Have you ever purchased a notebook that looked like the following? That notebook is called a composition notebook. The word composition means that you’re putting a bunch of things together into that single notebook - hence composition. Those of you familiar with music may recognize the word composition. Or, you may have been asked to write a composition in your English class! All of these compositions talk about putting together a bunch of things.
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Compositions
In Geometry, a composition of transformations is when we put together two or more transformations, one after another, into a single transformation. This will now bring in new notation! For instance, we are familiar with the image of point \(A\) is called \(A’\) (which we read as A ' prime'). But the image of \(A’\) will be called \(A''\) (which we read as A ‘double prime’), and the image of \(A”\) will be \(A'''\) (which we read as A ‘triple prime’). Let's take a look an example and a non-example below of a composition of transformations below.
Example of a composition:
The following is an example of a composition of a reflection, followed by a rotation, followed by a translation. Play around with the points, reflection line, center of rotation, and the vector.
In Geometry, a composition of transformations is when we put together two or more transformations, one after another, into a single transformation. This will now bring in new notation! For instance, we are familiar with the image of point \(A\) is called \(A’\) (which we read as A ' prime'). But the image of \(A’\) will be called \(A''\) (which we read as A ‘double prime’), and the image of \(A”\) will be \(A'''\) (which we read as A ‘triple prime’). Let's take a look an example and a non-example below of a composition of transformations below.
Example of a composition:
The following is an example of a composition of a reflection, followed by a rotation, followed by a translation. Play around with the points, reflection line, center of rotation, and the vector.
Notice that the pre-image for the above example originally starts with \(\triangle ABC\). The image after the reflection, \(\triangle A'B'C'\), then becomes the pre-image for the rotation. The image \(\triangle A''B''C''\), then becomes the pre-image of the translation to the image \(\triangle A'''B'''C'''\). This is what makes this an example of a composition!
Non-example of a composition:
Non-example of a composition:
Notice that \(\triangle A''B''C''\) after the rotation does not become the pre-image for the for the above non-example keeps going back to \(\triangle ABC\) as its pre-image. This is why the above it NOT an example of a composition, though it is an example of several individual transformations.
GLIDE REFLECTIONS
A glide reflection is a specific composition of transformations that is a reflection and a translation with the translation vector parallel to the line of reflection.
Create a vector on the reflection line. Go to the Transform Menu and Define a Reflection with the grey line as the reflection line. Define a Translation with the vector as the Point from and Point to. Select the Figure and Apply each transformation. You have just created a glide reflection.
GLIDE REFLECTIONS
A glide reflection is a specific composition of transformations that is a reflection and a translation with the translation vector parallel to the line of reflection.
Create a vector on the reflection line. Go to the Transform Menu and Define a Reflection with the grey line as the reflection line. Define a Translation with the vector as the Point from and Point to. Select the Figure and Apply each transformation. You have just created a glide reflection.
Example 1:
Complete the following translation followed by a reflection for : \((x, y)\rightarrow (x - 2, y - 3)\), line of reflection: \(y = -x\).
Complete the following translation followed by a reflection for : \((x, y)\rightarrow (x - 2, y - 3)\), line of reflection: \(y = -x\).
Solution:
Let's watch the video for the solution.
Let's watch the video for the solution.
SPECIAL COMPOSITIONS:
Certain compositions create another type of transformation. Use Desmos to create a reflection followed by another reflection. Go to the Transform menu. Select \(\overline{AB}\). Apply Reflection #2. Then Apply Reflection #1. Go to the Construct menu and label the first reflection \(\overline{A'B'}\) and the second reflection image \(\overline{A''B''}\). Now measure \(\angle CDE\) and \(\angle ADA''\). What do you notice about the two angle measures?
Certain compositions create another type of transformation. Use Desmos to create a reflection followed by another reflection. Go to the Transform menu. Select \(\overline{AB}\). Apply Reflection #2. Then Apply Reflection #1. Go to the Construct menu and label the first reflection \(\overline{A'B'}\) and the second reflection image \(\overline{A''B''}\). Now measure \(\angle CDE\) and \(\angle ADA''\). What do you notice about the two angle measures?
Notice, if you use \(\overline{AB}\) as the pre-image and \(\overline{A''B''}\) as the image, you will notice that it creates a rotation!
SPECIAL COMPOSITIONS: Double reflection over intersecting lines
A double reflection over intersecting lines creates a rotation. The angle of rotation is double the angle formed between the two lines of reflection.
A double reflection over intersecting lines creates a rotation. The angle of rotation is double the angle formed between the two lines of reflection.
Example 2: The following is a double reflection, first in the line \(g\) followed by the line \(h\). Given the diagram below, what is the angle of rotation from \(A\) to \(A''\) about point \(C\).
Solution: Since the above represents a double reflection into intersecting lines, the angle of rotation is double that of the angle between the lines:
Angle of rotation \(= 2\cdot40^{\circ} = 80^{\circ}\)
Angle of rotation \(= 2\cdot40^{\circ} = 80^{\circ}\)
SPECIAL COMPOSITIONS: Double reflection over parallel lines
A double reflection over parallel lines creates a translation. The distance between the pre-image and the double reflection is double that of the distance between the parallel lines.
A double reflection over parallel lines creates a translation. The distance between the pre-image and the double reflection is double that of the distance between the parallel lines.
Example 3: The following is a double reflection, first in the line \(g\) followed by the line \(h\). Given the diagram below, what is the distance between \(A\) & \(A''\) and \(B\) & \(B''\)s?
Solution: Since the distance between lines \(g\) and \(h\) is \(2.6\) units, \(AA'' = BB'' = 2\cdot2.6 = 5.2\) units.
Quick Check
1) For \(\triangle ABC\) below, complete the following compositions:
1) For \(\triangle ABC\) below, complete the following compositions:
a) Reflect \(\triangle ABC\) in the line \(x = 2\), followed by a \(90^{\circ}\) counter-clockwise rotation about point \((1, 0)\).
b) Rotate \(\triangle ABC\) \(90^{\circ}\) counter-clockwise about point \((1, 0)\), followed by a reflection in the line \(x = 2\).
c) Based on your results from above, are compositions commutative? In other words, if you have a reflection
followed by a rotation, will it be the same if you have the same rotation followed by the same reflection?
2) Given the diagram below, determine the angle between the two lines if the angle of rotation between \(A\) and \(A''\) is \(180^{\circ}\).
b) Rotate \(\triangle ABC\) \(90^{\circ}\) counter-clockwise about point \((1, 0)\), followed by a reflection in the line \(x = 2\).
c) Based on your results from above, are compositions commutative? In other words, if you have a reflection
followed by a rotation, will it be the same if you have the same rotation followed by the same reflection?
2) Given the diagram below, determine the angle between the two lines if the angle of rotation between \(A\) and \(A''\) is \(180^{\circ}\).