Today. CS-184: Computer Graphics. Introduction. Some Examples. 2D Transformations

Transcription

1 Toda CS-184: Comuter Grahics Lecture #3: 2D Transformations Prof. James O Brien Universit of California, Berkele V2005F D Transformations Primitive Oerations Scale, Rotate, Shear, Fli, Translate Homogenous Coordinates SVD Start thinking about rotations... 2 Introduction Some Eamles Transformation: An oeration that changes one configuration into another For images, shaes, etc. A geometric transformation mas ositions that define the object to other ositions Linear transformation means the transformation is defined b a linear function... which is what matrices are good for. 3 Original Images from Conan The Destroer, 1984 Shear Rotation Uniform Scale Nonuniform Scale 4

2 Maing Function Linear -vs- Nonlinear f () = in old image Nonlinear (swirl) c() = [195,120,58] c = c( f ()) Linear (shear) 5 6 Geometric -vs- Color Sace Instancing Color Sace Transform (edge finding) Linear Geometric (fli) 7 M.C. Escher, from Ghostscrit 8.0 Distribution RHW 8

3 RHW Instancing Reuse geometric descritions Saves memor Linear is Linear Polgons defined b oints Edges defined b interolation between two oints Interior defined b interolation between all oints Linear interolation 9 10 Linear is Linear Comosing two linear function is still linear Transform olgon b transforming vertices Linear is Linear Comosing two linear function is still linear Transform olgon b transforming vertices f () = a + b g( f ) = c + d f Scale g() = c + d f () = c + ad + bd g() = a + b 11 12

4 Points in Sace Reresent oint in sace b vector in Relative to some origin Relative to some coordinate aes Later we ll add something etra... = [4,2] T 2 R n Basic Transformations Basic transforms are: rotate, scale, and translate Shear is a comosite transformation Rotate Scale Uniform/isotroic Non-uniform/anisotroic 4 Origin, 0 13 Translate Shear -- not reall basic 14 Linear Functions in 2D Rotations = f (,) = c 1 + c 2 + c 3 = f (,) = d 1 + d 2 + d 3 [ ] [ ] [ ] [ t M M = + t M M ] Rotate 45 degree rotation & Cos( ( ) ' Sin ( )# ' = $ % Sin ( ) Cos( ( ) " = t + M

5 Rotations Rotations Rotations are ositive counter-clockwise Consistent w/ right-hand rule Don t be different... Note: rotate b zero degrees give identit rotations are modulo 360 (or ) 2 Preserve lengths and distance to origin Rotation matrices are orthonormal Det(R) = 1 1 In 2D rotations commute... But in 3D the won t Scales Scales Uniform/isotroic Non-uniform/anisotroic & s 0 # ' = $ % 0 s " Diagonal matrices Diagonal arts are scale in X and scale in Y directions Negative values fli Two negatives make a ositive (180 deg. rotation) Scale Reall, ais-aligned scales Not ais-aligned... 20

6 Shears Shears Shear & 1 H # ' = $ % H 1 " Shears are not reall rimitive transforms Related to non-ais-aligned scales More shortl Translation This is the not-so-useful wa: Translate & t ' = + $ % t # " Arbitrar Matrices For everthing but translations we have: = A Soon, translations will be assimilated as well Note that its not like the others. What does an arbitrar matri mean? 23 24

7 Singular Value Decomosition For an matri, A, we can write SVD: T A = QSR where Q and R are orthonormal and S is diagonal Can also write Polar Decomosition T A = QRSR Decomosing Matrices We can force Q and R to have Det=1 so the are rotations An matri is now: Rotation:Rotation:Scale:Rotation See, shear is just a mi of rotations and scales where Q is still orthonormal not the same Q Comosition Comosition Matri multilication comosites matrices ' = BA Al A to and then al B to that result. ' = B( A) = ( BA) = C Several translations comosted to one Translations still left out... ' = B( A + t) = BA + Bt = C + u shear shear shear Transformations built u from others SVD builds from scale and rotations Also build other was i.e. 45 deg rotation built from shears 27 28

8 Homogeneous Coordiantes Move to one higher dimensional sace Aend a 1 at the end of the vectors & = $ % # " & ~ = $ $ $ % 1 # " Homogeneous Translation & 1 0 ~ ' = $ $ 0 1 $ % 0 0 ~ ~ ' = A ~ t #& t $ $ 1 " $ % 1 # " 29 The tildes are for clarit to distinguish homogenized from non-homogenized vectors. 30 Homogeneous Others Comositing Matrices & ~ $ A A = $ % 0 0 0# 0 1" Rotations and scales alwas about the origin How to rotate/scale about another oint? -vs- Now everthing looks the same... Hence the term homogenized 31 32

9 Rotate About Arb. Point Ste 1: Translate oint to origin Translate (-C) Rotate About Arb. Point Ste 1: Translate oint to origin Ste 2: Rotate as desired Translate (-C) Rotate () Rotate About Arb. Point Scale About Arb. Ais Ste 1: Translate oint to origin Ste 2: Rotate as desired Ste 3: Put back where it was Translate (-C) Rotate () Translate (C) Diagonal matrices scale about coordinate aes onl: ~ ' = ( T) RT ~ = A ~ Not ais-aligned Don t negate the

10 Scale About Arb. Ais Ste 1: Translate ais to origin Scale About Arb. Ais Ste 1: Translate ais to origin Ste 2: Rotate ais to align with one of the coordinate aes Scale About Arb. Ais Ste 1: Translate ais to origin Ste 2: Rotate ais to align with one of the coordinate aes Ste 3: Scale as desired Scale About Arb. Ais Ste 1: Translate ais to origin Ste 2: Rotate ais to align with one of the coordinate aes Ste 3: Scale as desired Stes 4&5: Undo 2 and 1 (reverse order) 39 40

11 Order Matters The order that matrices aear in matters A B BA Some secial cases work, but the are secial But matrices are associative (A B) C = A (B C) Think about efficienc when ou have man oints to transform In general: Secial cases: Matri Inverses undoes effect of Translation: negate t and t Rotation: transose Scale: invert diagonal (ais-aligned scales) Others: A 1 Invert matri Invert SVD matrices A 42 Point Vectors / Direction Vectors Somethings Require Care Points in sace have a 1 for the w coordinate What should we have for a b? w = 0 Directions not the same as ositions Difference of ositions is a direction For eamle normals do not transform normall Position + direction is a osition Direction + direction is a direction Position + osition is nonsense 43 M(a b) (Ma) (Mb) M(Re) R(Me) 44