Elementary 3D Transformations a "Graphics Engine"

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Elementary 3D Transformations - a "Graphics
Engine"

• Transformation procedures
• Transformations of coordinate systems
• Translation
• Scaling
• Rotation

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Coordinate systems
left handed right handed
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Transformation procedures

• A scene is made up of objects
• Objects can be made of separately defined parts
• Each object / part defined by a list of points
  (vertices)
• Any part of the object can be moved or distorted
  by applying a transformation to the list of
  points which define it

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Transformations of coordinate system

• Separate coordinate systems for different objects
• Common (Word) coordinate system for the scene
• Building a scene - transformation to the Word
  coordinate system

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Basic transformations

• Translation (shift)
• Scaling
• Rotation

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Translation
x x Tx y y Ty z z Tz
T (Tx,Ty,Tz)
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Scalingabout the origin
x x Sx y y Sy z z Sz S gt 1 -
enlarge 0 lt S lt 1 - reduce S lt 0 - mirror
Y
Z
X
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Scalingabout an arbitrary point
Scaling about a fixed point ( xc, yc, zc ) x'
xc ( x xc ) Sx y' xc ( y yc )
Sy z' zc ( z zc ) Sz Can also be
achieved by a composite transformation.
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Rotation

• The direction of rotation in the left-handed
  system
• Positive angle of rotation
• when looking from a positive axis toward the
  origin
• a 90o clockwise rotation transforms one positive
  axis into the other.

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Positive angle of rotation for Z axis
Looking from the positive end of Z axis towards
the origin
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Rotation

• Axis of rotation is Direction of positive
  rotation is
• X from Y to Z
• Y from Z to X
• Z from X to Y

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Rotation about Z axis
Y
Z

• x' xcos ? - ysin ?
• y' xsin ? ycos ?
• z' z

X
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Rotation about X axis
Y
Z

• y' ycos ? - zsin ?
• z' ysin ? zcos ?
• x' x

X
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Rotation about Y axis
Y

• z' zcos? - xsin?
• x' zsin? xcos?
• y' y

Z
X
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Rotation about any fixed point

• 1. Translate the object so that the rotation axis
  coincides with the parallel coordinate axis
• 2. Perform the specified rotation
• 3. Translate the object so that the rotation axis
  is moved back to its original position

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Matrix representationHomogeneous coordinates

• common notation for ALL transformations
• common computational mechanism for ALL
  transformations
• simple mechanism for combining a number of
  transformations gt computational efficiency

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Common matrix operation for all transformations???

• Translate (shift) point P
• Scale point P
• Rotate point P
• Point (vector) P xp yp zp
• Matrix ???

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Homogeneous coordinates

• Point P (x, y, z ) represented by a vector
• P
• Transformations
• All represented by a 4 x 4 matrix T

T
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Point transformation in homogeneous coordinates

• Implemented by matrix multiplication
• P T P

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Transformation matrices for elementary
transformations

• 4 x 4 matrix
• Homogeneous coordinates
• Translation, scaling, rotation and perspective
  projection, all defined through matrices

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Translation

• x x Tx
• y y Ty
• z z Tz
• T

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Scaling

• x x Sx
• y y Sy
• z z Sz
• S

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Rotation about Z axis

• x' xcos ? - ysin ?
• y' xsin ? ycos ?
• z' z
• Rz

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Rotation about X axis

• y' ycos ? - zsin ?
• z' ysin ? zcos ?
• x' x
• Rx

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Rotation about Y axis

• z' zcos? - xsin?
• x' zsin? xcos?
• y' y
• Ry