Unit-3 overview of transformations
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This document provides an overview of transformations in computer graphics. It discusses various 2D and 3D transformations including translation, rotation, scaling, reflection, shear, and their applications. Transformation means changing the orientation, shape, and size of objects or images. Basic 2D transformations discussed are translation, rotation, scaling, reflection, and shear. Homogeneous coordinates and matrix representations are used to combine multiple transformations. The document also discusses general pivot point rotation, fixed point scaling, and 3D transformations.
Unit-3 overview of transformations
- 1. COMPUTER GRAPHICS by Amol S. Gaikwad Lecturer Government Polytechnic Gadchiroli
- 3. Welcome! Are you excited for a fun learning session? Hi!.
- 4. Unit Outcomes Perform the given operation in 2D transformation Perform the given operation in 3D transformation Solve the given problem based on composite transformations Apply the given type of projection on object
- 5. What is Transformation ? Transformation means changing orientation, shape and size of objects or images We use geometry to achieve transformation Coordinate descriptions of objects are changed for performing transformation Basic transformations are as follow : Translation Rotation Scaling
- 6. What is Transformation ? Other transformations are as follow : Reflection Shear
- 7. Why Transformation is Needed ? By using basic shapes we create various types of pictures and graphs But many applications requires that pictures or images should be altered and changed Design applications and layouts are created by arranging the orientation and sizes of parts Animations are created by moving objects as per the scene requirements
- 8. Why Transformation is Needed ? By using basic shapes we create various types of pictures and graphs But many applications requires that pictures or images should be altered and changed Design applications and layouts are created by arranging the orientation and sizes of parts Animations are created by moving objects as per the scene requirements
- 9. Categories of Transformation 2D (Two dimensional transformation) 3D (Three dimensional transformation) x-axis y-axis x-axis y-axis z-axis
- 10. Translation (2D) In translation we change the location of object from on coordinate position to other another position along straight line path let tx and ty are distances , where we want to move our point Consider (x,y) is the original coordinate position of the point and (x',y') are new coordinate position of the point, then x' and y' are calculated as follow : x' = x + tx y' = y + ty The pait (tx,ty) are called as translation distance or translation vector or shift vector ........... equation-1
- 11. Translation (2D) Translation equation-1 can also be expressed as matrix equation as follow : x y P = x' y' P' = tx ty T = Two dimensional translation equations in matrix form is as follow : P' = P + T Translation is a rigid body transformation - means it moves the objects without changing their shape and size
- 12. Translation (2D) 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 Before translation After translation
- 13. Rotation (2D) In two dimesional rotation we reposition the obect along circular path in the xy-plane For rotating the object we require angle of rotation 'a' and the position (xr,yr) of rotation point (pivot point) Positive angle of rotation means anti-clockwise rotation around rotation point (pivot point) Negative angle of rotation means clockwise rotation around rotation point (pivot point)
- 14. Rotation (2D) Two dimesnsional matrix equation for rotation about origin point (0,0) is as follow : P' = R.P Where P', R and P are as follow : x' y' P' = cos a sin a R = x y P = -sin a cos a Rotate is a rigid body transformation - means it rotates the objects without changing their shape and size
- 15. Rotation (2D) 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 Before rotation After rotation
- 16. Scaling (2D) Scaling transformation alters (changes) the size of an object Scaling transformation is achieved by multiplying coordinate values (x,y) of each vertex by scaling factor sx and sy The transformed coordinates are denoted by (x',y') x' = x.sx and y' = y.sy Scaling factors sx scales objects in x direction Scaling factors sy scales objects in y direction
- 17. Scaling (2D) The matrix equation for two-dimensional scaling transformation is as follow : P' = S.P x' y' P' = sx S = x P = 0 sy 0 y
- 18. Scaling (2D) If the value of scaling factor sx and sy is less than 1, then the size of the object decreases If the value of scaling factor sx and sy is greater than 1, then the size of the object increases If the value of scaling factor sx = sy = 1, then the size of the object remains unchanged If the value of scaling factor sx = sy, then uniform scaling is achieved which maintains relative proportion of the object If the value of scaling factor sx is not equal to sy then differential scaling is achieved
- 19. Scaling (2D) 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 Before scaling After scaling
