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Affine Transformations
6.2 34
Extending Affine Transformations to Vectors
Suppose we only have T defined on points.
Define T (~v ) as follows:
• There exist points Q and R such that ~v = Q − R.
• Define T (~v ) to be T (Q) − T (R).
Note that Q and R are not unique.
The definition works for P + ~v :
T (P + ~v ) = T (P + Q − R)
= T (P ) + T (Q) − T (R)
= T (P ) + T (~v )
Can now show that the definition is well defined.
Theorem: Affine transformations map parallel lines to parallel lines.
Mapping Through an Affine Transformation
Need to pick a numerical representation; use coordinates:
Let A and B be affine spaces.
• Let T : A 7→ B be an affine transformation.
• Let FA = (~v1 , ~v2 , OV ) be a frame for A.
• Let FB = (w
~ 1, w
~ 2 , OW ) be a frame for B.
• Let P be a point in A whose coordinates relative FA are (p1 , p2 , 1).
(P = p1~v1 + p2~v2 + 1OV )
OV and OW are called the origins of their respective frames.
Question: What are the coordinates (p01 , p02 , 1) of T (P ) relative to the frame FB ?
Fact: An affine transformation is completely characterized by the image of a frame in the domain:
T (P ) = T (p1~v1 + p2~v2 + OV )
= p1 T (~v1 ) + p2 T (~v2 ) + T (OV ).
If
T (~v1 ) = t1,1 w
~ 1 + t2,1 w
~2
T (~v2 ) = t1,2 w
~ 1 + t2,2 w
~2
T (OV ) = t1,3 w
~ 1 + t2,3 w
~ 2 + OW
then we can find (p01 , p02 , 1) by substitution and gathering like terms.