Transformations. Overview. Standard Transformations. David Carr Fundamentals of Computer Graphics Spring 2004 Based on Slides by E.

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1 INSTITUTIONEN FÖR SYSTEMTEKNIK LULEÅ TEKNISKA UNIVERSITET Transformations David Carr Fundamentals of Computer Graphics Spring 24 Based on Slides by E. Angel Feb-1-4 SMD159, Transformations 1 L Overview Standard transformations - Rotations, Translation, Scaling, Shear - Homogeneous coordinate transformation matrices - Constructing arbitrary transformation matrices from simple ones Transformations in OpenGL - Rotation, Translation, Scaling - OpenGL matrix modes + Model-view, Projection Feb-1-4 SMD159, Transformations 2 L INSTITUTIONEN FÖR SYSTEMTEKNIK LULEÅ TEKNISKA UNIVERSITET Standard Transformations Feb-1-4 SMD159, Transformations 3 L 1

2 General Transformations A transformation maps points to other points and/or vectors to other vectors v=t(u) Q=T(P) Feb-1-4 SMD159, Transformations 4 L Line preserving Affine Transformations Characteristic of many physically important transformations - Rigid body transformations: rotation, translation - Scaling, shear Importance in graphics is that: - Need only transform endpoints of line segments - Let implementation draw line segment between the transformed endpoints Feb-1-4 SMD159, Transformations 5 L Pipeline Implementation T (from application program) u v transformation T(u) T(v) rasterizer frame buffer v T(v) u T(u) T(u) vertices vertices pixels T(v) Feb-1-4 SMD159, Transformations 6 L 2

3 Notation Work with both: - Coordinate-free representations of transformations - Representations within a particular frame P,Q, R: points in an affine space u, v, w: vectors in an affine space a, b, g: 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 Feb-1-4 SMD159, Transformations 7 L Translation Move (translate, displace) a point to a new location P P d Displacement determined by a vector d - Three degrees of freedom - P =P+d Feb-1-4 SMD159, Transformations 8 L 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 object translation: every point displaced by same vector Feb-1-4 SMD159, Transformations 9 L 3

4 Translation Using Representations Using the homogeneous coordinate representation in some frame - p=[ x y z 1] T - p =[x y z 1] T - d=[dx dy dz ] 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 that point = vector + point Feb-1-4 SMD159, Transformations 1 L 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 ) = È1 Î 1 1 dx d y dz 1 This form is better for implementation because: - All affine transformations can be expressed this way - Multiple transformations can be concatenated together Feb-1-4 SMD159, Transformations 11 L Rotation (2D) Consider rotation about the origin by q degrees - radius stays the same, angle increases by q x = r cos (f + q) y = r sin (f + q) x =x cos q y sin q y = x sin q + y cos q x = r cos f y = r sin f Feb-1-4 SMD159, Transformations 12 L 4

5 Rotation About the Z-Axis Rotation about z axis in three dimensions leaves all points with the same z - Equivalent to rotation in 2 dimensions in planes of constant z x =x cos q y sin q y = x sin q + y cos q z =z - or in homogeneous coordinates p = R z (q)p Feb-1-4 SMD159, Transformations 13 L Rotation Matrix R = R z (q) = Ècos q sin q Î - sin q cos q 1 1 Feb-1-4 SMD159, Transformations 14 L 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 È1 cos q - sin q R = R x (q) = sin q cos q Î 1 È cos q R = R y (q) = 1 - sin q sin q cos q Î 1 Feb-1-4 SMD159, Transformations 15 L 5

6 Scaling Expand or contract along each axis (fixed point of origin) x =s x x y =s y x z =s z x p =Sp Èsx S = S(s x, s y, s z ) = Î s y 1 Feb-1-4 SMD159, Transformations 16 L s z Reflection Corresponds to negative scale factors s x = -1 s y = 1 original s x = -1 s y = -1 s x = 1 s y = -1 Feb-1-4 SMD159, Transformations 17 L Inverses Could compute inverse matrices by general formulas, But, simple geometric observations are easier - Translation: T -1 (d x, d y, d z ) = T(-d x, -d y, -d z ) - Rotation: R -1 (q) = R(-q) +Holds for any rotation matrix +Note that since cos(-q) = cos(q) and sin(- q)=-sin(q), R -1 (q) = R T (q) - Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z ) Feb-1-4 SMD159, Transformations 18 L 6

7 Concatenation Form arbitrary affine transformation matrices by multiplying - Rotation, translation, and scaling matrices Same transformation is applied to many vertices: - Cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p Difficulty is forming a desired transformation Feb-1-4 SMD159, Transformations 19 L 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 present points. In terms of column matrices p T = p T C T B T A T Feb-1-4 SMD159, Transformations 2 L General Rotation About the Origin A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes R(q) = R z (q z ) R y (q y ) R x (q x ) q x q y q z are called the Euler angles y v Note that rotations do not commute We can use rotations in another order but with different angles z Feb-1-4 SMD159, Transformations 21 L x 7

