1. PPT&Programs&Labcourse http://211.87.235.32/share/ComputerGraphics/ 1

2. 2 Transformations Shandong University Software College Instructor: Zhou Yuanfeng E-mail: yuanfeng.zhou@gmail.com

3. 3 Objectives Introduce standard transformations ­Rotation ­Translation ­Scaling ­Shear Derive homogeneous coordinate transformation matrices Learn to build arbitrary transformation matrices from simple transformations

4. Why transformation? 4 Before After Before

5. Why transformation? 5 Snow construction

6. 6 General Transformations A transformation maps points to other points and/or vectors to other vectors Q=T(P) v=T(u)

7. 7 Affine Transformations Linear transformation + translation However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations

8. 8 Affine Transformations Line preserving Characteristic of many physically important transformations ­Rigid body transformations: rotation, translation ­Scaling, shear Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

9. 9 Pipeline Implementation transformationrasterizer u v u v T T(u) T(v) T(u) T(v) vertices pixels frame buffer (from application program)

10. 10 Notation We will be working with both coordinate-free representations of transformations and representations within a particular frame P,Q, R: points in an affine space u, v, w : vectors in an affine space , ,  : scalars p, q, r : representations of points -array of 4 scalars in homogeneous coordinates u, v, w : representations of vectors -array of 4 scalars in homogeneous coordinates

11. 11 The World and Camera Frames When we work with representations, we work with n-tuples or arrays of scalars Changes in frame are then defined by 4 x 4 matrices In OpenGL, the base frame that we start with is the world frame Eventually we represent entities in the camera frame by changing the world representation using the model-view matrix Initially these frames are the same ( M=I )

12. 12 Moving the Camera If objects are on both sides of z=0, we must move camera frame M =

13. 13 Translation Move (translate, displace) a point to a new location Displacement determined by a vector v ­Three degrees of freedom ­P’=P+v P P’ v

14. 14 How many ways? Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way objecttranslation: every point displaced by same vector

15. 15 Translation Using Representations Using the homogeneous coordinate representation in some frame p=[ x y z 1] T p’=[x’ y’ z’ 1] T v=[vx vy vz 0] T Hence p’ = p + v or x’=x+ vx y’=y+ vy z’=z+ vz note that this expression is in four dimensions and expresses point = vector + point

16. 16 Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p ’= Tp where T = T ( v x, v y, v z ) = This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

17. 17 Rotation (2D) Consider rotation about the origin by  degrees ­radius stays the same, angle increases by  x’=x cos  –y sin  y’ = x sin  + y cos 

18. 18 Rotation about the z axis Rotation about z axis in three dimensions leaves all points with the same z ­Equivalent to rotation in two dimensions in planes of constant z ­or in homogeneous coordinates p ’= R z (  )p x’=x cos  –y sin  y’ = x sin  + y cos  z’ =z

19. 19 Rotation Matrix R = R z (  ) =

20. 20 Rotation about x and y axes Same argument as for rotation about z axis ­For rotation about x axis, x is unchanged ­For rotation about y axis, y is unchanged R = R x (  ) = R = R y (  ) =

21. 21 Scaling S = S(s x, s y, s z ) = x’=s x x y’=s y y z’=s z z p’=Sp Expand or contract along each axis (fixed point of origin)

22. 22 Reflection corresponds to negative scale factors original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

23. 23 Inverses Although we could compute inverse matrices by general formulas, we can use simple geometric observations ­Translation: T -1 (d x, d y, d z ) = T (-d x, -d y, -d z ) ­Rotation: R -1 (  ) = R(-  ) Holds for any rotation matrix Note that since cos(-  ) = cos(  ) and sin(-  )=-sin(  ) R -1 (  ) = R T (  ) ­Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z )

24. 24 Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The difficult part is how to form a desired transformation from the specifications in the application

25. 25 Order of Transformations Note that matrix on the right is the first applied Mathematically, the following are equivalent p’ = ABCp = A(B(Cp)) Note many references use column matrices to represent points. In terms of column matrices ((A B) T = B T A T ) p ’T = p T C T B T A T

