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2.  Transformation and object modeling is extended from 2D by including z coordinate  Object can translate by specifying 3D Vector  Can be moved in each of the three coordinates  Rotation – select any spatial orientation of the rotation axis  Express in 3*3 Matrix form 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 2

3.  In 3D Homogeneous representation a point is translated from position P to position P’ 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 3

4.  An object is translated in 3D dimensional by transforming each of the defining points of the objects. 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 4

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6.  Designate the axis of rotation and amount of angular rotation 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 6

7. Positive rotation angles produce counterclockwise rotations about a coordinate axis 12/24/2015 7

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13. A.Aruna/Assistant professor/IT/SNSCE  An object is to be rotated about an axis that is parallel to one of the coordinate axes 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 12/24/2015 13

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15. A.Aruna/Assistant professor/IT/SNSCE  An object is to be rotated about an axis that is not parallel to one of the coordinate axes 1. Translate the object so that the rotation axis passes through the coordinate origin. 2. Rotate the object so that the axis of rotation coincide with one of the coordinate axes. 3. Perform the specified rotation about that 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 position. 12/24/2015 15

16. Draw any shape, then moving translation matrix. Good Luck 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 16

17.  P is scaled to P' by S: Called the Scaling matrix S = 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 17

18.  Scaling with respect to the coordinate origin 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 18

19.  Scaling with respect to a selected fixed position (x f, y f, z f ) 1. Translate the fixed point to origin 2. Scale the object relative to the coordinate origin 3. Translate the fixed point back to its original position 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 19

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21.  About an axis: equivalent to 180˚rotation about that axis 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 21

22. 3D Reflections 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 22

23. 3D Shearing Modify object shapes Useful for perspective projections: –E.g. draw a cube (3D) on a screen (2D) –Alter the values for x and y by an amount proportional to the distance from z ref 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 23

24. 3D Shearing 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 24

25. Shears 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 25

26. PUZZLE 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 26

27. 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 27 4 = i

28. 12/24/2015A.Aruna/Assistant professor/IT/SNSCE 28 1 = h

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31. 3D Viewing  The steps for computer generation of a view of a three dimensional scene are somewhat analogous to the processes involved in taking a photograph. 12/24/2015 31 A.Aruna/Assistant professor/IT/SNSCE

32. Camera Analogy 1. Viewing position 2. Camera orientation 3. Size of clipping window Position Orientation Window (aperture) of the camera 12/24/2015 32 A.Aruna/Assistant professor/IT/SNSCE

33. Viewing Pipeline  The general processing steps for modeling and converting a world coordinate description of a scene to device coordinates 12/24/2015 33 A.Aruna/Assistant professor/IT/SNSCE

34. Viewing Pipeline 1. Construct the shape of individual objects in a scene within modeling coordinate, and place the objects into appropriate positions within the scene (world coordinate). 12/24/2015 34 A.Aruna/Assistant professor/IT/SNSCE

35. Viewing Pipeline 2. World coordinate positions are converted to viewing coordinates. 12/24/2015 35 A.Aruna/Assistant professor/IT/SNSCE

36. Viewing Pipeline 3. Convert the viewing coordinate description of the scene to coordinate positions on the projection plane. 12/24/2015 36 A.Aruna/Assistant professor/IT/SNSCE

37. Viewing Pipeline 4. Positions on the projection plane, will then mapped to the Normalized coordinate and output device. 12/24/2015 37 A.Aruna/Assistant professor/IT/SNSCE

38. Viewing Coordinates Camera Analogy Viewing coordinates system described 3D objects with respect to a viewer. A Viewing (Projector) plane is set up perpendicular to z v and aligned with (x v,y v ). 12/24/2015 38 A.Aruna/Assistant professor/IT/SNSCE

39. Specifying the Viewing Coordinate System (View Reference Point)  We first pick a world coordinate position called view reference point (origin of our viewing coordinate system).  P 0 is a point where a camera is located.  The view reference point is often chosen to be close to or on the surface of some object, or at the center of a group of objects. Position 12/24/2015 39 A.Aruna/Assistant professor/IT/SNSCE

40. Specifying the Viewing Coordinate System (Z v Axis)  Next, we select the positive direction for the viewing z v axis, by specifying the view plane normal vector, N.  The direction of N, is from the look at point (L) to the view reference point. Look Vector 12/24/2015 40 A.Aruna/Assistant professor/IT/SNSCE

41. Specifying the Viewing Coordinate System (y v Axis) up direction Vview up vector  Finally, we choose the up direction for the view by specifying a vector V, called the view up vector.  This vector is used to establish the positive direction for the y v axis.  V  V is projected into a plane that is perpendicular to the normal vector. Up Vector 12/24/2015 41 A.Aruna/Assistant professor/IT/SNSCE

42. Look and Up Vectors  Look Vector  the direction the camera is pointing  three degrees of freedom; can be any vector in 3-space  Up Vector  determines how the camera is rotated around the Look vector  for example, whether you’re holding the camera horizontally or vertically (or in between)  projection of Up vector must be in the plane perpendicular to the look vector (this allows Up vector to be specified at an arbitrary angle to its Look vector) Up vector Look vector Projection of up vector Position 12/24/2015 42 A.Aruna/Assistant professor/IT/SNSCE

43. Specifying the Viewing Coordinate System (x v Axis)  Using vectors N and V, the graphics package computer can compute a third vector U, perpendicular to both N and V, to define the direction for the x v axis. P0P0 P0P0 12/24/2015 43 A.Aruna/Assistant professor/IT/SNSCE

44. The View Plane  Graphics package allow users to choose the position of the view plane along the z v axis by specifying the view plane distance from the viewing origin.  The view plane is always parallel to the x v y v plane. 12/24/2015 44 A.Aruna/Assistant professor/IT/SNSCE

45. Obtain a Series of View  To obtain a series of view of a scene, we can keep the view reference point fixed and change the direction of N. 12/24/2015 45 A.Aruna/Assistant professor/IT/SNSCE

46. Simulate Camera Motion  To simulate camera motion through a scene, we can keep N fixed and move the view reference point around. 12/24/2015 46 A.Aruna/Assistant professor/IT/SNSCE

47. Transformation from World to Viewing Coordinates 12/24/2015 47 A.Aruna/Assistant professor/IT/SNSCE

48. Viewing Pipeline  Before object description can be projected to the view plane, they must be transferred to viewing coordinates.  World coordinate positions are converted to viewing coordinates. 12/24/2015 48 A.Aruna/Assistant professor/IT/SNSCE

49. Transformation from World to Viewing Coordinates  Transformation sequence from world to viewing coordinates: 12/24/2015 49 A.Aruna/Assistant professor/IT/SNSCE

50. Transformation from World to Viewing Coordinates  Another Method for generating the rotation- transformation matrix is to calculate unit uvn vectors and form the composite rotation matrix directly: 12/24/2015 50 A.Aruna/Assistant professor/IT/SNSCE

51. Projection 12/24/2015 51 A.Aruna/Assistant professor/IT/SNSCE

52. Viewing Pipeline  Convert the viewing coordinate description of the scene to coordinate positions on the projection plane.  Viewing 3D objects on a 2D display requires a mapping from 3D to 2D. 12/24/2015 52 A.Aruna/Assistant professor/IT/SNSCE

53. Projection  Projection can be defined as a mapping of point P(x,y,z) onto its image in the projection plane. projector  The mapping is determined by a projector that passes through P and intersects the view plane ( ). 12/24/2015 53 A.Aruna/Assistant professor/IT/SNSCE

54. Projection  Projectors are lines from center (reference) of projection through each point in the object.  The result of projecting an object is dependent on the spatial relationship among the projectors and the view plane. 12/24/2015 54 A.Aruna/Assistant professor/IT/SNSCE

55. Projection Parallel Projection Parallel Projection : Coordinate position are transformed to the view plane along parallel lines. Perspective Projection: Perspective Projection: Object positions are transformed to the view plane along lines that converge to the projection reference (center) point. 12/24/2015 55 A.Aruna/Assistant professor/IT/SNSCE

56. Parallel Projection  Coordinate position are transformed to the view plane along parallel lines.  Center of projection at infinity results with a parallel projection.  A parallel projection preserves relative proportion of objects, but dose not give us a realistic representation of the appearance of object. 12/24/2015 56 A.Aruna/Assistant professor/IT/SNSCE

57. Perspective Projection  Object positions are transformed to the view plane along lines that converge to the projection reference (center) point.  Produces realistic views but does not preserve relative proportion of objects. 12/24/2015 57 A.Aruna/Assistant professor/IT/SNSCE

58. Perspective Projection  Projections of distant objects are smaller than the projections of objects of the same size are closer to the projection plane. 12/24/2015 58 A.Aruna/Assistant professor/IT/SNSCE

