Geometric Transformations. Ceng 477 Introduction to Computer Graphics Fall 2007 Computer Engineering METU
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1 Geometric ranormation Ceng 477 Introdction to Compter Graphic Fall 7 Compter Engineering MEU
2 D Geometric ranormation
3 Baic Geometric ranormation Geometric tranormation are ed to tranorm the object and the camera in a cene (or animation or modelling and are alo ed to tranorm World Coordinate to View Coordinate Given the hape tranorm all the point o the hape? ranorm the point and/or vector decribing it. For eample: olgon: corner point Circle Ellipe: center point( point at angle Some tranormation preerve ome o the attribte like ie angle ratio o the hape.
4 ranlation Simpl move the object to a relative poition. ' +t ' +t ' t t ' ' ' ' +
5 Rotation A rotation i deined b a rotation ai and a rotation angle. For D rotation the parameter are rotation angle ( and the rotation point ( r r. We repoition the object in a circlar path arrond the rotation point (pivot point r r
6 Rotation When ( r r ( we have ' r co( φ + r coφ co r inφ in r in( φ + r coφ in + r inφ co r he original coordinate are: r coφ r inφ Sbtitting them in the irt eqation we get: co in + in co In the matri orm we have: R where R co in in co
7 Rotation Rotation arond an arbitrar point ( r r ' r r + ( r + ( r co ( in + ( r r in co r ( r r hi eqation can be written a matri operation (we will ee when we dic homogeneo coordinate.
8 Scaling Change the ie o an object. Inpt: caling actor ( ' ' ' S ' S non-niorm v. niorm caling
9 Homogeno Coordinate All tranormation can be repreented b matri operation. ranlation i additive rotation and caling i mltiplicative (+ additive i o rotate arond an arbitrar point or cale arond a ied point; making the operation complicated. Adding another dimenion to tranormation make tranlation alo repreentable b mltiplication. Carteian coordinate v homogeno coordinate. h h h h h h h h h h
10 Man point in homogeno coordinate can repreent the ame point in Carteian coordinate. In homogeno coordinate all tranormation can be written a matri mltiplication.
11 ranormation in Homogeno C. ranlation Rotation Scaling ( ( t t t t t t ' ( ( R R ' co in in co ( ( S S
12 Compoite ranormation Application o a eqence o tranormation to a point: M M M
13 Compoite ranormation Firt: compoition o imilar tpe tranormation I we appl to cceive tranlation to a point: ( ( ( t + t +t t t t t t t t t t t t t t + + } ( ( { } ( { ( t t t t t t t t
14 ( ( ( S S S Compoite ranormation ( ( ( +φ +φ +φ +φ +φ φ φ+ φ φ+ φ φ φ φ φ R R R co( in( in( co( co co in in in co co in co in in co in in co co co in in co co in in co ϕ ϕ ϕ ϕ
15 Rotation arond a pivot point ranlate the object o that the pivot point move to the origin Rotate arond origin ranlate the object o that the pivot point i back to it original poition ( ( ( ( ( in co co in in co in co co in in co + r r r r r r r r r r r r R ( r r ( R ( r r
16 Scaling with repect to a ied point ranlate to origin Scale ranlate back ( ( ( ( ( S ( ( S (
17 Order o matri compoition Matri compoition i not commtative. So be carel when appling a eqence o tranormation. pivot ame pivot
18 Other ranormation Relection
19 Shear: Deorm the hape like hited lice. ( ( ( (3 ' h ' h
20 ranormation Between the Coordinate Stem Between dierent tem: olar coordinate to carteian coordinate Between two carteian coordinate tem. For eample relative coordinate or window to viewport tranormation.
21 How to tranorm rom to ''? Sperimpoe '' to ' ranormation: ' ranlate o that ( move to ( o Rotate ' ai onto ai R ( ( ' '
22 Alternate method or rotation: Speci a vector V or poitive ' ai: nit vector in the v V V ' direction : ( v v V ' ' nit vector in the ' direction ( v v ( rotate v clockwie o 9
23 Element o an rotation matri can be epreed a element o a et o orthogonal nit vector: v v v v v v R ' ' v
24 Eample: ' ' ( ( ( ( ( ( ( M v R M v M Let triangle be deined a three colmn vector:
25 Aine ranormation Coordinate tranormation o the orm: ' a ' a +a +a +b +b ranlation rotation caling relection hear. An aine tranormation can be epreed a the combination o thee. Rotation tranlation relection: preerve angle length parallel line
26 3 DIMENSIONAL RANSFORMAIONS
27 3D ranormation coordinate. Ual notation: Right handed coordinate tem Analogo to D we have 4 dimenion in homogeno coordinate. Baic tranormation: ranlation Rotation Scaling
28 ranlation move the object to a relative poition. t t t
29 Rotation Rotation arrond the coordinate ae ai ai ai Conterclockwie when looking along the poitive hal toward origin
30 Rotation arond coordinate ae Arrond Arrond Arrond co in in co ( R R ( co in in co ( R R ( co in in co ( R R (
31 Rotation Arrond a arallel Ai Rotating the object arond a line parallel to one o the ae: ranlate to ai rotate tranlate back. ranlate Rotate ranlate back R ( ( ( p p p p
32 Figre rom the tetbook
33 Rotation Arond an Arbitrar Ai ranlate the object o that the rotation ai pae thogh the origin Rotate the object o that the rotation ai i aligned with one o the coordinate ae Make the peciied rotation Revere the ai rotation ranlate back
34 Rotation Arond an Arbitrar Ai
35 Rotation Arond an Arbitrar Ai ( V ( c b a V V i the nit vector along V: Firt tep: ranlate to origin: Net tep: Align with the ai we need two rotation: rotate arond ai to get onto the plane rotate arond ai to get aligned with ai.
