CS D Transformation. Junqiao Zhao 赵君峤

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1 CS D Transformation Junqiao Zhao 赵君峤 Department of Computer Science and Technology College of Electronics and Information Engineering Tongji University

2 Review Translation Linear transformation Affine transformation Composing transformations

3 3D Transformations Use homogeneous coordinates, just as in 2D case Transformations are now 4x4 matrices We will use a right-handed (world) coordinate system - ( z out of page ) Y Z X

4 Translation x y z 1 = dx 0 dy 1 dz 0 1 x y z 1

5 Scale x y z 1 = sx 0 0 sy sz x y z 1

6 Reflection x y z 1 = s s s x y z 1

7 Shear x y z 1 = 1 a 1 a 3 1 a 5 a a 2 0 a x y z 1

8 Rotation R z (θ) = x y z 1 = cosθ sinθ sinθ cosθ x y z 1 Y Z X

9 Rotation R x (θ) = x y z 1 = cosθ 0 sinθ sinθ 0 cosθ x y z 1 Y Z X

10 Rotation R y (θ) = x y z 1 = cosθ sinθ sinθ cosθ x y z 1 Y Z X

11 General Rotation Matrices Rotation in 2D? Around an arbitrary point Rotation in 3D? Around an arbitrary axis There are many more 3D rotations than 2D

12 Properties of Rotation Matrices Columns of R are mutually orthonormal RR T = R T R = I Square matrices with det(r)=1 Also the rules of determining whether a matrix is a rotation matrix

13 Specifying Rotations In 2D, just an angle θ In 3D, is more complex Basic rotation about origin: Axis and angle Convention: positive rotation is CCW (right handed) Many ways Directly through Euler angles: 3 angles about 3 axes Indirectly through frame transformations Quaternions

14 Euler Angles Any rotation may be described using three angles. If the rotations are written in terms of rotation matrices D, C, and B, then a general rotation A can be written as: A = BCD Gimbal

15 Euler Angles Roll(Φ) around X Pitch(Θ) around Y Yaw(Ψ) around Z

16 Extrinsic vs Intrinsic Rotations Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations: x-y -z = z-y-x

17 Rotation Not commutative if the axis of rotation are not parallel R ( ) R ( ) R ( ) R ( ) x y y x

18 Gimbal Lock Euler basic motions are never expressed in a frame, but a mixed axes of rotation R = cosα sinα 0 sinα cosα Set β = π/2? cosβ 0 sinβ sinβ 0 cosβ cosγ sinγ 0 sinγ cosγ

19 Matrices for Axis-angle Rotations Ruler angles are for coordinates axes What if we want rotation about some random axis? Compute by composing elementary transforms Transform rotation axis to align with x axis Apply rotation Inverse transform back into position

20 Building General Rotations Using elementary transforms Translate axis to pass through origin Rotate about y to get into x-y plane Rotate about z to get into x axis Alternative: construct a frame and change coordinates Choose p, u, v, w Apply transform T = FR x θ F 1 F = u v w p Y X v w p u Z

21 Build a 3D Frame Frame matrix F = u v w p Move point to and from frame P F = F 1 P P = FP F Move transformations using similarity transform T F = F 1 TF T = FT F F 1

22 Build a 3D Frame u, v, w, p Given a vector a and a secondary vector b u should be parallel to a: u = a a Then the u-v plane should contain b: w = u b u b Finally, v = w u Given a vector a only u should be parallel to a Then choose arbitrary b, which should not overlap a Do the same

23 Building General Rotations Build a frame with u, v, w, p F = u v w p Transform T = FR x θ F 1 Interpretation Move to the new frame, rotate, then move back Or, rotate in the new coordinate frame

24 Rotation about an Arbitrary Axis About (ux, uy, uz), a unit vector on an arbitrary axis Rodrigues' rotation formula y Rotate(k, θ) θ u z x x' y' z' 1 = uxux(1-c)+c uyux(1-c)+uzs uzux(1-c)-uys 0 uzux(1-c)-uzs uzux(1-c)+c uyuz(1-c)+uxs 0 uxuz(1-c)+uys uyuz(1-c)-uxs uzuz(1-c)+c x y z 1 where c = cos θ & s = sin θ

25 Quaternions Compare to Euler angles and Rotation matrices Can avoid the gimbal lock Can smoothly interpolate over a sphere Representation A quaternion is composed of one real element and three complex elements A rotation through an angle of θ around the axis defined by a unit vector u where u = u x, u y, u z = u x i + u y j + u z k can be represented by a quaternion: q = e θ 2 (u xi+u y j+u z k) = cos θ 2 + (u xi + u y j + u z k)sin θ 2 P = qpq 1

26 Transformations

27 View Transformation We have the world coordinates of all the vertices Now we want to convert the scene so that it appears in front of the camera