- 20. Scaling (2D) If the value of scaling factor sx and sy is less than 1, then the object moves closer to the coordinate origin (0,0) If the value of scaling factor sx and sy is greater than 1, then the object moves away from the coordinate origin (0,0)
- 21. Homogeneous Coordinates and Matrix Representations In many computer graphics applications we require to perform translations, rotations and scalings transformations sequencially With the help of homogeneous coordinates, we can combine various transformations so that final coordinates are obtained directly from the initial (starting) coordinates It eliminates the calculations of intermediate coordinate values By expanding the 2x2 matrix into 3x3 matrix, we combine the multiplicative and translational terms for 2D geometric transformations into single matrix
- 22. Homogeneous Coordinates and Matrix Representations Homogeneous coordinates are used to express any 2D transformation as a matrix multiplication We represent each cartesian coordinate position (x,y) with homogeneous coordinate triple (xh, yh, h) where we take value of h = 1 Each two-dimensional (2D) position is then represented with homogenous coordinates (x, y, 1)
- 23. Homogeneous Matrix Equations Homogeneous matrix equation for translation is written as follow : x' y' 1 = 1 0 0 0 1 0 tx ty 1 . x y 1 Homogeneous matrix equation for rotation is written as follow : x' y' 1 = cos a 0 -sin a 0 0 1 . x y 1 sin a cos a 0
- 24. Homogeneous Matrix Equations Homogeneous matrix equation for scaling is written as follow : x' y' 1 = sx 0 0 0 sy 0 0 1 . x y 1 0
- 25. Composite Transformations Composite transformation means performing various types of transformations in sequence Forming products of transformation matrices is often referred to as a concatenation or composition of matrices For column matrix representation of coordinate positions, we form composite transformations by multiplying matrices in order from right to left
- 26. Composite Transformations Homogeneous matrix equation for two successive translations is as follow : 1 0 0 0 1 0 tx1 ty1 1 . Homogeneous matrix equation for two successive scalings are as follow : 1 0 0 0 1 0 tx2 ty2 1 1 0 0 0 1 0 tx1 + tx2 ty1 + ty2 1 = sx1 0 0 0 sy1 0 0 0 1 . sx2 0 0 0 sy2 0 0 0 1 sx1.sx2 0 0 0 sy1.sy2 0 0 0 1 =
- 27. General Pivot Point Rotation It means performing rotations about any pivot point(xr, yr) . . ( xr, yr ) = pivot point ( 0, 0 ) Original position of object and pivot point Step-1 : Translate the object so that pivot point (xr, yr) is at origin (0, 0)
- 28. General Pivot Point Rotation ( xr, yr ) = pivot point . ( 0, 0 ) . Step-2 : Rotate the object about coordinate origin (0, 0) Step-3: Translate the object so that pivot point is returned to position (xr, yr)
- 29. General fixed Point Scaling It means performing scaling about any fixed point(xf, yf) . . ( xf, yf ) = fixed point ( 0, 0 ) Original position of object and fixed point Step-1 : Translate the object so that fixed point (xf, yf) is at origin (0, 0)
- 30. ( xf, yf ) = fixed point ( 0, 0 ) . Step-2 : Scale the object about coordinate origin (0, 0) Step-3: Translate the object so that fixed point is returned to original position (xf, yf) General fixed Point Scaling .
- 31. Other Transformations Translation, rotation and scaling are called as basic transformations Most computer graphics packages provide these basic transformations of translation, rotation and scaling Some computer graphics packages also provide additional transformations they are as follow : Reflection Shear
- 32. Reflection (2D) A reflection is a transformation that creates a mirror image of an object In two-dimensional reflection, a mirror image is generated by rotating the object by 180 degree around an axis called as 'axis of rotation' We can choose an axis of reflection in the xy plane or perpendicular to the xy plane The homogeneous matrix equation for reflection around line y = 0 (means x- axis) is as follow : 1 0 0 0 -1 0 1 0 0
- 33. Reflection (2D) Reflection around x-axis (means y = 0), keeps the x-values same but reverses the y values -1 0 0 0 1 0 1 0 0 It is same as rotating the object by 180 degree in three-dimensional space around x-axis The homogeneous matrix equation for reflection around line x= 0 (means y- axis) is as follow :
- 34. Reflection (2D) -1 0 0 0 -1 0 1 0 0 The homogeneous matrix equation for reflection around axis perpendicular to the xy plane and passing through the coordinate origin Original Position Reflected Position x y
- 35. Reflection (2D) Reflection of an object about the x axis Original Position Reflected Position x y Original Position Reflected Position Reflection of an object about the y axis x y
- 36. Shear (2D) Shear is a transformation that distorts (changes) the shape of object It transforms the object such that the object appears to be composed of internal layers and the layers slide over each other Two common shearing transformations are those that shift coordinate x values and those that shift y values Two-dimensional homogeneous matrix equation for x-direction shear is as follow : 1 0 0 shx 1 0 1 0 0 Where x' = x + shx.y y' = y
- 37. Shear (2D) a) Before shear b) After x-direction shear x y x y
- 38. Three-dimensional (3D) Transformations In two-dimension we require two coordinates (x, y) for every point In three-dimension we require three coordinates (x, y, z) for every point x-axis y-axis z-axis . (x, y, z) z y x
