Modeling Transformations

Transcription

1 Modeling Transformations Thomas Funkhouser Princeton Universit CS 426, Fall 2 Modeling Transformations Specif transformations for objects Allos definitions of objects in on coordinate sstems Allos use of object definition multiple times in a scene H&B Figure 9

2 Overvie 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ra casting 2D Modeling Transformations Modeling Coordinates Scale Translate Scale Rotate Translate World Coordinates 2

3 2D Modeling Transformations Modeling Coordinates Again? World Coordinates 2D Modeling Transformations Modeling Coordinates 3

4 2D Modeling Transformations Modeling Coordinates Scale.3,.3 2D Modeling Transformations Modeling Coordinates Scale.3,.3 Rotate -9 4

5 2D Modeling Transformations Modeling Coordinates Scale.3,.3 Rotate -9 Translate 5, 3 World Coordinates Basic 2D Transformations Translation: Scale: + t + t * s * s Rotation: *cos - *sin *sin + *cos Shear: + h* + h* 5

6 Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: *cos - *sin *sin + *cos Transformations can be combined (ith simple algebra) Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: *cos - *sin *sin + *cos 6

7 ! " # $ % & ' ( ) * +, -. / Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: (,) *cos - *sin *sin + *cos *s *s (,) Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: *cos - *sin *sin + *cos (*s)*cos (*s)*sin (*s)*sin + (*s)*cos 7

8 : ; < Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: *cos - *sin *sin + *cos ((*s)*cos (*s)*sin) + t ((*s)*sin + (*s)*cos) + t Basic 2D Transformations Translation: Scale: + t + t * s * s Shear: + h* + h* Rotation: *cos - *sin *sin + *cos ((*s)*cos (*s)*sin) + t ((*s)*sin + (*s)*cos) + t 8

9 A B C D E F G Overvie 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ra casting Matri Representation We can represent a 2D transformation b a matri a c b d Multipling a matri b a column vector corresponds to appling the transformation to a point a c b d a + b c + d 9

10 Matri Representation Transformations can be combined b matri multiplication a c be d g f i h k j l Matrices are a convenient and efficient a to represent a sequence of transformations 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Identit? a c b d 2D Scale around (,)? s* s * as c b ds

11 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Rotate around (,)? cos * sin * sin * + cos* cos a b sin sin c d cos 2D Shear? sh sh + * * + a bsh sh c d 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Mirror over Y ais? 2D Mirror over (,)? a c a c b d b d

12 H 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Translation? t + + t a NO! c b d Onl linear 2D transformations can be represented ith a 22 matri 2D Translation 2D translation can be represented b a 33 matri Point represented ith homogeneous coordinates t + + t t t 2

13 3 Basic 2D Transformations Basic 2D transformations as 33 matrices cos sin sin cos t t sh sh Translate Rotate Shear s s Scale Homogeneous Coordinates Add a 3rd coordinate to ever 2D point I (,, ) represents a point at location (/, /) J (,, ) represents a point at infinit K (,, ) is not alloed 2 2 (2,,) or (4,2,2) or (6,3,3) Convenient coordinate sstem to represent man useful transformations

14 4 Linear Transformations Linear transformations are combinations of L Scale, M Rotation, N Shear, and O Mirror Properties of linear transformations: P Satisfies: Q Origin maps to origin R Lines map to lines S Parallel lines remain parallel T Ratios are preserved U Closed under composition ) ( ) ( ) ( p p p p T s s T s s T + + d c b a Affine Transformations Affine transformations are combinations of V Linear transformations, and W Translations Properties of affine transformations: X Origin does not map to origin Y Lines map to lines Z Parallel lines remain parallel [ Ratios are preserved \ Closed under composition f e d c b a

15 ] ^ _ ` a b c d e f g h i j Projective Transformations Projective transformations Affine transformations, and Projective arps a b d e g h c f i Properties of projective transformations: Origin does not map to origin Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved (but cross-ratios are) Closed under composition Overvie 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ra casting 5

16 k l m Matri Composition Transformations can be combined b matri multiplication t t cos sin sin cos s s p T(t,t) R() S(s,s) p Matri Composition Matrices are a convenient and efficient a to represent a sequence of transformations General purpose representation Hardare matri multipl Efficienc ith premultiplication» Matri multiplication is associative p (T * (R * (S*p) ) ) p (T*R*S) * p 6

17 n o Matri Composition Be aare: order of transformations matters» Matri multiplication is not commutative p T * R * S * p Global Local Matri Composition Rotate b around arbitrar point (a,b) MT(-a,-b) * R() * T(a,b) (a,b) Scale b s,s around arbitrar point (a,b) MT(-a,-b) * S(s,s) * T(a,b) (a,b) 7

18 8 Overvie 2D Transformations p Basic 2D transformations q Matri representation r Matri composition 3D Transformations s Basic 3D transformations t Same as 2D Transformation Hierarchies u Scene graphs v Ra casting 3D Transformations Same idea as 2D transformations Homogeneous coordinates: (,,z,) 44 transformation matrices z p o n m l k j i h g f e d c b a z

19 9 Basic 3D Transformations z z z tz t t z z sz s s z z z Identit Scale Translation Mirror over X ais Basic 3D Transformations z z cos sin sin cos Rotate around Z ais: z z cos sin sin cos Rotate around Y ais: z z cos sin sin cos Rotate around X ais:

20 z { } ~ Overvie 2D Transformations Basic 2D transformations Matri representation Matri composition 3D Transformations Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ra casting Transformation Hierarchies Build scene ith hierarch of coordinate sstems Each level stores matri representing transformation from parents coordinate sstem Upper Arm [M 2 ] Base [M ] Loer Arm [M 3 ] Robot Arm Angel Figures 8.8 & 8.9 2

21 Transformation Eample Well-suited for humanoid characters Root Chest LHip RHip Neck LCollar LCollar LKnee RKnee Head LShld LShld LAnkle RAnkle LElbo LElbo LWrist LWrist Rose et al. 96 Transformation Eample Mike Marr, COS 426, Princeton Universit, 995 2

22 Transformation Eample 2 An object ma appear in a scene multiple times Dra same 3D data ith different transformations Transformation Eample 2 Building Floor Floor 2 Floor 3 Floor 4 Floor5 Floor Furniture Office Office N Bookshelf Office Furniture Desk Desk 2 Chair Chair K Bookshelf Desk Chair 22

23 ƒ Ra Casting With Hierarchies Transform ras, not primitives For each node...» Transform ra b inverse of matri» Intersect ith primitives» Transform hit b matri Upper Arm [M 2 ] Base [M ] Loer Arm [M 3 ] Robot Arm Angel Figures 8.8 & 8.9 Summar Coordinate sstems World coordinates Modeling coordinates Representations of 3D affine transformations Scale, rotate, translate, shear 44 Matrices Composition of 3D transformations Matri multiplication (order matters) Transformation hierarchies 23