Homogeneous representation in geometric transformation

Homogeneous representation in geometric transformation

• 1.
  GANDHINAGAR INSTITUTE OFTECHNOLOGY Computer Aided Design (2161903) Active Learning Assignment On Homogeneous representation in geometric transformation Branch : Mechanical Engineering Batch : 6 C-3 Prepared by: Guided By: Suthar Chandresh (140120119229) Prof. Jatin Patel
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  Contents • Definition &Motivation • Geometric Transformation :- Translation :- Scaling :- Shearing • Matrix Representation • Homogeneous Co-ordinates
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  Geometric transformation • Definition :-Translation , Scaling , Shearing • Motivation – Why do we need geometric transformations in CG? :- As a viewing aid :- As a modeling tool :- As an image manipulation tool
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  Example: 2D Translation Modeling Coordinates Translate(5,3) World Coordinates
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  EXAMPLE: 2D SCALING Modeling Coordinates WorldCoordinates Scale(0.3, 0.3)
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  Basic 2D Transformations •Translation – – • Scale – – • Shear – – txxx  tyyy  sxxx  syyy  yhxxx  xhyyy 
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  Matrix Representation • Representa 2D Transformation by a Matrix • Apply the Transformation to a Point                     y x dc ba y x dycxy byaxx         dc ba Transformation Matrix Point
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  Matrix Representation • Transformationscan be combined by matrix multiplication                                 y x lk ji hg fe dc ba y x Matrices are a convenient and efficient way to represent a sequence of transformations Transformation Matrix
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  2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Translation txxx  tyyy                      y x ty tx y x 0 0
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  2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix?2D Scaling ysyy xsxx                       y x sy sx y x 0 0
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  2×2 Matrices • Whattypes of transformations can be represented with a 2×2 matrix? 2D Shearing                     y x shy shx y x 1 1 yxshyy yshxxx  
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  Basic 2D Transformations •Basic 2D transformations as 3x3 Matrices                                  1100 10 01 1 y x ty tx y x                                  1100 00 00 1 y x sy sx y x                                  1100 01 01 1 y x shy shx y x Translate Shear Scale