28 View Transformation We want to know the positions in the camera coordinate system We can compute the camera-toworld transformation matrix using the orientation and translation of the camera from the origin of the world coordinate system M w c

29 View Transformation We want to know the positions in the camera coordinate system V w = M w c V c Camera-to-world transformation Point in the world coordinate V c = M 1 w c V w = M c w V w Point in the camera coordinate 29

30 Normal Vectors We also need to know the direction of the normal vectors in the world coordinate system This is going to be used at the shading operation We only want to rotate the normal vector Do not translate it

31 Normal Vectors We need to set elements of the translation part to zero r r r r r r r r r tx ty t z 1 r r r r r r r r r

32 Transformations in OpenGL gltranslatef (dx, dy, dz) glrotatef (theta, ux, uy, uz) glscalef (sx, sy, sz) glloadidentity() Transformations are modulated by OpenGL Matrices

33 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);

34 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 vertices p T CTM p =Tp vertices

35 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 CR Postmultiply by a scaling matrix: C CS

36 Rotation about a Fixed Point Start with identity matrix: C I Move fixed point to origin: C CT -1 Rotate: C CR Move fixed point back: C CT Result: C = T -1 R T which is backwards. This result is a consequence of doing postmultiplications.

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

38 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

39 Rotation, Translation, Scaling Load an identity matrix: glloadidentity() Multiply on right: glrotatef(theta, vx, vy, vz) theta in degrees, (vx, vy, vz) define axis of rotation gltranslatef(dx, dy, dz) glscalef( sx, sy, sz) Each has a float (f) and double (d) format (glscaled)

40 Example Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0) 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); Remember that last matrix specified in the program is the first applied Demo

41 gllookat(eye, center, up) glulookat creates a viewing matrix derived from an eye point, a reference point indicating the center of the scene, and an UP vector Let f = normalized center eye UP = normalized up X = f UP F = X UP f eye

42 Arbitrary Matrices Can load and multiply by matrices defined in the application program glloadmatrixf(m) glmultmatrixf(m) 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

43 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()

44 Matrix Stacks glpushmatrix pushes the current matrix stack down by one, duplicating the current matrix. That is, after a glpushmatrix call, the matrix on top of the stack is identical to the one below it.

45 Reading Back Matrices Can also access matrices (and other parts of the state) by query functions glgetintegerv glgetfloatv glgetbooleanv glgetdoublev glisenabled For matrices, we use as double m[16]; glgetfloatv(gl_modelview, m);

46 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

47 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(); }

48 Idle and Mouse callbacks void spincube() { theta[axis] += 2.0; if( theta[axis] > ) theta[axis] -= 360.0; glutpostredisplay(); } void mouse(int btn, int state, int x, int y) { if(btn==glut_left_button && state == GLUT_DOWN) axis = 0; if(btn==glut_middle_button && state == GLUT_DOWN) axis = 1; if(btn==glut_right_button && state == GLUT_DOWN) axis = 2; }

49 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 Camera information is in standard reshape callback

50 Animate Stars

51 Using the Model-view Matrix In OpenGL the model-view matrix is used to Position the camera Can be done by rotations and translations but is often easier to use glulookat Build models of objects The projection matrix is used to define the view volume and to select a camera lens

52 Model-view and Projection Matrices Although both are manipulated by the same functions, we have to be careful because incremental changes are always made by postmultiplication For example, rotating model-view and projection matrices by the same matrix are not equivalent operations. Postmultiplication of the model-view matrix is equivalent to premultiplication of the projection matrix

53 Smooth Rotation From a practical standpoint, we are often want to use transformations to move and reorient an object smoothly Problem: find a sequence of model-view matrices M 0,M 1,..,M n so that when they are applied successively to one or more objects we see a smooth transition For orientating an object, we can use the fact that every rotation corresponds to part of a great circle on a sphere Find the axis of rotation and angle Virtual trackball

54 Incremental Rotation Consider the two approaches For a sequence of rotation matrices R 0,R 1,..,R n, find the Euler angles for each and use R i = R iz R iy R ix Not very efficient Use the final positions to determine the axis and angle of rotation, then increment only the angle Quaternions can be more efficient than either

55 Interfaces One of the major problems in interactive computer graphics is how to use two-dimensional devices such as a mouse to interface with three dimensional objects Example: how to form an instance matrix? Some alternatives Virtual trackball 3D input devices such as the spaceball Use areas of the screen Distance from center controls angle, position, scale depending on mouse button depressed

56 Animate a bicycle

57 References Ed Angel, CS/EECE 433 Computer Graphics, University of New Mexico Steve Marschner, CS4620/5620 Computer Graphics, Cornell Tom Thorne, COMPUTER GRAPHICS, The University of Edinburgh Elif Tosun, Computer Graphics, The University of New York Lin Zhang, Computer Graphics, Tongji Unviersity

Order of Transformations

Order of Transformations 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 Note