Computer Graphics. Transformations. CSC 470 Computer Graphics 1
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1 Computer Graphics Transformations CSC 47 Computer Graphics 1
2 Today s Lecture Transformations How to: Rotate Scale and Translate 2
3 Introduction An important concept in computer graphics is Affine Transformations. Basically, these allow us to move objects around without deforming them. We will need to keep track of points and vectors as they do not transform in the same way. 3
4 Transformations One example of a transformation is the window to viewport transformation. Here we have seen an image in the world window scaled and translated (moved) into a viewport window. We will build on this transformation to allow us to move objects to more complex locations. 4
5 Transformations A Transformation consists of: a Rotation a Scaling and a Translation a Shearing They occur in 2D and 3D 5
6 Transformations Transformations allow for: 1. scene composition 6
7 Transformations Transformations allow for: 2. easily create symmetrical objects 7
8 Transformations Transformations allow for: 3. viewing objects at different angles 8
9 Transformations Transformations allow for: 4. computer animation where several objects need to move relative to one another 9
10 Transformations The OpenGL Pipeline OpenGL makes transformations easy. But that doesn t excuse you from learning about them. in detail!! 1
11 Transforming Points A transformation simply takes a point and maps it to another location. 11
12 Transforming Points In the 2D case this means. for P = (P( x, P y, 1) T and Q = (Q( x, Q y, 1) T P is at the location P = Pxi+Pyj + ξ (same for Q) then, Q = M (P x, P y, 1) T or Q = M(P) where M is some mapping matrix Q P 12
13 Transforming Points Exercise: if P = (3,4) and Q = (5, 7) what is M? (5,7,1) T = M (3,4,1) T We want to increase P x by 2 and increase P y by 3 what is M?? Q Q 1 x y = P P 1 x y 13
14 Affine Transformations As we have seen there are four types of transformations that can be combined to produce an affine transformation. We will look at these individually to begin 14
15 Translation Moving a point/object around in the x, y and/or z directions. In this example, the car does not rotate (it always remains upright) or scale in size. 15
16 Translation This means that values are being added or subtracted to the existing coordinates. dp x Q Q 1 x y = 1 1 dp dp 1 x y P P 1 x y Q P dp y 16
17 Translation Example: What is the translation matrix to move P=(4,6) to Q=(1,3)? Change in X = P Q Change in Y 17
18 18 18 Translation Translation Example: What is the Example: What is the translation matrix to move translation matrix to move P=(4,6,2) to Q=(1,3,5)? P=(4,6,2) to Q=(1,3,5)? P Q = Change in X Change in Y Change in Z
19 Scaling This means that the x,y and or z coordinates are being multiplied by a scalar. Q Q 1 x y = sp x sp y 1 P P 1 x y P 19
20 2 2 Scaling Scaling Example: What is the matrix that will scale a Example: What is the matrix that will scale a point P = (6,2) to Q = (3,4) point P = (6,2) to Q = (3,4) = P
21 21 21 Scaling Scaling Example: What is the matrix that will scale a Example: What is the matrix that will scale a point P = (6,2,9) to Q = (3,4,3) point P = (6,2,9) to Q = (3,4,3) = P
22 Rotation This means that the x,y and or z coordinates are rotated around a point. Q P Ө Ф P How do we calculate P rotating to Q? 22
23 Rotation Use right-angles and trig. We know: P(x,y) = (R cos(ф), R sin(ф) ) and Q(x,y) = (R cos(ө+ф), R sin(ө +Ф) ) y = R sin (Ф) From trigonometry we also know: cos(ө+ф) = cos(ө)cos(ф) sin(ө)sin(ф) sin(ө +Ф) = sin(ө)cos(ф) + cos(ө)sin(ф) x = R cos (Ф) 23
24 Rotation Use right-angles and trig. Q x = R cos(ө+ф) = R cos(ө)cos(ф) R sin(ө)sin(ф) Q y = R sin(ө +Ф) ) = R sin(ө)cos(ф) + R cos(ө)sin(ф) using P(x,y) = (R cos(ф), R sin(ф) ) we get Qx = P x cos(ө) P y sin(ө) Qy = P x sin(ө) + P y cos(ө) 24
25 25 25 Rotation Rotation This gives us the rotation matrix: This gives us the rotation matrix: = 1 1 ) cos( ) sin( ) sin( ) cos( 1 P P Q Q y x y x θ θ θ θ
26 Rotation What about 3D rotations? There are 3 types of 3D rotations: an x roll 26
27 Rotation What about 3D rotations? There are 3 types of 3D rotations: an x roll a y roll 27
28 Rotation What about 3D rotations? There are 3 types of 3D rotations: an x roll a y roll a z roll 28
29 29 29 Rotation Rotation A Z roll is the same A Z roll is the same as rolling in 2D as the as rolling in 2D as the object rolls between object rolls between the x and y axis. the x and y axis. = ) cos( ) sin( ) sin( ) cos( P P P Q Q Q z y x z y x θ θ θ θ Positive angle rotation occurs according to the right-hand rule!!!.
