Affine Transformations
These are illustrations of some affine transformations. For each transformations, the image of the square
is displayed and underneath is the parameters as entered in the
FractalPattern program.
See my notes on affine transformations for more details, and also Section 5.2 in
the text (where they explain the parameters r, s and the two angles that appear in the FractalPattern
menu; you can use either those parameters (and ask the program to 'compute parameters')
or directly enter the a,b,c,d in the menu yourself).
Recall that to determine the matrix M that combines two linear transformations, eg.,
first
rotate by 30o (=M1) and then shear in x-direction (=M2),
you multiply the two
matrices; M= M2M1 (in that order!). The rule for multiplying matrices is this;
.
Note that we are only multiplying the linear transformation parts of the affine transformations, i.e.,
we ignore the shift vectors. The shift vector of the final transformation will be determined after
the mulitplication.
Shrink in x-direction to 70% size, shrink in y-direction to 50% size, then shift up by 0.2 :
Rotate by 45 degrees CCW and shrink to 1/2 size, then shift right by 0.5 and up by 0.2:
Shear in x-direction (xshear):
Shear in y-direction and shrink by 50%:
Reflect across y-axis and shrink by 50%, shift right by 0.7 and shift up by 0.1:
