Geometry

Composition of Transformations

Big Picture

When transformations are combined, the resulting transformation is a composition of transformations. Having a thorough understanding of the individual transformations is key to dissecting compositions of transformations. Using visual aids also helps a lot on performing composition of transformations. When learning about transformations, picture each step along the way.

Key Terms

Transformation: An operation that moves, flips, or changes a figure to create a new figure.

Rigid: A transformation that preserves size and shape.

Isometry: Another word for rigid transformation, a transformation that does not change the shape or size of a figure.

Composition of Transformations: To perform more than one rigid transformation on a figure

Compositions

Theorem: A composition of two (or more) isometries is an isometry.

Each isometry is a rigid transformation, so after performing several isometries, the figure does not change the shape or size of a figure. We can think of the series of isometries as a single isometry.

Glide Reflections

A glide reflection consists of a reflection and a translation in a direction parallel to the line of reflection.

Glide Reflections

It doesn’t matter whether the reflection or the translation is done first. The composition can also be written as a single rule.

Reflections over Parallel Lines

A composition of reflections over parallel lines is equivalent to a translation.

Reflections over Parallel Lines

Reflections over Parallel Lines Theorem: If there are two reflections over parallel lines that are h units apart, it is the same as a single translation of 2h units

Geometry

Composition of Transformations Cont.

Reflections over the Axes

Reflection over the Axes Theorem: If there are two reflections over each axis, then the final image is a rotation of 180° of the original

Reflections over the Axes

Reflections over Intersecting Lines

Two reflections across two intersecting lines is equivalent to a rotation.

Reflection over Intersecting Lines Theorem: If there are two  reflections over lines that intersect at xº, then the resulting image is a rotation of 2xº where the point of intersection is the center of rotation.

Reflections over Intersecting Lines

Theorem: Any translation or rotation is equivalent to a composition of two reflections.

Notes