Using Matrix Transformations in Works of Visual Arts

Shears: Matrix Transformations with matrices of the form and are called horizontal and vertical shears, respectively. The sign of the value k determines the direction of the shear. The images shown below were generated by applying the shear matrix to a 563 pixel wide by 705 pixel tall digital image of one of Van Gogh's self portraits. This causes a horizontal shear, with the top moving to the right and the bottom moving to the left.

The Original Picture at left. Twenty applications of the horizontal shear matrix at center. One hundred applications of the matrix at right.
The figures below show the effects of a vertical shear with the matrix applied to the same image.

The original picture at left. Twenty applications of the vertical shear matrix at center. One hundred applications of the matrix at right.

Distant-Dependent Rotations: Matrix transformations with a matrix of the form

are called distant-dependent rotations. Multiplication by this matrix causes a counterclockwise rotation by θ degrees immediately about the origin, which grows linearly to an n Normal 0 false false false EN-US X-NONE AR-SA θ-degree rotation when x2+y2=k. The images shown below were generated by repeatedly applying the rotation matrix where rmax and cmax are as defined in the MathematicaTMcode, to a 400 pixel wide by 572 pixel tall digital image of Leonardo daVinci's painting "Mona Lisa."

The original picture at left. One application of the distant-dependent rotation matrix at center. One hundred eighty applications of the matrix at right.

It has often been said that "A picture is worth a thousand words." In our cases illustrated here, this idea could be extended to say "An animation is worth a thousand diagrams." Clearly then, there are many uses for matrices in various domains.

Data retrieved from: http://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=1017&context=math_fac_pub

By Michel Hamoush