8 Rotation About a Fixed Point, Not the Origin Move fixed point to origin Rotate Move fixed point back M = T(-p f ) R(q) T(p f ) Feb-1-4 SMD159, Transformations 22 L Instancing In modeling, we often start with: - A simple object - Centered at the origin - Oriented with the axes - At a standard size We apply an instance transformation to its vertices to - Scale - Orient - Locate Feb-1-4 SMD159, Transformations 23 L Shear Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions Feb-1-4 SMD159, Transformations 24 L 8

9 x = x + y cot q y = y z = z È1 H(q) = Î Shear Matrix Consider simple shear along the x-axis cot q Feb-1-4 SMD159, Transformations 25 L INSTITUTIONEN FÖR SYSTEMTEKNIK LULEÅ TEKNISKA UNIVERSITET Transformations in OpenGL Feb-1-4 SMD159, Transformations 26 L OpenGL Matrices In OpenGL matrices are part of the state Three types - Model-View (GL.GL_MODEL_VIEW) - Projection (GL.GL_PROJECTION) - Texture (GL.GL_TEXTURE) (ignore for now) Single set of functions for manipulation Select which to manipulated by - gl.glmatrixmode(gl_model_view); - gl.glmatrixmode(gl_projection); Feb-1-4 SMD159, Transformations 27 L 9

10 vertices 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 - Applied to all vertices that pass down the pipeline The CTM is defined in the user program and loaded into a transformation unit p C CTM p =Cp vertices Feb-1-4 SMD159, Transformations 28 L CTM operations The CTM can be altered either by - Loading a new CTM - 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 Feb-1-4 SMD159, Transformations 29 L 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 RT Each operation corresponds to one function call in the program. Note that the last operation specified is the first executed in the program Feb-1-4 SMD159, Transformations 3 L 1

11 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 matrix mode Feb-1-4 SMD159, Transformations 31 L Rotation, Translation, Scaling Load an identity matrix gl.glloadidentity() Multiply on right gl.glrotatef(theta, vx, vy, vz) - theta in degrees, (vx, vy, vz) define axis of rotation gl.gltranslatef(dx, dy, dz) gl.glscalef( sx, sy, sz) Each has a float (f) and double (d) format (e.g., gl.glscaled) Feb-1-4 SMD159, Transformations 32 L Example Rotation about z axis by 3 degrees with a fixed point of (1., 2., 3.) gl.glmatrixmode(gl_modelview); gl.glloadidentity(); gl.gltranslatef(1., 2., 3.); gl.glrotatef(3.,.,.,.1); gl.gltranslatef(-1., -2., -3.); Remember that last matrix specified in the program is the first applied Feb-1-4 SMD159, Transformations 33 L 11

12 Arbitrary Matrices Can load and multiply by matrices defined in the application program gl.glloadmatrixf(m) gl.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 Feb-1-4 SMD159, Transformations 34 L Matrix Stacks In many situations we want to save transformation matrices for use later - Traversing hierarchical data structures (Chapter 9) - Avoiding state changes when executing display lists OpenGL maintains stacks for each type of matrix - Access present type (as set by glmatrixmode) by gl.glpushmatrix() gl.glpopmatrix() Feb-1-4 SMD159, Transformations 35 L Reading Back Matrices Can also access matrices (and other parts of the state) by enquiry (query) functions glgetintegerv glgetfloatv glgetbooleanv glgetdoublev glisenabled For matrices, we use as double m[16]; glgetfloatv(gl_modelview, m); Feb-1-4 SMD159, Transformations 36 L 12

13 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 (Cube.java) in a standard way - Centered at origin - Sides aligned with axes - Will discuss modeling in next lecture Feb-1-4 SMD159, Transformations 37 L Method main() public class Cube { public static void main(string[] args) { Frame simpleframe = new Frame("Cube from Chapter 4"); SimpleRenderer renderer = new SimpleRenderer(); GLCapabilities glc = new GLCapabilities(); glc.setdoublebuffered(true); GLCanvas drawable = GLDrawableFactory.getFactory().createGLCanvas(glc); drawable.addgleventlistener(renderer); simpleframe.add(drawable); simpleframe.setlocation(1,1); simpleframe.setsize(5, 522); final Animator animator = new Animator(drawable); simpleframe.addwindowlistener(new WindowAdapter() { public void windowclosing(windowevent e) {animator.stop(); System.exit(); ); simpleframe.show(); animator.start(); Feb-1-4 SMD159, Transformations 38 L Renderer static class SimpleRenderer implements GLEventListener, MouseListener { final double XMIN = -2., XMAX = 2., YMIN = -2., YMAX = 2.; double vertices[][] = { {-1.,-1.,-1., {1.,-1.,-1., {1.,1.,-1., {-1.,1.,-1., {-1.,-1.,1., {1.,-1.,1., {1.,1.,1., {-1.,1.,1., colors[][] = { {.,.,., {1.,.,., {1.,1.,., {.,1.,., {.,.,1., {1.,.,1., {1.,1.,1., {.,1.,1., theta[] = {.,.,.; int axis = 2; boolean firstreshape = true; public void displaychanged(gldrawable drawable, boolean modechanged, boolean devicechanged) { Feb-1-4 SMD159, Transformations 39 L 13