- 39. Translation (3D) Consider P = (x, y, z) are old coordinates of point and P' = (x', y', z') are new coordinates of a point after translation tx, ty, and tz are translation distances in x, y and z direction respectively Then three-dimensional homogeneous coordinate equation for translation is as follow : P' = T.P x' y' 1 = 1 0 0 0 1 0 0 1 . x y 1 0 0 z' 0 0 ty 1 tx tz z
- 40. Then three-dimensional homogeneous coordinate equation for rotation around z-axis is as follow : = Rotation (3D) Also written as P' = Rz(a).P x' y' 1 cos a sin a 0 -sin a cos a 0 0 0 . x y 1 0 0 z' 0 1 0 1 0 0 z x-axis y-axis z-axis Rotation is positive if it is counterclockwise Rotation is negative if it is clockwise
- 41. Rotation (3D) Then three-dimensional homogeneous coordinate equation for rotation around x-axis is as follow : Also written as P' = Rx(a).P x' y' 1 = cos a sin a 0 -sin a cos a 0 0 0 . x y 1 0 0 z' 0 1 0 1 0 0 z x-axis y-axis z-axis
- 42. Rotation (3D) Then three-dimensional homogeneous coordinate equation for rotation around y-axis is as follow : Also written as P' = Ry(a).P x' y' 1 = cos a sin a 0 -sin a cos a 0 0 0 . x y 1 0 0 z' 0 1 0 1 0 0 z x-axis y-axis z-axis
- 43. Rotation about axis parrallel to any coordinate axes x y z a) Original position of object Rotation axis x y z b) Translate Rotation axis onto x-axis
- 44. Rotation about axis parrallel to any coordinate axes c) Rotate object through angle 'a' x y z a = angle of rotation x y z d) Translate rotation axis to original position Rotation axis
- 45. Scaling (3D) Consider P = (x, y, z) are old coordinates of point and P' = (x', y', z') are new coordinates of a point after scaling Sx, Sy and Sz are values of scaling parameters Where x' = x.Sx , y' = y.Sy and z' = z.Sz Then three-dimensional homogeneous coordinate equation for rotation around y-axis is as follow : x' y' 1 = Sx 0 0 0 Sy 0 0 0 . x y 1 0 0 z' 0 Sz z 0 1 0 0
- 46. Scaling (3D) x y z a) Before scaling x y z b) After scaling
- 47. Scaling (3D) x y z a) Before scaling x y z b) After scaling
- 48. Projection The three dimensional objects can be projected on onto two dimensional view plane, called as projection Projections Parallel Projection Perspective Projection Orthographics Oblique Multiview Axonometric Dimetric Isometric Trimetric General Cavalier Cabinet One point Two point Three point
- 49. Projection Parallel Projection : In parallel projection the lines of projection are parallel to each other Coordinate position are transformed to the view plane in along parallel lines It maintains the relative proportions of objects It is used for scale drawings of three-dimensional objects It gives us accurate view of various sides of an object But it doesn't give us realistic representation of appearance of three- dimensional objects
- 51. Projection Orthographic Projection : It is a type of parallel projection In this projection, the projection lines are perpendicular to the view plane Orthographic projections are used to produce the front, side, and top views of an object Front, side, and rear orthographic projections of an object are called as elevations Top orthographic projection is called as plan view Orthographic projections are commonly used by engineering and architectural drawings
- 53. Projection Axonometric Projection : It is a type of orthographic projection This projection displays more than one face of an object In axonometric projection scaling factor may be different for three axes - x, y and z Isometric Projection : It is a type of Axonometric projection In this projection, the projection plane is kept in such a position such that it intersects each coordinates axis (x, y and z axis) at same distance from the origin
- 54. Projection Fig: Axomometric Projection Fig: Isometric Projection
- 55. Projection Oblique Projection : It is a type of parallel projection In this projection, the projection lines are not perpendicular to the view plane The two types of oblique projections are cavalier projection and cabinet projection
- 57. Projection Perspective Projection : In perspective projection the lines of projections are not parallel to each other Object positions are transferred to the plane along lines that meet at a point called as projection reference point or center of projection The projected view of object is calculated from the intersection of the projected lines with the view plane It creates realistic view of an object It does not preserve relative proportion of the object
- 58. Projection Types of Perspective Projection : One-point In this any one principal axes intersect with the projection plane Projection plane is perpendicular to any one principal axis Two-point In this two principal axes intersect with the projection plane Three-point In this all the three principal axes (x, y, z) intersect with the projection plane
- 62. Additional Resources https://www.tutorialspoint.com/computer_graphics https://nptel.ac.in/courses/106/102/106102065 Computer Graphics - Donald Hearn, Baker M Pauline, Pearson Education
- 63. Summary of Class 2D Transformations Lesson Recap 1 3D Transformations Lesson Recap 2 Composite Transformations Lesson Recap 3 Types of projections Lesson Recap 4
- 64. Thank you for attending!