Coordinate Transformations and Homogeneous Coordinates

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• Misc
  • Math colloquia
    • Leary, “Axioms, Models, Infinity, and the Nature of the Universe”
      • Thursday, Feb. 5, 2:30 - 3:30, Newton 203
    • Dickerson, “Mathematics Majors’ Understandings of the Concept of Mathematical Definition”
      • Monday, Feb. 9 4:00 - 5:00, Newton 201
    • Extra credit for going and writing summary/reaction to either or both
  • Constant-valued functions in Matlab
    

    function bkgnd = BACKGROUND( )
    %BACKGROUND The background color (light blue) for ray traced images.
    bkgnd = [ 0.45 0.65 1.0 ];
    end

• Questions?
• Canonical Objects, Instancing, and Transformations
  • Define a “canonical” object in its own coordinate system, and then create actual “instances” in your scene by applying transformations to the canonical object
    
    • Object = ( transformation, color… )
    • Transformation = matrix
    • T(x,y,z) = (x′,y′,z′)
    • Linear transformation
      • x′ = a₁x + b₁y + c₁z
    • Affine transformation
      • x′ = a₁x + b₁y + c₁z + d₁
      • Could be represented as 2 matrices, M_3-by-3 + D_3-by-1 but there’s a better way
    • Translation
      • x′ = x + d₁
  • Homogeneous coordinates support affine transformations as matrix multiplication
    • 3D point/vector ≠ (x,y,z)
    • = (x,y,z,w) instead
      
• Next
  • More on homogeneous coordinates and affine transformations

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