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# Ambiguity

3 Types of Transformations:
1. (between two spaces)
2. (``warp'' an object within its own space)
3. T: change of coordinates

Changes of Coordinates:
• Given 2 frames:
• , orthonormal,
• , orthogonal.
• Suppose .
• Then .
Question:
What is the length of ?
Its length is       , regardless of its frame of representation.

Suppose
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'?

1. Change of Coordinates? \

2. Scale?

3. Transformations between spaces?

• With no other information, any of the above is a valid interpretation.

Do we care?
YES!
• 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.

Consider
the meaning of |P'-P|
1. |P'-P|=0
2. 3. |P'-P| has no meaning

We need
1. A matrix
2. A domain space
3. A range space
4. A coordinate frame in each space
to fully specify a transformation.

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

cs488@cgl.uwaterloo.ca