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3D transformation in computer graphics

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3D transformation in computer graphics

  1. 1. B-Tech CSE 6th 13/NR/UT/CS005 Submitted By: Shivani Soni Submitted To: Mr. Vinod Thakur
  2. 2.  Transformations are a fundamental part of the computer graphics. Transformations are the movement of the object in Cartesian plane .
  3. 3.  There are two types of transformation in computer graphics.  1) 2D transformation  2) 3D transformation  Types of 2D and 3D transformation  1) Translation  2) Rotation  3) Scaling  4) Shearing  5) Mirror reflection
  4. 4.  Transformation are used to position objects , to shape object , to change viewing positions , and even how something is viewed.  In simple words transformation is used for  1) Modeling  2) viewing
  5. 5.  When the transformation takes place on a 2D plane .it is called 2D transformation.  2D means two dimensional (x-axis and Y-axis) Object Transformation in 2D  Alter the coordinates descriptions an object  Translation , rotation , scaling , shearing,  reflection.  Coordinate system unchanged Coordinate transformation in 2D  Produce a different coordinate system
  6. 6.  Translation:- Rotation:-  Scaling:-
  7. 7. Shearing:- Reflection
  8. 8.  When the transformation takes place on a 3D plane .it is called 3D transformation.  Generalize from 2D by including z coordinate Straight forward for translation and scale, rotation more difficult Homogeneous coordinates: 4 components Transformation matrices: 4×4 elements             1000 z y x tihg tfed tcba
  9. 9.  Right Hand coordinate system:  Left Hand coordinate system
  10. 10. Point We will consider points as column Vectors . Thus, a typical point with coordinates (x, y, z) is represented as:           z y x 3D Point Homogenous Coordinate A 3D point P is represented in homogeneous coordinates by a 4-dim. The main advantage:- it is easier to compose translation and rotation              1 z y x P
  11. 11.  Moving of object is called translation.  In 3 dimensional homogeneous coordinate representation , a point is transformed from position P = ( x, y , z) to P’=( x’, y’, z’)  This can be written as:- Using P’ = T . P                                          11000 100 010 001 1 z y x t t t z y x z y x
  12. 12.  The matrix representation is equivalent to the three equation. x’=x+ tx , y’=y+ ty , z’=z+ tz Where parameter tx , ty , tz are specifying translation distance for the coordinate direction x , y , z are assigned any real value. Translate an object by translating each vertex in the object.
  13. 13. where an object is to be rotated about an axis that is parallel to one of the coordinate axis, we can obtain the desired rotation with the following transformation sequence. Coordinate axis rotation Z- axis Rotation(Roll) Y-axis Rotation(Yaw) X-axis Rotation(Pitch)
  14. 14. Z-axis rotation is same as the origin about in 2D for which we have the derived matrices already. x’ = x cosθ – y sinθ y’ = x sinθ – y cosθ z’ = z                                        11000 0100 00cossin 00sincos 1 ' ' ' z y x z y x  
  15. 15.  Obtain rotations around other axes through cyclic permutation of coordinate parameters:  xzyx 
  16. 16. the equation for X-axis rotaion x’ = x y’ = y cosθ – z sinθ z’ = y sinθ + z cosθ                                         11000 0cossin0 0sincos0 0001 1 ' ' ' z y x z y x
  17. 17. The equation for Y-axis rotaion x’ = x cosθ + z sinθ y’ = y z’ = z cosθ - x sinθ                                         11000 0cos0sin 0010 0sin0cos 1 ' ' ' z y x z y x
  18. 18. Roll:- Rotation Pitch :- rotation Yaw:- rotation around Front to around the side to around the verti- back (Z-axis) side (about x-axis) cal axis(y-axis)
  19. 19.  Changes the size of the object and repositions the object relative to the coordinate origin. PSP                                           11000 000 000 000 1 z y x s s s z y x z y x
  20. 20.  The equations for scaling x’ = x . sx Ssx,sy,sz  y’ = y . sy z’ = z . sz  The equations for scaling x’ = x . sx Ssx,sy,sz  y’ = y . sy z’ = z . sz
  21. 21.  Reflection in computer graphics is used to emulate reflective objects like mirrors and shiny surfaces  Reflection may be an x-axis y-axis , z-axis. and also in the planes xy-plane,yz-plane , and zx-plane. Reflection relative to a given Axis are equivalent to 180 Degree rotations
  22. 22.  Reflection about x-axis:-  x’=x y’=-y z’=-z  1 0 0 0  0 -1 0 0  0 0 -1 0  0 0 0 1 Reflection about y-axis:- y’=y x’=-x z’=-z
  23. 23.  The matrix for reflection about y-axis:-  -1 0 0 0  0 1 0 0  0 0 -1 0  0 0 0 1  Reflection about z-axis:-  x’=-x y’=-y z’=z -1 0 0 0  0 -1 0 0  0 0 1 0  0 0 0 1
  24. 24.  Modify object shapes  Useful for perspective projections  When an object is viewed from different directions and at different distances, the appearance of the object will be different. Such view is called perspective view. Perspective projections mimic what the human eyes see. 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 zref
  25. 25.  Matrix for 3d shearing  Where a and b can Be assigned any real Value.

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