Wednesday, February 1, 2017

Matrices

I enjoy linear algebra. I use it in making polyhedra and drawings. Here are a few visualizations of matrices. To keep it simple I use 2x2 matrices. 2x2 matrices act on vectors. The vertices of polygons in a plane can be described vectors, so these same transformations can be done to polygons that dwell in a plane.

Rotation Matrix


Rotating a polygon doesn't change it's area. The area remains the same. The determinant of this matrix is 1.


Proportional Scaling Matrix


Doubling size as well as height boosts a polygon's area by a factor of four. This determinant of this matrix is 4.


Non Proportional Scaling Matrix


This matrix stretches the width to twice the original and squeezes the height to half of what it was. Overall the area is unchanged. The determinant of this matrix is 1.


Shear or Skew Matrix

When I was using Macromedia Freehand, the graphics program called this transformation "skew". Then Adobe ate Macromedia and I was forced to use Adobe Illustrator. Illustrator calls it "shear".


This transformation transforms a horizontally aligned rectangle to a parallelogram with same base and height. Area remains unchanged. The determinant of this matrix is 1.

Flip Matrix


Making the first term in the main diagonal negative flips polygons about the y axis. Making the lower right term negative would flip polygons about the x axis.

Determinant is -1. Not sure what that means geometrically but absolute value of the area remains the same.

Illustrator Tool Box

The above matrices are various forms of tools from the Adobe Illustrator Tool Box. Here's a couple of screen caputres spliced together:


One tool compartment has the reflect and rotate transformations, the neighboring compartment has the scale and shear transformations. I'm not sure what the Reshape Tool is.

Hopefully I will have time and energy to add more to this matrix post soon.