Linear Transformations and Basic Computer Graphics

object

square

triangle

letter L

linear transformation

rotation angle

3π

4

θ

π

4

reflection line

x axis

y axis

y = x

y = -x

orthogonal projection line

x axis

y axis

y = x

y = -x

scaling factor

3

4

k

2

shear factor (off-diagonal entry)

horizontally

vertically

1

2

a

12

1

actions

add transformation

rot

ref

proj

scal

shear

remove transformation

all

first

last

order of transformations

no matrix transformation selected

standard matrix of the linear transformation

A = 

1	0
0	1



This Demonstration applies the concept of linear transformation to map graphical objects. It shows the standard matrix describing the given mapping and the image of the transformed object. You can choose a square, a triangle, or the letter L as the object. A list of linear transformations is available to apply to each of these graphical objects: rotation about the origin through an angle

θ

, reflection in a line, orthogonal projection onto a line, scaling with a given

k

factor, and a horizontal or vertical shear.

Choose up to three different actions with the "add transformation" buttons. This shows the ordered list of transformations and the corresponding image of the composite linear transformation. From there you can modify the values of the parameters with the sliders, or you can clear all or one of the actions with one of the "remove transformation" buttons.