Transformation as a change in coordinate system

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Transformation as a change in coordinate system
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Local versus world coordinate system

• Until now, we have defined all points, lines,
  etc, within the world coordinate system.
• Not ideal to have every object defined inside the
  world-coordinate system.
• Better (read easier, more re-usable) to have
  objects, each with their own local coordinate
  system which can then be placed into the world
  coordinate system.

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Local versus world coordinate system

• The idea is analogous to having several pieces of
  paper
• Each page with an object on it which is being
  translated, scaled, rotated.. Before being placed
  on the world coordinate system

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Local versus world coordinate system
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Local versus world coordinate system

• This requires a change in mentality when drawing
  our objects.
• Before, we drew an object in the world-coordinate
  system, scaled it, rotated it, and then
  transformed it to the required location.
• With a local coordinate system, the reverse is
  the case
• We translate our local coordinate system, then
  rotate, then scale, and finally draw our object
• The advantage of this, is that each object (or
  page) can in turn be made of several simliar
  local coordinate systems, which can be draw
  with out regard to the world coordinate system

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Local versus world coordinate system

• Let define the representation of a point
  in the coordinate system i, and define
  the representation of a point in the coordinate
  system j.
• Then let define the transformation the
  converts the representation of a point in the
  coordinate system j into its representation in
  the coordinate system i, such that

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Local versus world coordinate system

• If for coordinate system k, then
• so

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Local versus world coordinate system

• Computer graphics languages / APIs typically use
  this concept of local coordinate systems
• Cue processing example