26. 26 General Rotation About the Origin  x z y v A rotation by  about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes R(  ) = R y (-  y ) R x (-  x ) R z (  z ) R y (  y ) R x (  x )  x  y  z are called the Euler angles Note that rotations do not commute We can use rotations in another order but with different angles

27. 27 Rotation About a Fixed Point other than the Origin Move fixed point to origin Rotate Move fixed point back M = T(p f ) R(  ) T(-p f )

28. 28 Instance transformation In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size We apply an instance transformation to its vertices to Scale Orient Locate

29. 29 Shear Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions

30. 30 Shear Matrix Consider simple shear along x axis x’ = x + y cot  y’ = y z’ = z H(  ) =

31. Rotation around arbitrary vector R(Ax,Ay,Az) If then v=w×u 31 R = R z (  ) =

32. 32 OpenGL Matrices In OpenGL matrices are part of the state Multiple types ­Model-View ( GL_MODELVIEW ) ­Projection ( GL_PROJECTION ) ­Texture ( GL_TEXTURE ) (ignore for now) ­Color( GL_COLOR ) (ignore for now) Single set of functions for manipulation Select which to manipulated by ­glMatrixMode(GL_MODELVIEW); ­glMatrixMode(GL_PROJECTION);

33. 33 Current Transformation Matrix (CTM) Conceptually there is a 4 x 4 homogeneous coordinate matrix, the current transformation matrix (CTM) that is part of the state and is applied to all vertices that pass down the pipeline The CTM is defined in the user program and loaded into a transformation unit CTMvertices p p’=Cp C

34. 34 CTM operations The CTM can be altered either by loading a new CTM or by postmutiplication Load an identity matrix: C  I Load an arbitrary matrix: C  M Load a translation matrix: C  T Load a rotation matrix: C  R Load a scaling matrix: C  S Postmultiply by an arbitrary matrix: C  CM Postmultiply by a translation matrix: C  CT Postmultiply by a rotation matrix: C  C R Postmultiply by a scaling matrix: C  C S

35. 35 Rotation about a Fixed Point Start with identity matrix: C  I Move fixed point to origin: C  CT Rotate: C  CR Move fixed point back: C  CT -1 Result : C = TR T –1 which is backwards. This result is a consequence of doing postmultiplications. Let’s try again.

36. 36 Reversing the Order We want C = T –1 R T so we must do the operations in the following order C  I C  CT -1 C  CR C  CT Each operation corresponds to one function call in the program. Note that the last operation specified is the first executed in the program

37. 37 CTM in OpenGL OpenGL has a model-view and a projection matrix in the pipeline which are concatenated together to form the CTM Can manipulate each by first setting the correct matrix mode

38. 38 Rotation, Translation, Scaling glRotatef(theta, vx, vy, vz) glTranslatef(dx, dy, dz) glScalef(sx, sy, sz) glLoadIdentity() Load an identity matrix: Multiply on right: theta in degrees, ( vx, vy, vz ) define axis of rotation Each has a float (f) and double (d) format ( glScaled )

39. 39 Example Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0) Remember that last matrix specified in the program is the first applied Demo glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(1.0, 2.0, 3.0); glRotatef(30.0, 0.0, 0.0, 1.0); glTranslatef(-1.0, -2.0, -3.0);

40. 40 Arbitrary Matrices Can load and multiply by matrices defined in the application program The matrix m is a one dimension array of 16 elements which are the components of the desired 4 x 4 matrix stored by columns In glMultMatrixf, m multiplies the existing matrix on the right glLoadMatrixf(m) glMultMatrixf(m)

