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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 .
 Question:
 What is the length of ?
 Answer:
 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'?
 Change of Coordinates?
\
 Scale?
 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
 P'P=0

 P'P has no meaning
 We need

 A matrix
 A domain space
 A range space
 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