Lecture 72D Transformation

What is a transformation?• Exactly what it says - an operation that

transforms or changes a shape (line, shape, drawing etc.) They are best understood graphically first.

• There are several basic ways you can change a shape:

• Translation (moving it somewhere else).• Rotation (turning it round).• Scaling (making it bigger or smaller).

Translation• Essentially, we want to move the shape dx pixels

along the x-axis and dy pixels up the y-axis. In fact all this means is moving each constituent point by dx and dy.

• To move a point in this manner, simply add the values of dx and dy to its existing coordinates. Example 15 shows what a translate() method of the Point2D class might look like.

Translation

Rotation ( about the origin)• To rotate any shape about the origin requires rotating

each of its individual points. To work out how this is done, consider the coordinates of a point before and after the rotation: Both points will lie on the perimeter of a circle of radius r with its center on the origin.

Rotation ( about the origin)

Scaling• Scaling a shape simply means making a it bigger or

smaller. We can specify how much bigger or small be means of a “scale factor” - to double the size of an object we use a scale factor of 2, to half the size of an object we use a scale factor of 0.5.

Scaling• Again, scaling of a shape is achieved by applying an

operation to the individual points that make up the shape. In this case, the distance of a point from the origin changes by the scale factor.

• Simply multiplying the coordinates by the scale factor gives the new values of the coordinates.

Scaling• Note that the scaling can be different in different

directions: i.e. the x scale factor can be different to the y scale factor. By doing this we can stretch or squeeze a shape: Simply multiplying the coordinates by the scale factor gives the new values of the coordinates.

Identity Transformation

Order of Transformation• The order in which transformations are applied to a shape is

important. e.g. performing a translation followed by a rotation, will give an entirely different drawing to a performing the rotation followed by the same translation.

Rotation around local origin

Rotation around local origin

Rotation around local origin

Review

Exercises