41. Matrix multiply // 沿 Y 轴向上平移 10 个单位 glTranslatef(0,10,0); // 画第一个球体 DrawSphere(5); // 沿 X 轴向左平移 10 个单位 glTranslatef(10,0,0); // 画第二个球体 DrawSphere(5); 41 procedure RenderScene(); begin glMatrixMode(GL_MODELVIEW); // 沿 Y 轴向上平移 10 个单位 glTranslatef(0,10,0); // 画第一个球体 DrawSphere(5); // 加载单位矩阵 glLoadIdentity(); // 沿 X 轴向上平移 10 个单位 glTranslatef(10,0,0); // 画第二个球体 DrawSphere(5); end;

42. 42 Matrix Stacks In many situations we want to save transformation matrices for use later ­Traversing hierarchical data structures ­Avoiding state changes when executing display lists OpenGL maintains stacks for each type of matrix ­Access present type (as set by glMatrixMode) by glPushMatrix() glPopMatrix() GL_PROJECTION GL_MODEVIEW

43. Matrix Stacks procedure RenderScene(); begin glMatrixMode(GL_MODELVIEW); //push matrix stack glPushMatrix; //translate 10 along Y axis glTranslatef(0,10,0); //draw the first sphere DrawSphere(5); //come back to the last saved state glPopMatrix; // translate 10 along X axis glTranslatef(10,0,0); //draw the second sphere DrawSphere(5); end; 43

44. 44 Reading Back Matrices Can also access matrices (and other parts of the state) by query functions For matrices, we use as glGetIntegerv glGetFloatv glGetBooleanv glGetDoublev glIsEnabled float m[16]; glGetFloatv(GL_MODELVIEW, m);

45. 45 Using Transformations Example: use idle function to rotate a cube and mouse function to change direction of rotation Start with a program that draws a cube ( colorcube.c ) in a standard way ­Centered at origin ­Sides aligned with axes ­Will discuss modeling in next lecture

46. 46 main.c void main(int argc, char **argv) { glutInit(&argc, argv); glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH); glutInitWindowSize(500, 500); glutCreateWindow("colorcube"); glutReshapeFunc(myReshape); glutDisplayFunc(display); glutIdleFunc(spinCube); glutMouseFunc(mouse); glEnable(GL_DEPTH_TEST); glutMainLoop(); }

47. 47 Idle and Mouse callbacks void spinCube() { theta[axis] += 2.0; if( theta[axis] > 360.0 ) theta[axis] -= 360.0; glutPostRedisplay(); } void mouse (int btn, int state, int x, int y) { char *sAxis [] = { "X-axis", "Y-axis", "Z-axis" }; /* mouse callback, selects an axis about which to rotate */ if (btn == GLUT_LEFT_BUTTON && state == GLUT_DOWN) { axis = (++axis) % 3; printf ("Rotate about %s\n", sAxis[axis]); }

48. 48 Display callback void display() { glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glLoadIdentity(); glRotatef(theta[0], 1.0, 0.0, 0.0); glRotatef(theta[1], 0.0, 1.0, 0.0); glRotatef(theta[2], 0.0, 0.0, 1.0); colorcube(); glutSwapBuffers(); } Note that because of fixed from of callbacks, variables such as theta and axis must be defined as globals Demo

49. Polygonal Mesh 49

50. Polygonal mesh 50 Object Point cloud

51. Surface reconstruction Polygonal mesh PhD thesis of Hugues Hoppe 1994

52. Mesh smoothing Polygonal mesh Non-iterative, feature preserving mesh smoothing ACM Transactions on Graphics, 2003

53. Mesh simplification Polygonal mesh CGAL, manual

54. Parameterization Polygonal mesh

56. Mesh morphing Polygonal mesh Mean Value Coordinates for Closed Triangular Meshes Ju T., Schaefer S. and Warren J. ACM SIGGRAPH 2005

57. Remeshing&Optimization Polygonal mesh SGP 2009

58. Polygonal mesh Polygonal mesh in OpenGL 58 f=x^4-10*r^2*x^2+y^4-10*r^2*y^2+z^4-10*r^2*z^2 r=0.13

59. Representation 59

60. Representation 60