30 3 3 Rotation Rotation An X roll is a rotation between the y and z An X roll is a rotation between the y and z axes. axes. = ) cos( ) sin( ) sin( ) cos( 1 P P P Q Q Q z y x z y x θ θ θ θ
31 31 31 Rotation Rotation A Y roll is a rotation between the y and z axes. A Y roll is a rotation between the y and z axes. = ) cos( ) sin( 1 ) sin( ) cos( P P P Q Q Q z y x z y x θ θ θ θ
32 Shearing Shearing Shearing means that a point is dragged in a particular direction. This means that some coordinates are affected while others are not. 32
33 Shearing Shearing occurs along a line. In this example the shear occurs along the x axis. This gives us: Q x = P x + hpy; Q y = P y ; Q Q 1 x y = 1 h 1 1 P P 1 x y 33
34 Combining Transformations Rotation, Scaling, Translation and Shearing can be combined into the one matrix. For example, if you want to translate a shape, rotate it and then scale it, the transformation, T, would be: sp x sp y 1 cos( θ) sin( θ) sin( θ) cos( θ) dp dp 1 x y 34
35 Combining Transformations Transformation matrices are listed in reverse order and. Matrices are multiplied backwards. sp sp x y cos( θ) sin( θ) sp sp y x sin( θ) cos( θ) sp sp x y ( dp ( dp x x cos( θ) dp sin( θ) + dp 1 y y sin( θ)) cos( θ)) 35
36 Affine Transformations What we have been looking at are affine transformation. They have the following properties: Affine Transformations Preserve Affine Combinations of Points Affine Transformations Preserve Lines and Planes Affine Transformations Preserve Parallelism of Lines and Planes Relative Ratios are preserved The effect on areas can be predetermined. 36
37 Affine Transformations Affine Transformations Preserve Affine Combinations of Points. Given Q = a 1 P 1 + a 2 P 2 T(a 1 P 1 + a 2 P 2 ) = T(a 1 P 1 ) + T(a 2 P 2 ) 37
38 Affine Transformations Affine Transformations Preserve Affine Combinations of Points. Example: Show that point M which is the affine combination of.7*p(1,2) and.3*q(3,15) maintains the affine combination relationship when M, P and Q are transformed by the matrix, W:
39 Affine Transformations Affine Transformations Preserve Affine Combinations of Points. Example: Show that point M which is the affine combination of.7*p(1,2) and.3*q(3,15) maintains the affine combination relationship when M, P and Q are transformed by the matrix, W. [16,18.5,1] T =.7[1,2,1] T +.3[3,15] T WM =.7WP +.3WQ [49,41,1] T =.7[31,44,1] T +.3[91,34,1] T = [21.7, 3.8,.7] T + [27.3, 1.2,.3] T 39
40 Affine Transformations Affine Transformations Preserve Lines and Planes. preserves linearity and flatness lines stay as lines planes stay flat 4
41 Affine Transformations Affine Transformations Preserve Lines and Planes. Example: 41
42 Affine Transformations Affine Transformations Preserve Parallelism of Lines and Planes. if 2 parallel lines undertake the same transformation, they will remain parallel the same applies for planes. 42
43 Affine Transformations Relative Ratios are preserved - if P is a point some ratio of the way along a line, then after a transformation it will still be the same distance to line length ratio along the line. P t% P t% 43
44 Affine Transformations The effect on areas can be predetermined. area after transformation/area before transformation = det T Rotations and Translations do not affect areas or volumes. 44
45 Affine Transformations However, scaling does affect the area or volume. 45
46 Affine Transformations Example: If the volume of A is 1 units what will the volume of B be if B = T(A) where T = area(b)/area(a) ) = det T area(b) ) = det T * area(a) =.5 * 1 = B A 46
47 Affine Transformations How does this help us in computer graphics. An object will not lose its relative shape when transformed together. Objects will not lose their relative distances when transformed together. 47
48 Affine Transformations How does this help us in computer graphics. For example we can create a fly around of a scene where the whole scene is essentially being transformed. 48
