To Do. Course Outline. Course Outline. Goals. Motivation. Foundations of Computer Graphics (Fall 2012) CS 184, Lecture 3: Transformations 1

Transcription

1 Fondations of Compter Graphics (Fall 212) CS 184, Lectre 3: Transformations 1 Sbmit HW b To Do Start looking at HW 1 (simple, bt need to think) Ais-angle rotation and gllookat most sefl Probabl onl need final reslts, bt tr nderstanding derivations. Problems in tet help nderstanding material. Usall, we have review sessions per nit, bt this one before midterm 3D Graphics Pipeline Corse Otline Modeling Animation Rendering 3D Graphics Pipeline Corse Otline Modeling Animation Rendering Unit 1: Transformations Resizing and placing objects in the world. Creating perspective images. Weeks 1 and 2 Ass 1 de Sep 12 (DEMO) Motivation Man different coordinate sstems in graphics World, model, bod, arms, To relate them, we mst transform between them Also, for modeling objects. I have a teapot, bt Want to place 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 Demo of HW 1 Goals This nit is abot the math for these transformations Represent transformations sing matrices and matrivector mltiplications. Demos throghot lectre: HW 1 and Applet Transformations Game Applet Brown Universit Eploratories of Software Credit: Andries Van Dam and Jean Lalef 1

2 General Idea Otline Object in model coordinates Transform into world coordinates Represent points on object as vectors Mltipl b matrices Demos with applet Translation: Homogeneos Coordinates (net time) Transforming Normals (net time) (Nonniform) Scale Shear Scale(,s ) s S s Shear 1 a 1 S 1 a 1 s s z z s s z z Rotations 2D simple, 3D complicated. [Derivation? Eamples?] 2D? Linear ' ' Commtative sin θ R(X+Y)R(X)+R(Y) sin θ Otline Translation: Homogeneos Coordinates Transforming Normals 2

3 Composing Transforms E.g. Composing rotations, scales Often want to combine transforms E.g. first scale b 2, then rotate b 45 degrees Advantage of matri formlation: All still a matri Not commtative!! Order matters 3 R 2 2 S 1 3 R(S 1 ) (RS) 1 3 SR 1 Inverting Composite Transforms Sa I want to invert a combination of 3 transforms Option 1: Find composite matri, invert Option 2: Invert each transform and swap order Obvios from properties of matrices M M 1 M 2 M 3 M M 3 M 2 M 1 Otline Translation: Homogeneos Coordinates Transforming Normals M M M 3 (M 2 (M 1 M 1 )M 2 )M 3 Rotations Rotations in 3D Review of 2D case ' ' Orthogonal?, sin θ R T R I sin θ Rotations abot coordinate aes simple R z R cosθ sinθ sinθ cosθ 1 cosθ sinθ 1 sinθ cosθ Alwas linear, orthogonal Rows/cols orthonormal R 1 cosθ sinθ sinθ cosθ R T R I R(X+Y)R(X)+R(Y) 3

4 Geometric Interpretation 3D Rotations Rows of matri are 3 nit vectors of new coord frame Can constrct rotation matri from 3 orthonormal vectors z R vw v v w w z w z Rp v v w w z w p p z p X + Y + z Z p? v p w p Geometric Interpretation 3D Rotations z Rp v v w w z w Rows of matri are 3 nit vectors of new coord frame Can constrct rotation matri from 3 orthonormal vectors Effectivel, projections of point into new coord frame New coord frame vw taken to cartesian componentz Inverse or transpose takez cartesian to vw p p z p p v p w p Non-Commtativit Not Commtative (nlike in 2D)!! Rotate b, then is not same as then Order of appling rotations does matter Follows from matri mltiplication not commtative R1 * R2 is not the same as R2 * R1 Demo: HW1, order of right or p will matter Arbitrar rotation formla Rotate b an angle θ abot arbitrar ais a Homework 1: mst rotate ee, p direction Somewhat mathematical derivation bt sefl formla Problem setp: Rotate vector b b θ abot a Helpfl to relate b to X, a to Z, verif does right thing For HW1, o probabl jst need final formla Ais-Angle formla Step 1: b has components parallel to a, perpendiclar Parallel component nchanged (rotating abot an ais leaves that ais nchanged after rotation, e.g. rot abot z) Step 2: Define c orthogonal to both a and b Analogos to defining Y ais Use cross prodcts and matri formla for that Step 3: With respect to the perpendiclar comp of b Cos θ of it remains nchanged Sin θ of it projects onto vector c Verif this is correct for rotating X abot Z Verif this is correct for θ as, 9 degrees Ais-Angle: Ptting it together (b \ a) ROT (I 3 3 cosθ aa T cosθ)b + (A * sinθ)b (b a) ROT (aa T )b R(a,θ) I 3 3 cosθ + aa T (1 cosθ) + A * sinθ Unchanged (cosine) Component along a Perpendiclar (rotated comp) (hence nchanged) 4

5 Ais-Angle: Ptting it together (b \ a) ROT (I 3 3 cosθ aa T cosθ)b + (A * sinθ)b (b a) ROT (aa T )b R(a,θ) I 3 3 cosθ + aa T (1 cosθ) + A * sinθ R(a,θ) cosθ (1 cosθ) 2 z z 2 z + sinθ z z z z 2 ( z) are cartesian components of a 5