiY ev Abyev`(Translation) Acv‡ikb t Abyev` (Translation) 3x3 g¨vwUª· AvK…wZ t

x’ = x + tx

y’ = y + ty

Scaling ( ‡¯‹wjs )

‡¯‹wjs GKwU cÖwÎæqv hv, GKwU e¯‘i AvqZb †K cÖmvwiZ ev msKzwPZ K‡i| GKwU KzIwW©‡bU †¯‹wjs gv‡b †¯‹jvi Øviv Zvi cÖwZwU Dcvavb‡K ¸b Kiv nq|

GKiæc ev mgcÖK„wZ gv‡b me Dcvav‡bi Rb¨ GKB †¯‹jvi|

Tx = 2Ty =1

x2

‡¯‹wjs Acv‡ikb t 2 x 2 g¨vwUª· AvK…wZ t

x’ = x + Sx

y’ = y + Sy

BwZevPK ev c‡RwUf †¯‹wjs Gi aªeK Sx Ges Sy cwieZ©b m¤ú‡K© eb©bv Ki‡Z e¨envi Kiv nq h_vµ‡g X Ges Y wb‡`©k mv‡cÿ¨| GKwU †¯‹wjs aªeK GKwUi Zzjbvq Ab¨ GKwUi m¤úªmvi‡b Ges Kg GK ‰`‡N¨i ms‡KvP‡bi Bw½Z †`q|

wfbœiæc †¯‹wjs t wfbœiæc †¯‹wjs gv‡b cÖwZwU Dcv`v‡bi Rb¨ wewfbœ †¯‹jvi |

‡¯‹wjs iæcvšÍi m¤úbœ nËqvi ci bZzb e¯‘wU g~j we›`yi mv‡c‡ÿ Ab¨ GKwU Ae¯’v‡b Aew¯’Z nq| Ae‡kl we›`ywU-B nj w¯’i g~jwe›`y|

hw` Dfq †¯‹wjs Dcv`v‡bi gvb D`vnviY GKB nq Zvn‡j †¯‹wjs iæcvšÍi‡K mRvwZ() e‡j|

hw` s>1GwU GKwU e„nËixKiY hw` s<1GKwU n«vm|

wØgvwÎK N~Y©b (2D - Rotation)

N~Y©‡b, e¯‘ g~j we›`y †_‡K θ°‡h wgjb †h w`‡K Nyb©Y nq Zv Nwoi KvUvi gZ, hw` c‡RwUf ‡Kvb ev cÿ Zvn‡j,

x’ = x cos(θ) - y sin(θ), y’ = x sin(θ) + y cos(θ)

g~j we›`y †_‡K θ°Ges φ

x = r cos (φ)

Sx = 2Sy = 0.5

(x, y)

(x’, y’)

(x, y)

(x’, y’)

y = r sin (φ)x’ = r cos (φ + θ)y’ = r sin (φ + θ)¯úó fv‡e… 2 x 2 g¨vwUª· AvK…wZ t

x’ = r cos(φ) cos(θ) – r sin(φ) sin(θ),y’ = r sin(φ) sin(θ) + r cos(φ) cos(θ)

weKí fv‡e.....x’ = x cos(θ) - y sin(θ)y’ = x sin(θ) + y cos(θ)

hw`I sin(θ)Ges cos(θ) ‰iwLK (nonlinear) dvskb Gi θ nq|Ñ x’ GKwU x Ges y Gi ˆiwLK mgbœq| Ñ y’ GKwU x Ges y Gi ˆiwLK mgbœq|

mgm¨v -1 tA) wKfv‡e GKwU Ae‡R± †gwUª·‡K g~j we›`y †_‡K 300 `~‡i N~Y©b ev AveZ©b Ly‡R †ei Kiv hvq| B) †hLv‡b bZzb ¯’vbv¼ c‡q›U P (2, -4).

mgvavb t B) c‡q›U P (2, -4) bZzb Ae‡R‡±I ev e¯‘i ¯’vbv¼ n‡e|A)

mgm¨v -2 t hw` GKwU wÎfz‡Ri A(0, 0), B (1, 1) Ges B (5, 2) c‡q›U ¸‡jv AvevZ©b ev N~Y©b K‡i, Zvn‡j1| DrcwËi m¤úK© wK Ges 2| c‡q›U P(-1, -1) Gi m¤úK© wK?

mgvavbtA) Avgiv, mRvwZ ¯’vbv¼ Gi †Q`we›`y †_‡K GKwU wÎfzR AvK…wZ‡Z g¨vwUª· Øviv wPwÎZ Kwi t

GLb...

