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- 3 Types of Transformations:
(between two spaces)
(``warp'' an object within its own space)
- T: change of coordinates
- Changes of Coordinates:
- Given 2 frames:
- , orthonormal,
- , orthogonal.
- Suppose .
- Then .
- What is the length of ?
- Its length is ,
regardless of its frame of representation.
we have and
- P,Q relative to ,
- We are given a matrix representation of a transformation T:
- Consider P'=TP and Q'=TQ.
- How do we interpret P' and Q'?
- Change of Coordinates?
- Transformations between spaces?
- With no other information, any of the above is a valid
- Do we care?
- In (1) nothing changes except the representation.
- In (1) distances are preserved while they change in
(2) and the question has no meaning in (3).
- In (3), we've completely changed spaces.
- the meaning of |P'-P|
- |P'-P| has no meaning
- We need
to fully specify a transformation.
- A matrix
- A domain space
- A range space
- A coordinate frame in each space
- Most of the time not all will be specified!
...So be careful out there.
CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab