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Computer Graphics - Transformation - Hanyang University Jong-Il Park

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Division of Electrical and Computer Engineering, Hanyang UniversityObjectives Introduce standard transformations Rotation Translation Scaling Shear Derive homogeneous coordinate transformation matrices Learn to build arbitrary transformation matrices from simple transformations Learn how to carry out transformations in OpenGL Introduce OpenGL matrix modes Model-view Projection

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Division of Electrical and Computer Engineering, Hanyang University General Transformations A transformation maps points to other points and/or vectors to other vectors Q=T(P) v=T(u)

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University Pipeline Implementation transformationrasterizer u v u v T T(u) T(v) T(u) T(v) vertices pixels frame buffer (from application program)

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Division of Electrical and Computer Engineering, Hanyang University 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 points -array of 4 scalars in homogeneous coordinates

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Division of Electrical and Computer Engineering, Hanyang University Translation Move (translate, displace) a point to a new location Displacement determined by a vector d Three degrees of freedom P=P+d P P d

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University Translation Using Representations Using the homogeneous coordinate representation in some frame p=[ x y z 1] T d=[dx dy dz 0] T Hence p = p + d or x=x+d x y=y+d y z=z+d z note that this expression is in four dimensions and expresses point = vector + point

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Division of Electrical and Computer Engineering, Hanyang University Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p = Tp where T = T (d x, d y, d z ) = This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

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Division of Electrical and Computer Engineering, Hanyang University 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 x = r cos y = r sin x = r cos ( y = r sin (

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University Rotation Matrix R = R z ( ) =

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Division of Electrical and Computer Engineering, Hanyang University 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 ( ) =

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Division of Electrical and Computer Engineering, Hanyang University Non-rigid-body Transformations

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Division of Electrical and Computer Engineering, Hanyang University Scaling S = S(s x, s y, s z ) = x=s x x y=s y x z=s z x p=Sp Expand or contract along each axis (fixed point of origin)

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Division of Electrical and Computer Engineering, Hanyang University Reflection corresponds to negative scale factors original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

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Division of Electrical and Computer Engineering, Hanyang University 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 )

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Division of Electrical and Computer Engineering, Hanyang University Linear Transformations x' = Ax + By + Cz + D y' = Ex + Fy + Gz + H z' = Ix + Jy + Kz + L Transform the geometry – leave the topology as is

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Division of Electrical and Computer Engineering, Hanyang University Geometry vs. Topology

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Division of Electrical and Computer Engineering, Hanyang University Transformations in H.C.

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Division of Electrical and Computer Engineering, Hanyang University Identity Matrix

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Division of Electrical and Computer Engineering, Hanyang University Matrix Inverse

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Division of Electrical and Computer Engineering, Hanyang University Translation

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Division of Electrical and Computer Engineering, Hanyang University 2D Rotation

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Division of Electrical and Computer Engineering, Hanyang University 3D Rotation about Z

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Division of Electrical and Computer Engineering, Hanyang University 3D Rotation about X & Y About X About Y

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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 p T = p T C T B T A T

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Division of Electrical and Computer Engineering, Hanyang University 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 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

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Division of Electrical and Computer Engineering, Hanyang University General Rotation Read Sect. 4.9.4.

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Division of Electrical and Computer Engineering, Hanyang University Rotation About a Fixed Point other than the Origin 1. Move fixed point to origin 2. Rotate 3. Move fixed point back M = T(p f ) R( ) T(-p f )

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Division of Electrical and Computer Engineering, Hanyang University Instancing 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

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Division of Electrical and Computer Engineering, Hanyang University Shear Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions

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Division of Electrical and Computer Engineering, Hanyang University Shear Matrix Consider simple shear along x axis x = x + y cot y = y z = z H( ) =

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Division of Electrical and Computer Engineering, Hanyang University Matrix Multiplication Is NOT Commutative

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Division of Electrical and Computer Engineering, Hanyang University Combining Translations, Rotations

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Division of Electrical and Computer Engineering, Hanyang University Combining Translations, Rotations

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Division of Electrical and Computer Engineering, Hanyang University Matrix Multiplication Is Associative

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Division of Electrical and Computer Engineering, Hanyang University Transformation Game transformation_game.jar http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/ repositoryApplets.html

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Division of Electrical and Computer Engineering, Hanyang University 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 be manipulated by glMatrixMode(GL_MODELVIEW); glMatrixMode(GL_PROJECTION);

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Division of Electrical and Computer Engineering, Hanyang University 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 CTM vertices pp=Cp C

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Division of Electrical and Computer Engineering, Hanyang University Concatenation of Transformations

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Division of Electrical and Computer Engineering, Hanyang University Hardware can do this very quickly!

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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. Lets try again.

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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 )

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Division of Electrical and Computer Engineering, Hanyang University 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 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);

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Division of Electrical and Computer Engineering, Hanyang University Compound Transformation

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Division of Electrical and Computer Engineering, Hanyang University 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)

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Division of Electrical and Computer Engineering, Hanyang University Matrix Stacks In many situations we want to save transformation matrices for use later Traversing hierarchical data structures (Chapter 10) 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()

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Division of Electrical and Computer Engineering, Hanyang University 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 double m[16]; glGetFloatv(GL_MODELVIEW, m);

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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(); }

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Division of Electrical and Computer Engineering, Hanyang University 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) { 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; }

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Division of Electrical and Computer Engineering, Hanyang University 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 form of callbacks, variables such as theta and axis must be defined as globals Camera information is in standard reshape callback

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University 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 (see text)

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Division of Electrical and Computer Engineering, Hanyang University 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

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Division of Electrical and Computer Engineering, Hanyang University Quaternions Extension of imaginary numbers from two to three dimensions Requires one real and three imaginary components i, j, k Quaternions can express rotations on sphere smoothly and efficiently. Process: Model-view matrix quaternion Carry out operations with quaternions Quaternion Model-view matrix q=q 0 +q 1 i+q 2 j+q 3 k

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Division of Electrical and Computer Engineering, Hanyang University 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

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