How Do We Use Other Definitions of Transformations?

There are many types of transformations in Geometry. Some of them are Glide Reflections, Orientation, and Isometry, Direct Isometry and Invariant.

• Glide Reflection is the combination of a reflection in a line and a translation along that line

• Orientation is the arrangement of points relative to one another after a transformation has occurred.

• Isometry is a type of transformation of the plane that preserves length. For Example: a transformation that preserves everything except dilation.

• Direct Isometry preserves orientation or order - the letters on the diagram go in  the same clockwise or counterclockwise direction on the figure.

• Invariant is a figure that remains unchanged under a transformation of the plane. No variations have occurred.

How Do We Solve Composition of Transformation Problem?

Knowing the rules of doing the second transformation first and the first transformation second, we are set to solve compositions of transformations.

Rules:


Examples:

 is the image of   under a glide reflection
that is a composition of a reflection over the line l
and a translation through the vector v.

Examples:

*Images and and captions were provided by Google Image

How Do We Identify Composition of Transformation?

We can identify composition of transformation in one way, which for my opinion is the easiest one.

• When two or more transformations are combined to form a new transformation is called the composition of transformations.

For Example:

1.  R(x-axis)  º T(3,4)

Rules:

• Do the coordinates with the "T" first and then do the "R" second.

How Do We Identify Transformations?

A Transformation is when you move a geometric figure. There's four types of transformations:

• Translation
• Rotation
• Reflection
• Dilation

A Translation is when every point is moved the same distance in the same direction.

In this example, every point in the star has moved the same distance in the same distance

A  Rotation is when a figure is turned around a single point (See "How Do We Graph Rotations?" for more info)

In this picture, point A has been rotated 180 degrees.

A Reflection is when a figure is flipped over a line of symmetry.

In this picture, the line of symmetry is X-axis and the mountains are been reflected  on the lake.

A Dilation is an enlargement or reduction in size of an image.

In this example, triangle ABC has been enlarged.

HOW DO WE GRAPH ROTATIONS?

There's 3 steps for you to graph rotations:

• Find the angle of rotation
• Find the direction (clockwise or counterclockwise)
• Find the center of rotation 

A rotation is a type of transformation in which you rotate the figure. This is mostly use in Geometry. You may think rotation is hard, but it really isn't. If you know the rules of rotation you'll be great in a test and/or homework the teacher gives you. The rules are:

• For 90 degrees:

                                  (A, B) -> (-B, A)

• For 180 degrees:

                                    (A, B) -> (-A, -B)

• For 270 degrees:

                                    (A, B) -> (B, -A)

As you can see, you only have to change the signs and/or switch the positions of the coordinates. Its pretty simple.

In the figure above,  the triangle P has been moved 90 degrees counterclockwise.

In this figure, it shows the Triangle in the grid rotating clockwise in every quadrant.