14 Method reshape( ) public void reshape(gldrawable drawable, int x, int y, int w, int h) { GL gl = drawable.getgl(); double scale = (w <= h)? (double) h / (double) w : (double) w / (double) h; gl.glviewport(x, y, w, h); gl.glmatrixmode(gl.gl_projection); // switch matrix mode gl. glloadidentity(); if(w<=h) gl.glortho(xmin,xmax,ymin*scale,ymax*scale,-1.,1.); else gl.glortho(xmin*scale,xmax*scale,ymin,ymax,-1.,1.); gl.glmatrixmode(gl.gl_modelview); // return to modelview mode gl.glclear(gl.gl_color_buffer_bit); if (firstreshape) firstreshape = false; else drawable.display(); Calling display on first pass causes an exception! Feb-1-4 SMD159, Transformations 4 L Method init( ) public void init(gldrawable drawable) { GL gl = drawable.getgl(); // get interfaces drawable.addmouselistener(this); gl.glclearcolor(f,f,f,f); // connect mouse // set bg and fill color gl.glmatrixmode(gl.gl_projection); // view intialization gl.glloadidentity(); gl.glortho(xmin, XMAX, YMIN, YMAX, -1., 1.); gl.glenable(gl.gl_depth_test); // enable z-buffer Feb-1-4 SMD159, Transformations 41 L Method display( ) public void display(gldrawable drawable) { GL gl = drawable.getgl(); gl.glclear(gl.gl_color_buffer_bit GL.GL_DEPTH_BUFFER_BIT); gl.glloadidentity(); gl.glrotated(theta[], 1.,.,.); gl.glrotated(theta[1],., 1.,.); gl.glrotated(theta[2],.,., 1.); colorcube(gl); spincube(); // draw cube // rotate for next draw gl.glflush(); Feb-1-4 SMD159, Transformations 42 L 14

15 Drawing the Cube private void cubeface(gl gl, int a, int b, int c, int d) { gl.glbegin(gl.gl_polygon); gl.glcolor3dv(colors[a]); gl.glvertex3dv(vertices[a]); gl.glcolor3dv(colors[b]); gl.glvertex3dv(vertices[b]); gl.glcolor3dv(colors[c]); gl.glvertex3dv(vertices[c]); gl.glcolor3dv(colors[d]); gl.glvertex3dv(vertices[d]); gl.glend(); Feb-1-4 SMD159, Transformations 43 L Drawing the Cube, private void colorcube(gl gl) { cubeface(gl,,3,2,1); cubeface(gl,2,3,7,6); cubeface(gl,,4,7,3); cubeface(gl,1,2,6,5); cubeface(gl,4,5,6,7); cubeface(gl,,1,5,4); private void spincube() { theta[axis] +=.2; if( theta[axis] > 36. ) theta[axis] -= 36.; Feb-1-4 SMD159, Transformations 44 L Mouse Interface public void mouseentered(mouseevent e) { public void mouseexited(mouseevent e) { public void mouseclicked(mouseevent e) { public void mousereleased(mouseevent e) { public void mousepressed(mouseevent e) { int button = e.getbutton(); switch (button) { case MouseEvent.BUTTON1: axis = ; break; case MouseEvent.BUTTON2: axis = 1; break; case MouseEvent.BUTTON3: axis = 2; break; Feb-1-4 SMD159, Transformations 45 L 15

16 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 (Chapter 5) - Build models of obejcts The projection matrix is used to: - Define the view volume - Select a camera lens Although both are manipulated by the same functions, - Be careful because incremental changes are always made by post-multiplication - For example, rotating model-view and projection matrices by the same matrix are not equivalent operations. - Post-multiplication of the model-view matrix is equivalent to pre-multiplication of the projection matrix Feb-1-4 SMD159, Transformations 46 L Smooth Rotation From a practical standpoint, - Often want to use transformations to move and reorient an object smoothly - Problem: find a sequence of model-view matrices M,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 - 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) Feb-1-4 SMD159, Transformations 47 L Incremental Rotation Consider the two approaches - For a sequence of rotation matrices R,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 Feb-1-4 SMD159, Transformations 48 L 16

17 Quaternions Extension of imaginary numbers from two to three dimensions Requires 1 real and 3 imaginary components i, j, k q=q +q 1 i+q 2 j+q 3 k Quaternions can express rotations on sphere smoothly and efficiently. Process: - Model-view matrix Æ quaternion - Carry out operations with quaternions - Quaternion Æ Model-view matrix Feb-1-4 SMD159, Transformations 49 L Interfaces One of the major problems in interactive computer graphics is how to - Use 2D devices such as a mouse to interface with 3D obejcts 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 Feb-1-4 SMD159, Transformations 5 L Questions Feb-1-4 SMD159, Transformations 51 L 17