Computer Graphics. Chapter 5 Geometric Transformations. Somsak Walairacht, Computer Engineering, KMITL

Transcription

1 Chapter 5 Geometric Transformations Somsak Walairacht, Computer Engineering, KMITL 1

2 Outline Basic Two-Dimensional Geometric Transformations Matrix Representations and Homogeneous Coordinates Inverse Transformations Two-Dimensional Composite Transformations Other Two-Dimensional Transformations Raster Methods for Geometric Transformations OpenGL Raster Transformations Transformations between Two-Dimensional Coordinate Systems 2

3 Outline (2) Geometric Transformations in Three-Dimensional Space Translation Rotation Scaling Composite Three-Dimensional Transformations Other Three-Dimensional Transformations Transformations between Three-Dimensional Coordinate Systems Affine Transformations 3

4 Introduction Operations that are applied to the geometric description of an object to change its position, orientation, or size are called geometric transformations Sometimes geometric-transformation operations are also referred to as modeling transformations A distinction between the two Modeling transformations are used to construct a scene or to give the hierarchical h description of a complex object that is composed of several parts Geometric transformations describe how objects might move around in a scene during an animation sequence or simply to view them from another angle 4

5 Basic Two-Dimensional Geometric e Transformations a o s Available in all graphics packages are Translation Rotation Scaling Other useful transformation routines are Reflection Shearing 5

6 Two-Dimensional so Translation sa To translate a two-dimensional position, we add translation distances t x and t y to the original coordinates (x, y) to obtain the new coordinate position (x, y ) Translation equations in the matrix Translation without deformation is a rigid-body transformation that moves objects การทาใหผดร ป 6

7 Polygon Translation A polygon is translated similarly Adding a translation vector to the coordinate position of each vertex and then regenerate the polygon using the new set of vertex coordinates 7

8 Two-Dimensional Rotation A rotation transformation is generated by specifying a rotation axis and a rotation angle Parameters are the rotation angle θ and a position (x r, y r ) called the rotation ti point (or pivot point), จ ดหม น about which the object is to be rotated A positive value for the angle θ defines a counterclockwise rotation about the pivot point ทวนเข มนาฬ กา 8

9 Two-Dimensional Rotation (2) Let r is the constant distance of the point from the origin, angle is the original angular position of the point from the horizontal, and θ is the rotation angle 9

10 Rotation about Arbitrary Point Transformation equations for rotation of a point about any specified rotation position (x r, y r ) Rotations are rigid body transformations that move objects without deformation 10

11 Two-Dimensional Scaling A simple two-dimensional scaling operation is performed by multiplying object positions (x, y) by scaling factors s x and s y to produce the transformed coordinates (x, y ) In matrix format, where S is a 2x22 scaling matrix 11

12 Scaling by a Fixed Point Coordinates for the fixed point, (x f,y f ), are often chosen at some object position, such as its centroid Objects are now resized by scaling the distances between object points and the fixed point where the additive terms x f (1 s x ) and y f (1 s y ) are constants for all points in the object 12

13 Matrix Representations and Homogeneous oge eous Coordinates Many graphics applications involve sequences of geometric transformations An animation might require an object to be translated and rotated at each increment of the motion The viewing transformations involve sequences of translations and rotations to take us from the original scene specification to the display on an output device 13

14 Matrix Representations and Homogeneous oge eous Coordinates Matrix M 1 is a 2x2 array containing multiplicative factors, and M 2 is a two-element column matrix (2x1 matrix) containing translational terms Multiplicative and translational terms can be combined into a single matrix if we expand the representations to 3x3 A three-element representation (x h, y h, h), called homogeneous coordinates, where the homogeneous parameter h is a nonzero value such that t 14

15 Matrix Representations and Homogeneous oge eous Coordinates Two-Dimensional Translation Matrix Rotation Matrix Scaling Matrix 15

16 Inverse esetransformations so a o s Inverse translation matrix Translate in the opposite direction Inverse rotation matrix Rotate in the clockwise direction Inverse scaling matrix 16