49 Programming Affine Transformations OpenGL works with 4 dimensional matrices. However, you don t have to deal with them directly. OpenGL does all the hard work for you. 49
50 Programming Affine Take a green square produced with glrecti(-5, 5,- 5,5,5); Transformations 5
51 Programming Affine Now we can translate it with: gltranslated(5,5,); Transformations gltranslated(x,y,z); 51
52 Programming Affine or we can rotate it with: glrotate(3,,,1); Transformations glrotate(degrees,, x, y, z); 52
53 Programming Affine or we can scale it with: glscaled(.5,.5,1); Transformations glscaled(x,, y, z); 53
54 Programming Affine or we can combine them: Transformations glrotated(3,,,1); glscaled(.5,.5,1); gltranslated(5,5,); 54
55 Programming Affine Transformations And as we have to multiply the transformation matrices in reverse order we also have to specify them in reverse in OpenGL. E.g. To draw the rectangle which we translate, scale and then rotate the code is: glrotated(3,,,1); glscaled(.5,.5,1); gltranslated(5,5,); glrecti(-5, 5,-5,5,5); 5,5,5); 55
56 3D Programming At first it will seem complex, but its really quite easy. 56
57 3D Programming The OpenGL pipeline: modelview matrix, projection matrix, viewport matrix. 57
58 3D Programming Viewport Matrix set with glviewport() 58
59 3D Programming Projection Matrix set with glortho(left,, right, bottom, top, near, far) far near top left right bottom 59
60 3D Programming Modelview Matrix besides gltranslated, glscaled and glrotated glulookat(eye.x, eye.y, eye.z, lookat.x, lookat.y, lookat.z, up.x, up.y, up.z); eye position (x, y, z) up direction (which way is up?) 6
61 3D Programming void myinit() { glmatrixmode(gl_projection); glloadidentity(); glortho(-1, 1, -1, 1,, 2); glmatrixmode(gl_modelview); glloadidentity(); glulookat(1., 1., 1.,.,.,.,., 1.,.); } void main(int argc,, char **argv argv) { glutinitwindowsize(64,48); glutcreatewindow( 3D Window"); glutdisplayfunc(displaywire); glviewport(,, 64, 48); myinit(); glutmainloop(); } Set the Viewport Matrix Set the Projection Matrix Set the Modelview Matrix 61
62 3D Programming Elementary Shapes the following are drawn with the myinit() function set up thus: myinit() { } glmatrixmode(gl_projection); glloadidentity(); glortho(-1,1, 1,1,-1,1,,2); 1,1,,2); glmatrixmode(gl_modelview); glloadidentity(); glulookat(.,., 15.,.,.,.,., 1.,.); 62
63 3D Programming Elementary Shapes Wireframe Solids Cubes, Spheres etc 63
64 3D Programming glutwirecube(gldouble size); size is the length of a side 64
65 3D Programming glutwiresphere(gldouble radius, GLint slices, GLint stacks); slices are the vertical cuts (slices of cake), stacks are the horizontal cuts (stacks of pancakes) 65
66 3D Programming glutwiretorus(gldouble inrad, GLdouble outrad, GLint slices, GLint stacks); inrad is the centre radius, outrad is the outer radius 66
67 3D Programming glutwiretetrahedron(); unit in size (use glscaled to make bigger) 4 planes 67
68 3D Programming glutwireoctahedron(); unit in size (use glscaled to make bigger) 8 planes 68
69 3D Programming glutwiredodecahedron(); unit in size (use glscaled to make bigger) 1 planes 69
70 3D Programming glutwireicosahedron(); unit in size (use glscaled to make bigger) 2 planes 7
71 3D Programming glutwireicosahedron(); unit in size (use glscaled to make bigger) 2 planes 71
72 3D Programming And the best shape of them all. 72
73 3D Programming glutwireteapot(glfloat size); 73
74 Programming Affine Transformations We can also transform 3D objects such as the glutwireteapot(), thus: glrotated(3,,1,1); glscaled(.5,1.5,.2); gltranslated(1,5,); 74
75 3D Programming What about solid shapes? glutsolidsphere glutsolidcube etc. and you guessed it glutsolidteapot However, drawing these isn t as straight forward as the 2D shapes and the wireframes.. Lets look at the result of changing from glutwireteapot to glutsolidteapot in the previous program. 75
76 3D Programming What happened? Why can t we see a solid teapot? 76
77 3D Programming For many reasons, but in short LIGHTING!!! Without lighting, there are no: shadows or highlights These provide us with the illusion of depth and 3 dimensions. 77
