To Do. Computer Graphics (Fall 2004) Course Outline. Course Outline. Motivation. Motivation

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1 Comuter Grahics (Fall 24) COMS 416, Lecture 3: ransformations 1 htt:// o Do Start (thinking about) assignment 1 Much of information ou need is in this lecture (slides) Ask A NOW if comilation roblems, visual C++ etc. Not that much coding [solution is aro. 2 lines, but ou ma need more to imlement basic matri/vector math], but some thinking and debugging likel involved Secifics of HW 1 Ais-angle rotation derivation and glulookat most useful (essential?). hese are not covered in tet (look at slides). You robabl onl need final results, but tr understanding derivation. Understanding it ma make it easier to debug/imlement Problems in tet hel understanding material. Usuall, we have review sessions er unit, but this one before midterm Course Course 3D Grahics Pieline 3D Grahics Pieline Modeling (Creating 3D Geometr) Rendering (Creating, shading images from geometr, lighting, materials) Modeling (Creating 3D Geometr) Rendering (Creating, shading images from geometr, lighting, materials) Unit 1: ransformations Resizing and lacing objects in the world. Creating ersective images. Weeks 1 and 2 simlestglut.ee Ass 1 due Se 23 (Demo) Motivation Man different coordinate sstems in grahics World, model, bod, arms, o relate them, we must transform between them Also, for modeling objects. I have a teaot, but Want to lace it at correct location in the world Want to view it from different angles (HW 1) Want to scale it to make it bigger or smaller Motivation Man different coordinate sstems in grahics World, model, bod, arms, o relate them, we must transform between them Also, for modeling objects. I have a teaot, but Want to lace it at correct location in the world Want to view it from different angles (HW 1) Want to scale it to make it bigger or smaller his unit is about the math for doing all these things Reresent transformations using matrices and matri-vector multilications. Demo: HW 1, alet

2 General Idea Object in model coordinates ransform into world coordinates Reresent oints on object as vectors Multil b matrices Demos with alet ranslation: Homogeneous Coordinates (net time) ransforming Normals (net time) Chater 5 in tet. We cover most of it essentiall as in the book. Worthwhile (but not essential) to read whole chater (Nonuniform)) Scale 1 s s 1 Scale( s, s) S 1 s s s s s s s z z szz Shear 1 a 1 1 Shear S a 1 1 Rotations 2D simle, 3D comlicated. [Derivation? Eamles?] 2D? ' sinθ ' sinθ Linear R(X+Y)R(X)+R(Y) Commutative ranslation: Homogeneous Coordinates ransforming Normals

3 Comosing ransforms Often want to combine transforms E.g. first scale b 2, then rotate b 45 degrees Advantage of matri formulation: All still a matri Not commutative!! Order matters E.g. Comosing rotations, scales R S R( S ) ( RS) SR Inverting Comosite ransforms Sa I want to invert a combination of 3 transforms Otion 1: Find comosite matri, invert Otion 2: Invert each transform and swa order Obvious from roerties of matrices M M M M M M M M M M M M M M M M ( 2 ( 1 1) 2) 3 ranslation: Homogeneous Coordinates ransforming Normals Review of 2D case Orthogonal?, Rotations ' sinθ ' sinθ RR I Rotations in 3D Rotations about coordinate aes simle sinθ 1 Rz sinθ R sinθ 1 sinθ sinθ R 1 sinθ Alwas linear, orthogonal Rows/cols orthonormal R R I R(X+Y)R(X)+R(Y)

4 Geometric Interretation 3D Rotations Rows of matri are 3 unit vectors of new coord frame Can construct rotation matri from 3 orthonormal vectors u u zu Ruvw v v zv u u X + uy + zuz w w z w u u zu R v v zv? w w z w z u v w Geometric Interretation 3D Rotations u u zu u R v v zv v w w z w z w Rows of matri are 3 unit vectors of new coord frame Can construct rotation matri from 3 orthonormal vectors Effectivel, rojections of oint into new coord frame New coord frame uvw taken to cartesian comonents z Inverse or transose takes z cartesian to uvw Non-Commutativit Not Commutative (unlike in 2D)!! Rotate b, then is not same as then Order of aling rotations does matter Follows from matri multilication not commutative R1 R2 is not the same as R2 R1 Demo: HW1, order of right or u will matter simlestglut.ee Arbitrar rotation formula Rotate b an angle θ about arbitrar ais a Not in book. (see bottom age 98, but not ver comlete) Homework 1: must rotate ee, u direction Somewhat mathematical derivation, but useful formula Problem setu: Rotate vector b b θ about a Helful to relate b to X, a to Z, verif does right thing For HW1, ou robabl just need final formula simlestglut.ee Warnings and Caveats he derivation is quite involved mathematicall Don t focus on math details (but the are here for those who are articularl interested). Instead, see if ou can understand the ver basic stes his section is more for ou, if ou are interested. his material was covered quickl in class and won t be tested Common oeration In ractice, such as in HW 1, ou do often need to rotate b an arbitrar vector. So, the final formula is good to know hough in ractice, ou ll likel use a canned routine like setrot or glrotate that imlements it directl Ais-Angle Angle formula Ste 1: b has comonents arallel to a, erendicular Parallel comonent unchanged (rotating about an ais leaves that ais unchanged after rotation, e.g. rot about z) Ste 2: Define c orthogonal to both a and b Analogous to defining Y ais Use cross roducts and matri formula for that Ste 3: With resect to the erendicular com of b Cos θ of it remains unchanged Sin θ of it rojects onto vector c Verif this is correct for rotating X about Z Verif this is correct for θ as, 9 degrees

5 Ais-Angle Angle formula 1(derive on board) Ste 1: b has comonents arallel to a, erendicular Parallel comonent unchanged (rotating about an ais leaves that ais unchanged after rotation, e.g. rot about z) b a ( a b) a a( a b) aa b ( aa ) b b\ a b ( aa ) b ( I aa ) b Ais-Angle Angle formula 2(from last lecture) Ste 2: Define c orthogonal to both a and b Analogous to defining Y ais Use cross roducts and matri formula for that a a c a b A b A za a a z a Dual matri of vector a Ais-Angle Angle formula 3 Ste 3: With resect to the erendicular com of b Cos θ of it remains unchanged Sin θ of it rojects onto vector c ( b\ a) ( b\ a) + csinθ RO b\ a ( I33 aa ) b c Ab ( \ ) RO ( cos θ) + ( sin θ) b a I aa b A b Ais-Angle: Angle: Putting it together (derive) ( \ ) RO ( cos θ) + ( sin θ) RO b a I aa b A b ( b a) ( aa ) b Raθ I θ aa θ A θ (, ) cos + (1 cos ) + sin 2 1 z z 2 Ra (, θ) 1 + (1 cos θ) z+ sinθ z 2 1 z z z