17 Two-Dimensional Composite Transformations a o s Forming products of transformation matrices is often referred to as a concatenation, or composition, of matrices We do premultiply the column matrix by the matrices representing any transformation sequence Since many positions in a scene are typically transformed by the same sequence, it is more efficient i to first multiply l the transformation ti matrices to form a single composite matrix 17

18 Composite Two-Dimensional Translations If two successive translation vectors (t 1x, t 1y) ) and (t 2x, t 2y) ) are applied to a 2-D coordinate position P, the final transformed location P is The composite transformation matrix for this sequence of translations is 18

19 Composite Two-Dimensional Rotations Two successive rotations applied to a point P We can verify that two successive rotations are additive: The composition matrix 19

20 Composite Two-Dimensional Scaling For two successive scaling operations in 2-D produces the following composite scaling matrix 20

21 General Two-Dimensional Pivot-Point Rotation Graphics package provides only a rotate function with respect to the coordinate origin To generate a 2-D rotation about any other pivot point (x r,y r ), follows the sequence of translate-rotate-translate operations 1. Translate the object so that the pivot-point position is moved to the coordinate origin 2. Rotate the object about the coordinate origin 3. Translate the object so that t the pivot point is returned to its original position 21

22 General Two-Dimensional Pivot-Point Rotation (2) 22

23 General Two-Dimensional Fixed-Point Scaling 1. Translate the object so that the fixed point coincides with the coordinate origin 2. Scale the object with respect to the coordinate origin 3. Use the inverse of the translation in step (1) to return the object to its original position 23

24 General Two-Dimensional Scaling Directions To accomplish the scaling without changing the orientation of the object 1. Performs a rotation so that the directions for s 1 and s 2 coincide with the x and y axes 2. Scaling transformation S(s 1, s 2 ) is applied 3. An opposite rotation to return points to their original orientations ti 24

25 Example of Scaling Transformation Turn a unit square into a parallelogram by stretching it along the diagonal from (0, 0) to (1, 1) Rotate the diagonal onto the y axis using θ = 45 Double its length with the scaling values s 1 =1 and s 2 =2 Rotate again to return the diagonal to its original orientation 25

26 Matrix Concatenation Properties Multiplication of matrices is associative Depending upon the order in which the transformations Premultiplying, concatenated transformations are applied from right to left Postmultiplying, concatenated transformations are applied from left to right Premultiplying V = [M1 * M2 * M3] * V Postmultiplying V T = V T * [M3 * M2 * M1] T 26

27 Matrix Concatenation Properties (2) Transformation products may not be commutative M 2 M 1 is not equal to M 1 M 2 Be careful about the order to translate and rotate an object Some special cases the multiplication of transformation matrices is commutative Two successive rotations give the same final position 27

28 General Two-Dimensional Composite Transformations and Computational Efficiency From (5.41), it requires 9 multiplications and 6 additions Actually, only 4 multiplications and 4 additions Once matrix is concatenated, it is maximum number of computations required Without concatenation, individual transformations would be applied one at a time, and the number of calculations could be significantly increased 28

29 Computational Efficiency (2) Rotation calculations require trigonometric evaluations and several multiplications Computational efficiency can become an important consideration in rotation transformations For small enough angles (less than 10), cosθ is approximately 1.0 and sinθ has a value very close to the value of θ in radians 29

30 Computational Efficiency (3) Composite transformations often involve inverse matrices Operations are much simpler than direct inverse matrix calculations Inverse translation matrix is obtained by changing the signs of the translation distances Inverse rotation ti matrix is obtained by performing a matrix transpose 30

31 Two-Dimensional Rigid-Body Transformation All angles and distances between coordinate positions are unchanged by the transformation Upper-left 2x2 submatrix is an orthogonal matrix Two row vectors (r xx, r xy ) and (r yx, r yy ) (or the two column vectors) form an orthogonal set of unit vectors Set of vectors is also referred to as an orthonormal vector set 31