78 3D Programming Lets turn on the lights. glenable() is used to turn many OpenGL features on.. One of these is lighting, thus: glenable(gl_lighting); this can be called in the main or the myinit when the window is being initialised. 78
79 3D Programming Now, what does the teapot look like? Not exactly the effect we were hoping for! So far we have enabled lighting, but we haven t actually turned any lights on!! 79
80 3D Programming We have to call glenable() again to turn on a light, thus: glenable(gl_light); There are a maximum of 8 lights you can use, GL_LIGHT, GL_LIGHT1,.., GL_LIGHT7 And the result. 8
81 3D Programming Better. But not great if you know what is possible with OpenGL. 81
82 3D Programming How can we improve this? First, we select a Shade Model. This is set using glshademodel(); There are two types of shading: GL_FLAT GL_SMOOTH 82
83 3D Programming glshademodel(gl_flat); 83
84 3D Programming glshademodel(gl_smooth); 84
85 3D Programming This improves the teapot s surface, but it still isn t quite right. There are parts of the teapot that shouldn t be displaying. e.g. the hidden parts. We need a depth test! glenable(gl_depth_test); 85
86 3D Programming glenable(gl_depth_test); 86
87 3D Programming glenable(gl_normalize); 87
88 3D Programming Initialisation of the drawing environment for 3D. main( int argc,, char **argv argv) { glutinit(&argc, argv); glutinitdisplaymode(glut_single GLUT_RGB GLUT_DEPTH ); glutinitwindowsize(screenwidth,, SCREENHEIGHT); glutinitwindowposition(1, 1); glutcreatewindow("solid"); "); glutdisplayfunc(displaysolid); glenable(gl_lighting); glenable(gl_light); glshademodel(gl_smooth); glenable(gl_depth_test); glenable(gl_normalize); glclearcolor(.1f,.1f,.1f,.f); glviewport(,, SCREENWIDTH, SCREENHEIGHT); glutmainloop(); } return 1; 88
89 3D Programming To display the teapot. This function is registered with mydisplayfunc(). void displaysolid(void) { glmatrixmode(gl_projection); glloadidentity(); glortho(-1, 1, -1, 1,.1, 1.); glmatrixmode(gl_modelview); glloadidentity(); glulookat(2.3,1.3,2,,.25,,.,1.,.); glclear(gl_color_buffer_bit GL_DEPTH_BUFFER_BIT); } glcolor3f(,1,); glpushmatrix(); gltranslated(.4,.5,.4); glrotated(3,,1,); glutsolidteapot(.5); glpopmatrix(); 89
90 3D Programming Positioning the Light lightposition = x, y, z, w GLfloat lightposition[]={2.f,3.f,3.f,.f}; gllightfv(gl_light, GL_POSITION, lightposition); 9
91 3D Programming lightposition[]={2.f,6.f,3.f,.f}; lightposition[]={1.f,2.f,4.f,.f}; 91
92 3D Programming Turning up the Lights lightintensity = red, green, blue, alpha GLfloat lightintensity[] = {.7f,.7f,.7f, 1.f}; gllightfv(gl_light, GL_DIFFUSE, lightintensity); 92
93 3D Programming lightintensity[] = {1.f, 1.f, 1.f, 1.f}; lightintensity[] = {.2f,.2f,.2f, 1.f}; 93
94 3D Programming lightintensity[] = {.2f, 1.f,.2f, 1.f}; lightintensity[] = {.2f,.f,.8f, 1.f}; 94
95 3D Programming Types of Light GL_AMBIENT scattered light, difficult to determine origin GL_DIFFUSE direct light with an origin GL_SPECULAR same as shininess, light comes from one direction and bounces off the surface 95
96 3D Programming Surface Colours Light can make an object appear in a different colour, but the actual colour of the object needs to be set differently. For this we use glmaterialfv() The material of an object can be ambient, diffuse and specular,, just as for light. It also has a factor of shininess. 96
97 3D Programming Surface Colours For example: GLfloat mat_ambient[]= {.5f,.8f,.6f, 1.f}; GLfloat mat_diffuse[] = {.6f,.8f,.6f, 1.f}; GLfloat mat_specular[] = {1.f,.8f, 1.f, 1.f}; GLfloat mat_shininess[] = {1.f}; glmaterialfv(gl_front,, GL_AMBIENT, mat_ambient); glmaterialfv(gl_front,, GL_DIFFUSE, mat_diffuse); glmaterialfv(gl_front,, GL_SPECULAR, mat_specular); glmaterialfv(gl_front,, GL_SHININESS, mat_shininess); 97
98 3D Programming Surface Colours 98
99 The End Next Two Weeks More about transformation Creating complex 3D shapes using polygon meshes 99
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