CS430 Computer Graphics

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CS430 Computer Graphics

• Transformations of Objects 3D

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Topics

• 3D Affine Transformations
• Composition of 3D Affine Transformations
• Properties of Affine Transformations
• Changing Coordinate Systems

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3D Affine Transformations

• Translation
• T(dx, dy, dz)

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3D Affine Transformations

• Scaling
• S(Sx, Sy, Sz)

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3D Affine Transformations

• Rotations Three elementary rotations
• Rotation about the x-axis (x-roll)
• Rotation about the y-axis (y-roll)
• Rotation about the z-axis (z-roll)
• Positive values of rotation angel cause a
  counterclockwise (CCW) rotation about an axis as
  one looks inward from a point on the positive
  axis toward the origin

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3D Rotations

• Let c cos(?) and s sin(?)

? x-roll
? z-roll
? y-roll
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Composition of 3D Affine Transformations

• Similar to 2D
• If a sequence of transformations, represented by
  M1, M2, , Mn, is applied to a 3D point P, then P
  is transformed to MP, where M Mn ? ? ? M2 ? M1
• Important distinction between 2D and 3D rotations
• 3D rotation matrices do not commute
• 3D rotation can be about different axis

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Combining 3D Rotations

• Eulers Theorem
• Any rotation (or sequence of rotations) about a
  point is equivalent to a single rotation about
  some axis through that point
• Calculation is complex
• Good news OpenGL can do it for you
• glRotated(angle, ux, uy, uz),
• rotates angle degrees about an axis with unit
  vector (ux, uy, uz)

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Properties of Affine Transformations

• Affine transformation preserve affine
  combinations of points
• Affine transformation preserve lines and planes
• Parallelism of lines and planes is preserved
• Relative ratios are preserved
• Every affine transformation is composed of
  fundamental transformations

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Properties of Affine Transformations

• If M is the 2D transformation matrix, then
• Rotation, translation, and shearing do not change
  the area (why?)
• For scaling, new area (old area)?Sx?Sy
• If M is the 3D transformation matrix, then

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Properties of Affine Transformations

• The columns of the matrix M reveal the
  transformed coordinate frame
• In 2D,
• The axes of the new coordinate frame are m1 and
  m2, and the origin is m3
• The axes of new coordinates frame are not
  necessarily perpendicular nor must they be of
  unit length

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Changing Coordinate Systems

• Suppose coordinate system 2 is formed from
  coordinate system 1 by affine transformation M ,
  then for a point P in system 2, its coordinates
  are MP.

System 2
P
b
System 1
c
d
j
i
a
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Changing Coordinate Systems

• Successive changes in coordinate system

System 3
P
b
f
System 2
e
System 1
c
d

• System 1
• System 2
• System 3

M1
a
M2
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Changing Coordinate Systems

• Transforming points
• If a sequence of transformations, represented by
  M1, M2, , Mn, is applied to a point P, then P is
  transformed to MP, where
• M Mn ? ? ? M2 ? M1
• Transforming coordinate system
• If a sequence of transformations represented by
  M1, M1, , Mn, is applied to the coordinate
  system, then a point P expressed in the
  transformed system has coordinates MP in the
  original system, where
• M M1 ? M2 ? ? ? Mn