Image Transformation is a learning aid that enables you see the effect of using Linear Transformations to transform graphical images. It relies on two key ideas -- an original image and a transformed image. The image above is an original image. The coordinates of this original image are denoted by the variables x and y. The range of the variable x is from -2 to 2 and the range of the variable y also runs from -2 to 2. We use the variables u and v to denote the coordinates of the transformed image. The values of these variables also range from -2 to 2. The simplest use of Image Transformation defines the transformed image by means of a transformation:
defined by a pair of functions u(x, y) and v(x, y). For example, the image below can be turned upside down (i.e. reflected about the x-axis) by using the functions:
The resulting 2X2 matrix T = | 1 0 | since T| x | = | x |
| 0 -1 | | y | | -y |
Another very interesting linear transformation is a Rotation about the origin. The drawing below demonstarates what happens when the vector determined by the point (x,y), which forms an angle of alpha with the positive x-axis, is rotated tghrough an angle of beta to a new location at the point (u,v), whose vector now forms an angle of alpha + beta with the positive x-axis. If the corresponding vector for each point in the picture is subjected to the same rotational transformation, the entire picture will will be rotated counter clockwise through an angle of beta.
Using the form below you can experiment with transformations like this. Simply enter two expressions defining the functions u(x, y) and v(x, y) in the appropriate boxes and press the Try it!! button. If you click the Try it!! button without changing the default expressions, the image will be turned upside down (i.e. reflected about the x-axis).