59. Parallel and Perspective Projection 12/24/2015 59 A.Aruna/Assistant professor/IT/SNSCE

60. Parallel Projection 12/24/2015 60 A.Aruna/Assistant professor/IT/SNSCE

61. Parallel Projection  Projection vector: Defines the direction for the projection lines (projectors).  Orthographic Projection  Orthographic Projection : Projectors (projection vectors) are perpendicular to the projection plane.  Oblique Projection not  Oblique Projection : Projectors (projection vectors) are not perpendicular to the projection plane. 12/24/2015 61 A.Aruna/Assistant professor/IT/SNSCE

62. Orthographic Parallel Projection 12/24/2015 62 A.Aruna/Assistant professor/IT/SNSCE

63. Orthographic Parallel Projection  Orthographic projection used to produce the front, side, and top views of an object. 12/24/2015 63 A.Aruna/Assistant professor/IT/SNSCE

64. Orthographic Parallel Projection  Frontsiderear elevations  Front, side, and rear orthographic projections of an object are called elevations.  Topplan  Top orthographic projection is called a plan view. 12/24/2015 64 A.Aruna/Assistant professor/IT/SNSCE

65. Orthographic Parallel Projection Multi View Orthographic 12/24/2015 65 A.Aruna/Assistant professor/IT/SNSCE

66. Orthographic Parallel Projection  Axonometric orthographic  Axonometric orthographic projections display more than one face of an object. 12/24/2015 66 A.Aruna/Assistant professor/IT/SNSCE

67. Orthographic Parallel Projection  Isometric Projection  Isometric Projection : Projection plane intersects each coordinate axis in which the object is defined (principal axes) at the same distant from the origin.  Projection vector makes equal angles with all of the three principal axes. Isometric projection is obtained by aligning the projection vector with the cube diagonal. 12/24/2015 67 A.Aruna/Assistant professor/IT/SNSCE

68. Orthographic Parallel Projection  Dimetric Projection  Dimetric Projection : Projection vector makes equal angles with exactly two of the principal axes. 12/24/2015 68 A.Aruna/Assistant professor/IT/SNSCE

69. Orthographic Parallel Projection  Trimetric Projection  Trimetric Projection : Projection vector makes unequal angles with the three principal axes. 12/24/2015 69 A.Aruna/Assistant professor/IT/SNSCE

70. Orthographic Parallel Projection 12/24/2015 70 A.Aruna/Assistant professor/IT/SNSCE

71. Orthographic Parallel Projection Transformation 12/24/2015 71 A.Aruna/Assistant professor/IT/SNSCE

72.  Convert the viewing coordinate description of the scene to coordinate positions on the Orthographic parallel projection plane. Orthographic Parallel Projection Transformation 12/24/2015 72 A.Aruna/Assistant professor/IT/SNSCE

73.  Since the view plane is placed at position z vp along the z v axis. Then any point (x,y,z) in viewing coordinates is transformed to projection coordinates as: Orthographic Parallel Projection Transformation 12/24/2015 73 A.Aruna/Assistant professor/IT/SNSCE

74. Oblique Parallel Projection 12/24/2015 74 A.Aruna/Assistant professor/IT/SNSCE

75. Oblique Parallel Projection  Projection are not perpendicular to the viewing plane.  Angles and lengths are preserved for faces parallel the plane of projection.  Preserves 3D nature of an object. 12/24/2015 75 A.Aruna/Assistant professor/IT/SNSCE

76. Oblique Parallel Projection Transformation 12/24/2015 76 A.Aruna/Assistant professor/IT/SNSCE

77.  Convert the viewing coordinate description of the scene to coordinate positions on the Oblique parallel projection plane. Oblique Parallel Projection Transformation 12/24/2015 77 A.Aruna/Assistant professor/IT/SNSCE

78. Oblique Parallel Projection  Point (x,y,z) is projected to position (x p,y p ) on the view plane. L  Projector (oblique) from (x,y,z) to (x p,y p ) makes an angle with the line ( L ) on the projection plane that joins (x p,y p ) and (x,y). L  Line L is at an angle with the horizontal direction in the projection plane. 12/24/2015 78 A.Aruna/Assistant professor/IT/SNSCE

