06a Transformation

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06a Transformation Powered By Docstoc
					        CPSC 453:
      Transformations
            (Chapter 4.7)



                             Mark Matthews
               matthews@cpsc.ucalgary.ca
      Office Hours: 3:15-4:00PM TR (same)
                        Office: 680J (same)
www.cpsc.ucalgary.ca/~sheelagh/courses/453
                 Review
• 2P-Q okay
• P-3Q+2v not okay
• Why? Substitute:
  • POINTS = 1
  • vector = 0
• 2P-Q = 2(1) – 1 = 1 …. A valid point
• P-3Q+2v = (1) – 3(1) + 2(0) = -2 … invalid
• Result must be 0 or 1
Transformations
             Transformations
• Essential concept of computer graphics

• Translation



• Rotation



• Scaling
              Transformations
• Adjusts objects for proper size, situation and
  direction
• Change transformation properties over time for
  animation
• Compose a scene out of similar objects




• Viewing is also a transformation (discussed later)
                 Rotation




• Can be used for points or vectors
• Affine transformation
  • Transforming endpoints is enough
                  Translation
• Point Translation

• P’ = P + v


• Is there a matrix form for this?

• Is it affine? Line preserving?

• Does it work for points and vectors?
                    Answers
• There is no matrix operation for translation

       x' a b   x 
       y '   c d    y 
                    
• Translation is affine.

• Different interpretations for points and vectors

• We need a new method
Homogenous Coordinates
         Useful Representation


• Good separation between points and vectors

• Shows directly our method for Affine
  transformations

• A method for translation using matrices
                  Translation
• Point Translation




•   Points: moved by translation.
•   Vectors: unaffected by translation.
•   2D translation in homogenous coordinates
•   3D translation in homogenous coordinates
                 New Rotation
• Rotation:

     x' cos         sin    0  x 
     y '   sin    cos          y
                                 0  
      
    1  0
                      0       1  1 
                                    
                    Scaling
• Scale about the origin
• Changes the size of the object, uniformly
  or non-uniformly
• Matrix operation:




• Negative values produce reflection
 Criteria for Affine Transformations
• Is the transformation affine?
• What is a simple criteria
• Matrix form of an affine transform:
   • 2D




   • 3D




• So scaling is affine
     Review of Affine Transforms
• Affine Transform
• A maps points to points
• A maps vectors to vectors
         Shear Transformation?
• Along the x-axis:
• 2D Matrix




• 3D Matrix




• Affine Transformation
                Rotation in 3D
• Pick an axis to rotate about
• Simple extension of planar rotation

• For z-axis, matrix is:




• Where c denotes cos and s denotes sin
• Also given as P’ = Rz() . P
  Rotation about the X and Y axis
• For x-axis, the matrix
  is:




• For y-axis, the matrix
  is:

				
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