Geometrical Transformations 2

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Geometrical Transformations 2

• Adapted from Fundamentals of Interactive Computer
  Graphics, Foley and van Dam, pp. 245-315, by Geb
  Thomas

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Learning Objectives

• Learn how 2D transformations are represented in
  3D.
• Recognize the inverse matrix for a homogeneous
  transformation.
• Understand how the transformations also represent
  coordinate frame transformations.
• Understand the concept of a matrix stack.

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2D Translation and 3D Translation
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2D Rotation
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3D Rotation Z Axis
About the Z axis
z
y
x
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3D Rotation X Axis
About the Z axis
z
y
x
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3D Rotation Y Axis
About the Z axis
z
y
x
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Inverse Matrices

• The 3x3 rotation submatrix is orthogonal.
• The inverse of the 3x3 matrix is the transform
  of the original matrix
• The inverse of the translation component is just
  the reverse translation.

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Composition of Transforms

• Various motions can be tacked, one after the
  other, in a long sequence of matrices T1T2T3T4
• These combinations will maintain the relative
  shape of the vectors processed, but will shift
  them around the original coordinate frame.

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Thinking of Reference Frames

• Another way to think of this mathematics, is to
  imagine transforming the coordinate frame to a
  new place

Transformation, T, moves things 5 to the right
and 2 up. The whole coordinate frame moves to a
new position
z1
y1
z0
y0
T(5,0,2)
x1
x0
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Coordinate Rotation
y0
y1
R(y-35)
z0
z1
x1
x0
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The Matrix Stack

• While drawing a world, you often want to draw
  with respect to a convenient coordinate frame
• The graphics card need only keep track of the
  current position (current transformation)
• If you want to shift to the left, multiply the
  current frame by a left translation
• When done, shift back to the right

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The Matrix Stack, FILO
T(3,2,5)
R(x35)
R(y15)
Programmers Next desired Shift of
reference frames
Current position of Drawing reference frame
T(2,1,1)
R(z12)
T(15,20,35)
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Learning Objectives

• Learn how 2D transformations are represented in
  3D.
• Recognize the inverse matrix for a homogeneous
  transformation.
• Understand how the transformations also represent
  coordinate frame transformations.
• Understand the concept of a matrix stack.

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