Fall CSCI 420: Computer Graphics. 2.2 Transformations. Hao Li.

Transcription

1 Fall 2017 CSCI 420: Computer Graphics 2.2 Transformations Hao Li 1

2 OpenGL Transformations Matrices Model-view matrix (4x4 matrix) Projection matrix (4x4 matrix) vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 2

3 4x4 Model-view Matrix (this lecture) Translate, rotate, scale objects Position the camera vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 3

4 4x4 Model-view Matrix (next lecture) Projection from 3D to 2D vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 4

5 OpenGL Transformation Matrices Model-view Projection Manipulated separately in OpenGL (must set matrix mode) : glmatrixmode (GL_MODELVIEW); glmatrixmode (GL_PROJECTION; 5

6 Setting the Current Model-view Matrix Load or post-multiply glmatrixmode (GL_MODELVIEW); glloadidentity(); // very common usage float m[16] = { }; glloadmatrixf(m); // rare, advanced glmultmatrixf(m); // rare, advanced Use library functions gltranslatef(dx, dy, dz); glrotatef(angle, vx, vy, vz); glscalef(sx, sy, sz); 6

7 Translated, rotated, scaled object world 7

8 The rendering coordinate system Initially (after glloadidentity()) : rendering coordinate system = world coordinate system world 8

9 The rendering coordinate system gltranslatef(x, y, z); world [x, y,z] rendering coordinate system 9

10 The rendering coordinate system glrotatef(angle,ax, ay, az); world rendering coordinate system 10

11 The rendering coordinate system glscalef(sx, sy, sz); world rendering coordinate system 11

12 OpenGL code glmatrixmode (GL_MODELVIEW); glloadidentity(); gltranslatef(x, y, z); glrotatef(angle, ax, ay, az); glscalef(sx, sy, sz); renderbunny(); world rendering coordinate system 12

13 Rendering more objects How to obtain this frame? world rendering coordinate system 13

14 Solution 1 How to obtain this frame? Find gltranslate( ), glrotatef( ), glscalef( ) world 14

15 Solution 2: gl{push,pop}matrix glmatrixmode (GL_MODELVIEW); glloadidentity(); // render first bunny glpushmatrix(); // store current matrix gltranslate3f( ); glrotatef( ); renderbunny(); glpopmatrix(); // pop matrix // render second bunny glpushmatrix(); // store current matrix gltranslate3f( ); glrotatef( ); renderbunny(); glpopmatrix(); // pop matrix 15

16 Recall: Linear Algebra 16

17 Scalars Scalars,, from a scalar field () 1 Operations,,,,, Expected laws apply Examples: rationals or reals with addition and multiplication 17

18 Vectors Vectors u, v, w from a vector space u + v u v Vector addition, subtraction Zero vector 0 Scalar multiplication v 18

19 Euclidean Space Vector space over real numbers Three-dimensional in computer graphics Dot product: = u > v = u 1 v 1 + u 2 v 2 + u 3 v 3 0 > 0 =0, u, v are orthogonal if u > v =0 kvk 2 = v > v kvk v defines, the length of 19

20 Lines and Line Segments Parametric form of line: p( ) =p 0 + d Line segment between q and r : for p( ) =(1 )q + r 0 apple apple 1 20

21 Convex Hull Convex hull defined by p = 1 p n p n n =1 for and 0 apple 1 apple 1,i=1...n 21

22 Projection Dot product projects one vector onto another vector u > v = u 1 v 1 + u 2 v 2 + u 3 v 3 = kukkvk cos( ) v (u) =(u > v)v/kvk 2 22

23 Cross Product ka bk = a b sin( ) Cross product is perpendicular to both Right-hand rule a and b 23

24 Plane Plane defined by point and vectors u and v u and v should not be parallel Parametric form: p 0 t(, )=p 0 + u + v n = u v/ku vk is the normal n > (p p 0 )=0if and only if p lies in plane 24

25 Coordinate Systems v 1, v 2, v 3 Let be three linearly independent vectors in a 3-dimensional vector space Can write any vector w as w = 1 v v v 3 for some scalars 1, 2, 3 25

26 Frames p 0 Frame = origin + coordinate system Any point p = p v v v 3 26

27 In Practice, Frames are Often Orthogonal 27

28 Change of Coordinate System u 1, u 2, u 3 v 1, v 2, v 3 Bases { } and { } u i Express basis vectors in terms of v j u 1 = 11 v v v 3 u 2 = 21 v v v 3 u 3 = 31 v v v 3 Represent in matrix form: 2 4 u > 1 u > = M 4 v 1 > v 2 > 3 5 M u > 3 v > 3 28

29 Representing 3D transformations (and model-view matrices) 29

30 Linear Transformations 3 x 3 matrices represent linear transformations a = Mb Can represent rotation, scaling, and reflection Cannot represent translation M 30

31 Homogeneous Coordinates In order to represent rotations, scales AND translations [ 1, 2, 3 ] > Augment by adding a fourth component (1): p =[ 1, 2, 3, 1] > Homogeneous property:, for any scalar p =[ 1, 2, 3, 1] > =[ 1, 2, 3 ] > =0 31

32 Homogeneous Coordinates Homogeneous coordinates are transformed by 4x4 matrices p q q = Ap world 4-vector 4x4 matrix 4-vector 32

33 Affine Transformations (4x4 matrices) Translation Rotation Scaling Any composition of the above Later: projective (perspective) transformations -Also expressible as 4 x 4 matrices! 33

34 Translation q = p + d d =[ x, y, z, 0] > where, p =[x, y, z, 1] >, q =[x 0,y 0,z 0, 1] >, q = Tp Express in matrix form and solve for T 34

35 Scaling x 0 = y 0 = x x y y z 0 = z z Express as q = Sp and solve for S 35

36 Rotation in 2 Dimensions Rotation by about the origin x 0 = x cos( ) y sin( ) y 0 = x sin( )+y cos( ) Express in matrix form: Note that the determinant is 1 36

37 Rotation in 3 Dimensions Orthogonal matrices: RR > = R > R = I det(r) =1 Affine transformation: 37

38 Affine Matrices are Composed by Matrix Multiplication A = A 1 A 2 A 3 Applied from right to left Ap =(A 1 A 2 A 3 )p = A 1 (A 2 (A 3 p)) When calling gltranslate3f, glrotatef, or glscalef, OpenGL forms the corresponding 4x4 matrix, and multiplies the current modelview matrix with it. 38

39 Summary OpenGL Transformation Matrices Vector Spaces Frames Homogeneous Coordinates Transformation Matrices 39

40 Next Time: Viewing & Projection 40

41 Thanks! 41