Notes. University of British Columbia

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1 Notes Drop-bo is no. 14 You can hand in our assignments Assignment 0 due Fri. 4pm Assignment 1 is out Office hours toda 16:00 17:00, in lab or in reading room Uniersit of

2 Uniersit of Chapter 4 - Reminder Transformations transformations

3 Uniersit of Uniersit of Reminder Linear transformation combinations of Shear, scale, rotate, reflect Affine transformation - Add translations Closed under composition Use homogeneous coordinates to keep in matri form General forms: cos sin 1 sin cos 1 z z t t t s s s

4 Uniersit of Uniersit of Clarification Wh is this a rotation matri? cos sin sin cos R s.t. 2 b a a,b R cos sin sin cos b a br ar R cos cos cos cos sin sin cos R

5 Uniersit of Uniersit of Clarification Wh does this matri transform between frames? z z z w u w u w u R ( ) ( ) ( ) ( ) ( ) ( ) z z z z z z UVW XYZ z z z z z z UVW w u w u w u w u w u w u R Z w u Y w u X w u w Z w Y X w Z Y X Z u Y u X u W V U γ β γ β γ β γ β γ β γ β γ β γ β γ β γ β w Z w Y X w W Z Y X V Z u Y u X u U z z z

6 Uniersit of Chapter 5: Transformations- Transforming Normals, Hierarchies and OpenGL, Assignment 1

7 Uniersit of Transforming Normals

8 Computing Normals polgon: N P 3 N ( P2 P1 ) ( P3 P1 ) P 1 P 2 assume ertices ordered CCW when iewed from isible side of polgon normal for a erte used for lighting N supplied b model (i.e., sphere), or computed from neighboring polgons Uniersit of

9 Transforming Normals What is a normal? Vector Orthogonal (perpendicular) to plane/surface Do standard transformations presere orthogonalit? Uniersit of

10 Uniersit of Uniersit of Planes and Normals Plane - all points where Implicit form D z C B A Plane N P 0 D C B A N z P, 1

11 Finding Correct Normal Transform Uniersit of transform a plane P N N' T P' 0 ( QN) T ( MP) 0 N T Q T MP 0 Q T M Q I ( ) M 1 T P ' MP N ' QN N T P 0 Gien M, find Q sta perpendicular substitute from aboe T T T (AB) B A Normal transformed b transpose of the inerse of the modeling transformation

12 Uniersit of Transformations in OpenGL

13 Transformations in OpenGL glmatrimode(gl_modelview); glloadidentit(); glbegin(gl_line_loop); glverte2f(0,0); glverte2f(2,0); glverte2f(2,2); glverte2f(1,3); glverte2f(0,2); glend(); DrawHouse() j O F w i Uniersit of

14 Transformations in OpenGL z 1 w z 1 1 obj Uniersit of GLfloat T[16] { 2,0,0,0, 0,2,0,0, 0,0,2,0 3,1,0,1}; glmatrimode(gl_modelview); glloadmatrif(t); DrawHouse(); j O j F obj F w i O i

15 Transformations in OpenGL An easier wa to do the same thing... glmatrimode(gl_modelview); glloadidentit(); gltranslatef(3,1,0); glscale(2,2,2); DrawHouse(); Uniersit of j O j F obj F w i O i

16 Matri Operations in OpenGL 2 Matrices: Model/iew matri M Projectie matri P Eample: glmatrimode( GL_MODELVIEW ); glloadidentit(); // MId glrotatef( angle,,, z ); // M R()*Id gltranslatef(,, z ); // M T(,,z)*R()*Id glmatrimode( GL_PROJECTION ); glrotatef( ); // P Uniersit of

17 Composing Transformations suppose we want Rotate(z,-90) Translate(2,3,0) j F W i F h i j F W F h F W F h P Rot( z, 90) A P h P Trans(2,3,0) W P A F h Uniersit of P Trans( 2,3,0) Rot( z, 90) W P h

18 Composing Transformations P Trans( 2,3,0) Rot( z, 90) W P h R-to-L: interpret operations wrt fied coords moing object L-to-R: interpret operations wrt local coords changing coordinate sstem OpenGL (L-to-R, local coords) gltranslatef(2,3,0); glrotatef(-90,0,0,1); DrawHouse(); M M MV MV Trans(2,3,0) M Rot( z, 90) M MV MV Uniersit of updates current transformation matri b postmultipling

