# R2-D2, Rotations and Dilations in Two Dimensions

`R2_D2` is is the name I've given a new MATLAB program that provides animations of 2-by-2 rotation and dilation matrices. I admit I chose "dilations" so the acronym would be memorable, but otherwise the code has little to do with the famous Star Wars droid.

### Contents

#### House

This outline of a house is featured in *Experiments with MATLAB*. The data are the 11 blue dots. The coordinates of each dot form a 2-by-1 vector; the 2-by-2 rotation and dilation matrices multiply each of these vectors separately. (The lines between the dots complete the picture and are not involved in any computation.)

The house also appears in several editions of Gil Strang's textbooks. The cover of the third edition of Strang's *Introduction to Linear Algebra* features nine houses on a quilt made by Gil's friend Chris Curtis.

#### Hand

This outline of a hand, which I made from measurements of my own hand, is also used in *Experiments with MATLAB*. There are 37 points, so `Hand` is a 2-by-37 matrix.

#### Rotation

Multiplication by this matrix produces a two-dimensional rotation by an angle `theta`.

$$ R = \left( \begin{array}{rr} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{array} \right) $$

`R` is displayed in the first panel. If the house is not rotating in your browser, try this link: https://blogs.mathworks.com/cleve/files/house_rotate.gif

#### Dilation

Dilation is the process of making objects larger or smaller. Multiplication by this diagonal matrix produces a dilation by a factor `sigma`.

$$ S = \left( \begin{array}{rr} \sigma & 0 \\ 0 & \sigma \end{array} \right) $$

`S` is displayed in the second panel. The animation is also available at: https://blogs.mathworks.com/cleve/files/hand_dilate.gif

#### Both

Here rotation and dilation are combined. The product of the rotation and dilation matrices drives the action. If you are missing all the action, try: https://blogs.mathworks.com/cleve/files/hand_both.gif

`R2_D2`

When you run `R2_D2` on your own computer, you can drive the rotations and dilations yourself. Mousing outside of the object creates rotation and mousing inside produces dilation.

The `R2_D2` program is available from https://blogs.mathworks.com/cleve/files/R2_D2_14.m

#### Further Reading

If you are not familiar with matrices, or just want a quick refresher, check out the Matrices chapter of *Experiments with MATLAB*. Exercise 4.14 is particularly handy.

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