36 Rotation Arond an Arbitrar Ai Align with the ai rotate arond ai to get into the plane rotate arond ai to get aligned with the ai ' α β α
37 Dot prodct and Cro rodct v dot v * + v * + v *. hat eqal alo to v * *co(a i a i the angle between v and vector. Dot prodct i ero i vector are perpendiclar. v i a vector that i perpendiclar to both vector o mltipl. It length i v * *in(a that i an area o parallelogram bilt on them. I v and are parallel then the prodct i the nll vector.
38 Rotation Arond an Arbitrar Ai Align with the ai rotate arond ai to get into the plane rotate arond ai to get aligned with the ai We need coine and ine o α or rotation ' α ( b c rojection o on plane coα b d inα c d d b + inα c coα inα d b d b c R ( α c d b d b d c d
39 Rotation Arond an Arbitrar Ai '' (ad Align with the ai rotate arond ai to get into the plane rotate arond ai to get aligned with the ai d co β ( in a β ( in co d a a d a d β β β R ( ( ( ( ( ( ( ( R R R R R R α β β α β
40 Rotation... Alternative Method An rotation arond origin can be repreented b 3 orthogonal nit vector: r r r3 r r r3 R r 3 r3 r33 onto and So to align a given rotation ai onto the hi matri can be thoght o a rotating the nit r * r * and r 3* vector ae. ai we can deine an (orthogonal coordinate tem and orm thi R matri Deine a new coordinate tem ( with the given rotation ai ing:
41 Rotation... Alternative Method + / / ( ( / / ( / / ( ( ( ( ( c b a d b d c d c a d b a d d c a d b a d c b a d b d c d b d c c b b c c b a c b a R Check i thi i eqal to ( ( α β R R
42 Scaling Change the coordinate o the object b caling actor. ' ' ' S
43 Scaling with repect to a Fied oint ranlate to origin cale tranlate back ranlate Scale ranlate back S ( (
44 Scaling with repect to a Fied oint ( ( ( ( ( ( ( S S
45 Relection Relection over plane line or point
46 Shear Deorm the hape depending on another dimenion SH b a and vale depend on vale o the hape SH a b and vale depend on vale o the hape
47 OpenGL Geometric-ranormation Fnction In the core OpenGL librar a eparate nction i available or each baic tranormation (tranlate rotate cale all tranormation are peciied in 3D arameter ranlation: tranlation amont in ae Rotation: angle orientation o the rotation ai that pae throgh the origin Scaling: caling actor or three coordinate
48 Baic OpenGL ranormation glranlate* (t t t; For D application et t glrotate* (theta v v v; theta in degree he rotation ai i deined b the vector (vvv i.e. ( (vvv glscale* ( ; Ue negative vale to get relection tranormation
49 OpenGL Matri Operation glmatrimode (GL_MODELVIEW; modelview mode to tell OpenGL that we will be peciing geometric tranormation. he command impl a that the crrent matri operation will be applied on the 4 b 4 modelview matri. the other mode i the projection mode which peciie the matri that i ed o projection tranormation (i.e. how a cene i projected onto the creen here are alo color and tetre mode that we will dic later
50 OpenGL Matri Operation Once o are in the modelview mode a call to a tranormation rotine generate a matri that i mltiplied b the crrent matri or that mode Whatever object deined i mltiplied with the crrent matri he content o the crrent matri can alo be maniplated eplicitl glloadidentit(; glloadmatri* (element6; where element6 i a ingle bcripted arra that peciie a matri in colmn-major order
51 OpenGL Matri Operation Eample: or (int k; k<6;k++ element6[k](loatk; glloadmatri(element6; will prodce the matri M
52 OpenGL Matri compoition glmltmatri* (otherelement6 he crrent matri i potmltiplied with the matri peciied in otherelement6 M M M crr crr what doe thi impl? In a eqence o tranormation command the lat one peciied in the code will be the irt tranormation to be applied.
53 OpenGL Matri Stack OpenGL maintain a matri tack or all the or matri mode When we appl geometric tranormation ing OpenGL nction the 4 b 4 matri at the top o the matri tack i modiied he top i alo called the crrent matri I we want to create mltiple tranormation eqence and ave the compoition relt we can make e o the OpenGL matri tack
54 OpenGL Matri Stack Initiall there i onl the identit matri in the tack o ind ot how man matrice are crrentl in the tack: glgetintegerv(gl_modelview_sack_dehnmmat glhmatri (; he crrent matri i copied and tored in the econd tack poition glopmatri (; Detro the matri at the top and the econd matri in the tack become the crrent matri
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