32 Two-Dimensional Rigid-Body Transformation Slide #22 Each vector has unit length Their dot product is 0 If these unit vectors are transformed by the rotation sub-matrix, then Example, 32

33 Constructing Two-Dimensional Rotation Matrices The orthogonal property p of rotation matrices is useful for constructing the matrix when we know the final orientation of an object In modeling application, we can obtain the transformation matrix within object s co-or system when knowing its orientation within overall word co-or system 33

34 Other Two-Dimensional Transformations a o s Reflection Shear 34

35 Reflection Reflection about the line A reflection about the y = 0 (the x axis) is line x = 0 (the y axis) accomplished with the flips x coordinates while transformation matrix keeping y coordinates the same 35

36 Reflection (2) Reflection relative to Reflection axis as the the coordinate origin diagonal line y = x 36

37 Shear A transformation that distorts the shape of an object The transformed shape appears as if the object were composed of internal layers that had been caused to slide over each other An x-direction shear relative to the x axis 37

38 Shear (2) x-direction shears relative to other reference lines y-direction shear relative to the line x = x ref 38

39 Raster Methods for Geometric Transformations a o s All bit settings in the rectangular area shown are copied as a block into another part of the frame buffer Rotate a two-dimensional object or pattern 90 counterclockwise by reversing the pixel values in each row of the array, then interchanging rows and columns 39

40 Raster Methods for Geometric Transformations a o s (2) For array rotations that are not multiples of 90, we need to do some extra processing Similar methods to scale a block of pixels 40

41 Transformations between Two- Dimensional Coordinate Systems Computer-graphics p applications involve coordinate transformations from one reference frame to another during various stages of scene processing To transform object descriptions from xy co-or or to x y xy co-or or 1. Translate so that the origin (x 0, y 0 ) of the x y system is moved to the origin (0, 0) of the xy system 2. Rotate the x axis onto the x axis 41

42 Transformations between 2-D Coordinate Systems s (2) Alternative method Specify a vector V that indicates the direction for the positive y axis Obtain the unit vector u along the x axis by applying a 90 clockwise rotation to vector v Rotation matrix could be expressed essed as elements of a set of orthonormal vectors 42

43 Geometric Transformations in Three-Dimensional e so Space A position P=(x, y, z) in 3-D is translated to a location P =(x, y, z ) by adding translation distances t x, t y, and t z 43

44 3-Dimensional Rotation z-axis rotation cos sin 0 0 sin cos

45 3-Dimensional Rotation (2) Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters x, y, and z x y z x 45

46 3-Dimensional Rotation (3) ( ) x-axis rotation y-axis rotation x axis rotation y axis rotation cos sin 0 0 sin cos cos 0 sin sin 0 cos

47 Composite Three-Dimensional Transformations a o s A rotation matrix for any axis that does not coincide with a coordinate axis 1. Translate the object so that the rotation axis coincides with the parallel coordinate axis 2. Perform the specified rotation about that axis 3. Translate the object so that the rotation axis is moved back to its original position 47

48 Composite Three-Dimensional Transformations a o s (2) A coordinate position P is transformed with the sequence where 48

49 Composite Three-Dimensional Transformations a o s (3) Rotate about an axis that is not parallel to one of the coordinate axes Transformation requires 5 steps 1. Translate the object so that the rotation axis passes through the coordinate origin 2. Rotate the object so that the axis of rotation coincides with one of the coordinate axes 3. Perform the specified rotation about the selected coordinate axis 4. Apply inverse rotations to bring the rotation axis back to its original orientation 5. Apply the inverse translation to bring the rotation axis back to its original spatial position 49

50 Composite Three-Dimensional Transformations a o s (4) 50

51 Composite Three-Dimensional Transformations a o s (5) The components of the rotation-axis vector The unit rotation-axis vector u Move the point P1 to the origin x-axis rotation gets u into the xz plane y-axis rotation swings u around to the z axis 51

