Modeling Transformations

Transcription

1 Transformations Transformations Specif transformations for objects o Allos definitions of objects in on coordinate sstems o Allos use of object definition multiple times in a scene Adam Finkelstein Princeton Universit COS 426, Spring 23 H&B Figure 9 Overvie 2D Transformations o Matri representation o Matri composition o Ra casting Scale Rotate Translate Scale Translate World 2D Transformations 2D Transformations Let s look at this in detail World

2 2D Transformations 2D Transformations Scale.3,.3 Rotate -9 Translate 5, 3 Scale.3,.3 Rotate -9 Translate 5, 3 2D Transformations Scale.3,.3 Rotate -9 Translate 5, 3 World o + t o + t o * s o * s o + h* o + h* o *cos -*sin o *sin + *cos Transformations can be combined (ith simple algebra) o + t o + t o * s o * s o + h* o + h* o *cos -*sin o *sin + *cos o + t o + t o * s o * s o + h* o + h* (, ) o *cos -*sin o *sin + *cos *s *s (,)

3 o + t o + t o * s o * s o + h* o + h* o *cos - *sin o *sin + *cos (, ) (*s)*cos (*s)*sin (*s)*sin + (*s)*cos o + t o + t o * s o * s o + h* o + h* o *cos -*sin o *sin + *cos (, ) ((*s)*cos (*s)*sin) + t ((*s)*sin + (*s)*cos) + t o + t o + t o * s o * s o + h* o + h* o *cos -*sin o *sin + *cos ((*s)*cos (*s)*sin) + t ((*s)*sin + (*s)*cos) + t Overvie o Matri representation o Matri composition o Ra casting Matri Representation Represent 2D transformation b a matri Multipl matri b column vector appl transformation to point a c a c b d b d a + b c + d Matri Representation Transformations combined b 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!

4 22 Matrices What tpes of transformations can be represented ith a 22 matri? 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Identit? a c b d 2D Rotate around (,)? cos * sin * sin * + cos* cos a b sin sin c d cos 2D Scale around (,)? s* s * as c b ds 2D Shear? + sh* sh * + a bsh sh c d 22 Matrices What tpes of transformations can be represented ith a 22 matri? 22 Matrices What tpes of transformations can be represented ith a 22 matri? 2D Mirror over Y ais? a b c d 2D Translation? t + + t a NO! c b d 2D Mirror over (,)? a c b d Onl linear 2D transformations can be represented ith a 22 matri Linear Transformations Linear transformations are combinations of o Scale, o Rotation, a b o Shear, and c d o Mirror Properties of linear transformations: o Satisfies: T s p + s2p2) st ( p) + s2t ( o Origin maps to origin o Lines map to lines o Parallel lines remain parallel o Ratios are preserved o Closed under composition ( p2 ) 2D Translation 2D translation represented b a 33 matri o Point represented ith homogeneous coordinates + t + t t t

5 Homogeneous Add a 3rd coordinate to ever 2D point o (,, ) represents a point at location (/, /) o (,, ) represents a point at infinit o (,, ) is not alloed 2 2 (2,,) or (4,2,2) or (6,3,3) Convenient coordinate sstem to represent man useful transformations Basic 2D transformations as 33 matrices cos sin sin cos t t sh sh Translate Rotate Shear s s Scale Affine Transformations Affine transformations are combinations of o Linear transformations, and o Translations Properties of affine transformations: o Origin does not necessaril map to origin o Lines map to lines o Parallel lines remain parallel o Ratios are preserved o Closed under composition f e d c b a Projective Transformations Projective transformations o Affine transformations, and o Projective arps Properties of projective transformations: o Origin does not necessaril map to origin o Lines map to lines o Parallel lines do not necessaril remain parallel o Ratios are not preserved (but cross-ratios are) o Closed under composition i h g f e d c b a Overvie o Matri representation o Matri composition o Ra casting Matri Composition Transformations can be combined b matri multiplication s s t t cos sin sin cos p T(t,t) R() S(s,s) p

6 Matri Composition Matrices are a convenient and efficient a to represent a sequence of transformations o General purpose representation o Hardare matri multipl o Efficienc ith premultiplication» Matri multiplication is associative p (T * (R * (S*p) ) ) p (T*R*S) * p 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) o MT(a,b) * R() * T(-a,-b) Scale b s,s around arbitrar point (a,b) o MT(a,b) * S(s,s) * T(-a,-b) (a,b) (a,b) The trick: First, translate (a,b) to the origin. Net, do the rotation about origin. Finall, translate back. (Use the same trick.) Overvie o Matri representation o Matri composition o Ra casting 3D Transformations Same idea as 2D transformations o Homogeneous coordinates: (,,z,) o 44 transformation matrices p o n m l k j i h g f e d c b a Basic 3D Transformations tz t t sz s s Identit Scale Translation Mirror over X ais

7 Basic 3D Transformations Rotate around Z ais: cos sin sin cos z Rotate around Y ais: cos z sin Rotate around X ais: cos z sin sin cos sin cos z z z Overvie o Matri representation o Matri composition o Ra casting Transformation Hierarchies Scene ma have hierarch of coordinate sstems o Each level stores matri Base representing transformation [M ] from parent s coordinate sstem Upper Arm [M 2 ] Transformation Eample Well-suited for humanoid characters Root Chest Neck LCollar LHip RHip LCollar LKnee RKnee Loer Arm [M 3 ] Head LShld LElbo LShld LElbo LAnkle RAnkle LWrist LWrist Robot Arm Angel Figures 8.8 & 8.9 Rose et al. `96 Transformation Eample Transformation Eample 2 An object ma appear in a scene multiple times Mike Marr, COS 426, Princeton Universit, 995 Dra same 3D data ith different transformations

8 Transformation Eample 2 Building Floor Floor 2 Floor 3 Floor 4 Floor Furniture Office Office N Office Furniture Floor5 Instances Ra Casting With Hierarchies Transform ras, not primitives o For each node...» Transform ra b inverse of matri» Intersect ith primitives» Transform hit b matri Loer Arm [M 3 ] Upper Arm [M 2 ] Base [M ] Bookshelf Desk Desk 2 Chair Chair K Bookshelf Desk Chair Definitions Robot Arm Angel Figures 8.8 & 8.9 Summar Coordinate sstems o World coordinates o coordinates Representations of 3D modeling transformations o 44 Matrices» Scale, rotate, translate, shear, projections, etc.» Not arbitrar arps Composition of 3D transformations o Matri multiplication (order matters) o Transformation hierarchies