B) Avgiv, mRvwZ ¯’vbv¼ Gi †Q`we›`y †_‡K GKwU wÎfzR AvK…wZ‡Z g¨vwUª· Øviv wPwÎZ Kwi t

GLb...

wÎgvwÎK iæcvšÍiY ( 3D Transformation )

wØgvwÎK iæcvšÍi‡bi gZB, wÎgvwÎK iæcvšÍi‡b 4 × 4 g¨vwUª· GKBiæc ev mRvwZ ¯’vbv¼ e¨envi K‡i Dc¯’vcb Kiv hvq| Avgiv GKBiæc ev mRvwZ ¯’vbv¼ e¨envi K‡i c‡q›U ¸‡jv Dc¯’vcb Ki‡Z cvwi| c‡q›U ¸‡jv n‡jv- P = (x, y, z) ----> P ' = (x, y, z, W).

Abyiæc we›`y ev c‡q›U nj P = (x/W, y/W, z/W),

1| GKwU GKiæc ev mRvwZ we›`yi W ¯’vbv¼ nj mvaviYZ 1|

2| W = 0 mv‡_ GKiæc ev mRvwZ c‡q›U ev we›`y n‡jv GKwU we‡kl w`‡K Amxg `yi‡Z¡ Aew¯’Z|

wÎgvwÎK ¯’vbvšÍiY ( 3D Translation )

GKwU we›`y ev c‡q›U-G ¯’vbvšÍiY(Translation) t 4x4 g¨vwUª· AvK…wZ t

x'= x + t , y'= y + t , z'= z + tz

wÎgvwÎK ‡¯‹wjs ( 3D Scaling)

GKiæc †¯‹wjs 4x4 g¨vwUª· AvK…wZ t

x'= x* s , y'= y * s , z'= z * sz

D`vnviYmiƒc, hLb Sx= Sy=Sz = 0.5 Ges eo NbK (sides = 1) Ges †Qvb NbK(sides = 0.5) nq Zvn‡j

GKwU wbw`©ó wbe©vwPZ Ae¯’v‡b †¯‹wjs|

mgm¨v-1 t wKfv‡e y ‡ivj mgxKiY e¨envi K‡i, 300 Øviv N~Y©b ev AveZ©b Kiv hvq|mgvavbt c = cos(30) = 0.866, s = sin(30) = 0.5, Ges

cÖ_g jvBbt 3*c + 1*0 + 4*s + 1*0 = 4.62q jvBbt 3*0 + 1*1 + 4*0 + 1*0 = 1, 3q jvBbt 3*(-s) + 1*0 + 4*c + 1*0 = 1.964,4_© jvBbt 3*0 + 1*0 + 4*0 + 1*1 = 1

mgm¨v-2 t wKfv‡e z ‡ivj mgxKiY e¨envi K‡i, 300 Øviv N~Y©b ev AveZ©b Kiv hvq|mgvavbt c = cos(30) = 0.866, s = sin(30) = 0.5, Ges

cÖ_g jvBbt 3*c + 1*(-s) + 4*0 + 1*0 = 2.0982q jvBbt 3*s + 1*c + 4*0 + 1*0 = 2.366, 3q jvBbt 3*0 + 1*0 + 4*1 + 1*0 = 4,4_© jvBbt 3*0 + 1*0 + 4*0 + 1*1 = 1

References:

Books:1. Computer Graphics Principles and Practice, Foley, van Dam, Feiner, and Hughes,

Addison-Wesley © 1997 2. Computer Graphics, Schaum's Outlines, Plastock and Kalley, McGraw-Hill © 1986

3. Principles of Interactive Computer Graphics, Newman and Sproull, McGraw-Hill ©1979