19 Composing Transformations Rotate(z,-90) Translate(-3,2,0) in local coords F h F W F W F h Uniersit of P Rot( z, 90) Trans( 3,2,0) W P h glrotatef(-90,0,0,1); gltranslatef(-3,2,0); draw_house();

20 Rotation About a Point: Moing Object rotate about p b : θ translate p to origin rotate about origin translate p back θ p (, ) F W Uniersit of T(,,z)R(z,θ)T(,, z)

21 Rotation: Changing Coordinate Sstems same eample: rotation around arbitrar center Uniersit of

22 Rotation: Changing Coordinate Sstems rotation around arbitrar center step 1: translate coordinate sstem to rotation center Uniersit of

23 Rotation: Changing Coordinate Sstems rotation around arbitrar center step 2: perform rotation Uniersit of

24 Rotation: Changing Coordinate Sstems rotation around arbitrar center step 3: back to original coordinate sstem Uniersit of

25 General Transform Composition transformation of geometr into coordinate sstem where operation becomes simpler tpicall translate to origin perform operation transform geometr back to original coordinate sstem Uniersit of

26 Rotation About an Arbitrar Ais ais defined b two points translate point to the origin rotate to align ais with z-ais (or or ) perform rotation undo aligning rotations undo translation Uniersit of

27 Uniersit of Transformation Hierarchies

28 Transformation Hierarchies scene ma hae a hierarch of coordinate sstems stores matri at each leel with incremental transform from parent s coordinate sstem scene graph road stripe1 stripe2... car1 car2... Uniersit of w1 w2 w3 w4

29 Transformation Hierarchies world torso LUleg RUleg LUarm RUarm head LLleg RLleg LLarm RLarm Lfoot Rfoot Lhand Rhand Uniersit of θ rot(z,, ) trans(0.30,0,0)

30 Demo: Brown Applets freesoftware/catalogs/scenegraphs.html Uniersit of

31 Composing Transformations OpenGL eample F h F h F h F 1 F h F h F h glloadidentit(); gltranslatef(4,1,0); glpushmatri(); glrotatef(45,0,0,1); gltranslatef(0,2,0); glscalef(2,1,1); gltranslate(1,0,0); glpopmatri(); F W Uniersit of

32 Transformation Hierarchies Matri Stack D C scale(2,2,2) trans(1,0,0) C D DrawSquare() C C C C glpushmatri() B A B A B A B A glscale3f(2,2,2) gltranslate3f(1,0,0) DrawSquare() glpopmatri() Uniersit of

33 Matri Stacks Means of returning to preiousl-used coordinate sstem Support seeral models or model parts Natural hierarchical structure depth of matri stacks limited in hardware tpicall: 16 for ModelView, 4 for Projection Uniersit of

34 Transformation Hierarchies Uniersit of Eample θ 2 θ 4 θ θ 5 θ 3 1 gltranslatef(,,0); glrotatef( ( θ 1,0,0,1); DrawBod(); glpushmatri(); gltranslatef(0,7,0); DrawHead(); glpopmatri(); glpushmatri(); gltranslate(2.5,5.5,0); glrotatef( ( θ 2,0,0,1); DrawUArm(); gltranslate(0,-3.5,0); glrotatef( ( θ 3,0,0,1); DrawLArm(); glpopmatri();... (draw other arm)

35 Uniersit of Assignment 1

36 Assignment 1 Out toda, due 4pm Fri Oct 15, 2005 Start er soon! Build dinosaur out of spheres and 44 matrices think cartoon, not beaut Template code - program shell, Makefile a1/a1.tar.gz Uniersit of

37 Dinosaurs Uniersit of

38 Articulated Dino Uniersit of

39 Articulated Dino Uniersit of

40 Demo Mabe in a couple weeks Ask prof. Can iew last ears demos of dogs and birds Uniersit of

41 Adice Build then animate one section at a time Ensure ou re constructing hierarch correctl Use bod as scene graph root Continue with attached parts Finish all required parts before Going for etra credit Plaing with lighting or iewing Uniersit of

42 More Adice OK to use glrotate, gltranslate, glscale OK to use glutsolidsphere, or build our own where to put origin? our choice center of object, range -.5 to.5 corner of object, range 0 to 1 Uniersit of

43 More Adice Visual debugging Color sphere faces differentl Draw the current coord sstem Transformations - intuition moe phsical objects around pla with demos Brown scenegraph applets Uniersit of

44 More Adice Transitions safe to linearl interpolate parameters for glrotate/gltranslate/glscale do not interpolate indiidual elements of 44 matri! Uniersit of