79. Oblique Parallel Projection 12/24/2015 79 A.Aruna/Assistant professor/IT/SNSCE

80. Oblique Parallel Projection Orthographic Projection: 12/24/2015 80 A.Aruna/Assistant professor/IT/SNSCE

81. Oblique Parallel Projection  Angles, distances, and parallel lines in the plane are projected accurately. 12/24/2015 81 A.Aruna/Assistant professor/IT/SNSCE

82. Cavalier Projection Cavalier Projection:  Preserves lengths of lines perpendicular to the viewing plane.  3D nature can be captured but shape seems distorted.  Can display a combination of front, and side, and top views. 12/24/2015 82 A.Aruna/Assistant professor/IT/SNSCE

83. Cabinet Projection Cabinet Projection: ½  Lines perpendicular to the viewing plane project at ½ of their length.  A more realistic view than the cavalier projection.  Can display a combination of front, and side, and top views. 12/24/2015 83 A.Aruna/Assistant professor/IT/SNSCE

84. Cavalier & Cabinet Projection CavalierCabinet 12/24/2015 84 A.Aruna/Assistant professor/IT/SNSCE

85. Perspective Projection 12/24/2015 85 A.Aruna/Assistant professor/IT/SNSCE

86. Perspective Projection 12/24/2015 86 A.Aruna/Assistant professor/IT/SNSCE

87. Perspective Projection  In a perspective projection, the center of projection is at a finite distance from the viewing plane.  Produces realistic views but does not preserve relative proportion of objects  The size of a projection object is inversely proportional to its distance from the viewing plane. 12/24/2015 87 A.Aruna/Assistant professor/IT/SNSCE

88. Perspective Projection vanishing point  Parallel lines that are not parallel to the viewing plane, converge to a vanishing point.  A vanishing point is the projection of a point at infinity. 12/24/2015 88 A.Aruna/Assistant professor/IT/SNSCE

89. Vanishing Points  Each set of projected parallel lines will have a separate vanishing points.  There are infinity many general vanishing points. 12/24/2015 89 A.Aruna/Assistant professor/IT/SNSCE

90. Perspective Projection  The vanishing point for any set of lines that are parallel to one of the principal axes of an object is referred to as a principal vanishing point.  We control the number of principal vanishing points (one, two, or three) with the orientation of the projection plane. 12/24/2015 90 A.Aruna/Assistant professor/IT/SNSCE

91. Perspective Projection  The number of principal vanishing points in a projection is determined by the number of principal axes intersecting the view plane. 12/24/2015 91 A.Aruna/Assistant professor/IT/SNSCE

92. Perspective Projection One Point Perspective (z-axis vanishing point) z 12/24/2015 92 A.Aruna/Assistant professor/IT/SNSCE

93. Perspective Projection Two Point Perspective (z, and x-axis vanishing points) 12/24/2015 93 A.Aruna/Assistant professor/IT/SNSCE

94. Perspective Projection Two Point Perspective 12/24/2015 94 A.Aruna/Assistant professor/IT/SNSCE

95. Perspective Projection ThreePoint Perspective Three Point Perspective (z, x, and y-axis vanishing points) 12/24/2015 95 A.Aruna/Assistant professor/IT/SNSCE

96. Perspective Projection 12/24/2015 96 A.Aruna/Assistant professor/IT/SNSCE

97. Perspective Projection Transformation 12/24/2015 97 A.Aruna/Assistant professor/IT/SNSCE

98.  Convert the viewing coordinate description of the scene to coordinate positions on the perspective projection plane. Perspective Projection Transformation 12/24/2015 98 A.Aruna/Assistant professor/IT/SNSCE

99.  Suppose the projection reference point at position z prp along the z v axis, and the view plane at z vp. Perspective Projection Transformation 12/24/2015 99 A.Aruna/Assistant professor/IT/SNSCE

100. Perspective Projection Transformation On the view plane: 12/24/2015 100 A.Aruna/Assistant professor/IT/SNSCE

101. Perspective Projection Transformation On the view plane: 12/24/2015 101 A.Aruna/Assistant professor/IT/SNSCE

102. Perspective Projection Transformation Special Cases: 12/24/2015 102 A.Aruna/Assistant professor/IT/SNSCE

103. Perspective Projection Transformation Special Cases: Special Cases: The projection reference point is at the viewing coordinate origin: Z prp =0 12/24/2015 103 A.Aruna/Assistant professor/IT/SNSCE

104. Summery 12/24/2015 104 A.Aruna/Assistant professor/IT/SNSCE

105. Summary 12/24/2015 105 A.Aruna/Assistant professor/IT/SNSCE

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