52 Composite Three-Dimensional Transformations a o s (6) Rotate around the x axis to get u into the xz plane Rotation matrix cos sin 0 0 sin cos u z =1, u =d 52

53 Composite Three-Dimensional Transformations a o s (7) Resulting from the rotation about x axis is a vector labeled u Rotation angle β The transformation matrix for rotation of u about the y axis is cos 0 sin sin 0 cos

54 Composite Three-Dimensional Transformations a o s (8) The rotation axis is aligned with the positive z axis Apply the specified rotation angle θ To complete the required rotation about the given axis, we need to transform the rotation axis back to its original position 54

55 Quaternion Methods for Three-Dimensional Rotations A more efficient method Require less storage space than 4x4 matrices Important in animations, which often require complicated motion sequences and motion interpolations between two given positions of an object q=(a,b,c) M R () = R x -1 () R y -1 () R z () R y () R x () 55

56 ควอเทอเน ยน ควอเทอเน ยนท แทนการหมนเป ควอเทอเนยนทแทนการหม นเปนม ม นมม รอบแกน (x, y, z) ค อ cos ; xsin, 2 2 y sin, 2 z sin 2 ระว งว า (x, y, z) ต องเป นเวกเตอร หน งหน วย 56

57 ต วอย าง จงหาควอเทอเน ยนท แทนการหมนเป จงหาควอเทอเนยนทแทนการหม นเปนม ม นมม 60 องศา รอบแกน (1,1,1) เวกเตอร หน งหน วยของแกนค อ 1 3,1 3,1 3 ค านวณค า cos และ sin cos30 3 2,sin 30 และจะได ว าควอเทอเน ยนค อ ; 2 1 3, 2 1 3,

58 ต วอย าง ควอเทอเน ยนต อไปน แทนการหมนก องศา ควอเทอเนยนตอไปนแทนการหม นกองศา รอบแกนอะไร? 1 ;0, 2 6 4, 6 4 เราได ว า ฉะน น ฉะนน 1 cos sin แกนท หม นรอบค อ 1 sin ,, 0,, 0,,

59 Other Three-Dimensional Transformations a o s Scaling 59

60 Other Three-Dimensional Transformations a o s (2) Reflection Shear 60

61 Transformations between Three- Dimensional Coordinate Systems An xyz x y z system is defined with respect to an xyz system To transfer the xyz coordinate descriptions to the x y z system Translation that brings the x y z coordinate origin to the xyz origin A sequence of rotations that align corresponding coordinate axes A scaling transformation may also be necessary T(-x 0, -y 0, -z 0 ) The composition matrix is RT 61

62 Two main classes of Transformations a o s การแปลงส มพรรค Affine transformations can be represented by a matrix A set of affine transformations ti can be combined into a single overall affine transformation Structure deforming transformations are nonlinear, and cannot be represented by a matrix Ex. tapering, twisting, bending 62

63 Affine Transformations a o s An affine transformation is a form of coordinate transformation An affine transformation preserves angles and lengths Parallel lines are transformed into parallel lines and finite points map to finite points Examples Translation, rotation, scaling, reflection, and shear Conversion of coordinate descriptions i from one reference system to another 63

64 Affine Transformations To represent affine transformations with matrices, we must use homogeneous coordinates a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions Translation can be expressed with matrix multiplication The functional form x' = x + tx; y' = y + ty Any linear transformation can be represented by a general transformation matrix Translation can be seamlessly intermixed with all other types of transformations 64

65 OpenGL Geometric- Transformation a o Functions Basic OpenGL Geometric Transformations gltranslate* (tx, ty, tz); gltranslatef (25.0, -10.0, 0.0); glrotate* (theta, vx, vy, vz); glrotatef (90.0, 0.0, 0.0, 1.0); glscale* (sx, sy, sz); glscalef (2.0, -3.0, 1.0); OpenGL Matrix Operations glmatrixmode Mode (GL_MODELVIEW); glmatrixmode (GL_PROJECTION); 65

66 End of Chapter 5 66