Glide Reflections

You are enjoying yourself on a beach and taking a stroll alongside the shore. Suddenly, you noticed your footprints on the sand and the patterns they left behind. Your mathematical brain starts to analyze it and notice some Symmetry and mirror-like images in the footprints. Is there some possible logic hidden in this or just some coincidence?

The prints and an excellent example of a transformation called glide reflections. This article will discuss glide reflections and learn how to glide and reflect objects.

Glide reflection meaning

The glide reflection meaning is actually in its name. Glide reflection has a glide and the reflect effect when applied to any image.

There are only two pieces of information one needs to know when performing glide reflection operations: the translation rule and the line to reflect your figure over. A simple example of how this works is demonstrated in the figure below.

Fig. 1. Glide reflection on footsteps.

Glide reflection pattern

A glide reflection is a Symmetry that follows a pattern of transformations. The glide reflection pattern consists of two transformations - Reflection over a line and translation along the taken line. Hence, there is a reflection of any figure and translation (or glide) of that figure.

This Composition is commutative, so it doesn't matter whether an image reflects and then glides or vice versa. Also, the shape and size remain the same after the glide reflection composition.

In the below example of the footsteps image, we can see both the possible glide reflection patterns.

Fig. 2. Glide reflection patterns.

The resulting image in both the case of composition - \(T\circ r\) or \(r\circ T\) will always be the same. In short, in glide reflection, you can either see first reflection and then translation or vice versa.

Glide reflection formula

As mentioned earlier, glide reflection involves the process of mapping the figure \(P\) to \( {P}''\). However, this process takes two steps:

• Translating figure \(P\rightarrow {P}'\).
• Reflecting figure \( {P}'\rightarrow {P}''\) .

The glide reflection formula is the Composition sequence of translation and reflection transformation.

\[\Rightarrow \text{Glide reflection} = T\circ r \; \text{or}\; r\circ T\]

Translation formula

When performing the translation it uses the following translation formulas:

• Translating positively on the x-axis would shift the image to the right whilst translating negatively on the x-axis would shift the image to the left. \[\text{Positive translation}: (x,y) \rightarrow (x+h,y)\] \[\text{Negative translation}: (x,y) \rightarrow (x-h,y)\]
• Translating positively in the y-axis would shift the image to the image upward whilst translating negatively in the y-axis would shift the image downward. \[\text{Positive translation}: (x,y) \rightarrow (x,y+k)\] \[\text{Negative translation}: (x,y) \rightarrow (x,y-k)\]
• Combined translation occurs by shifting the x-axis and y-axis simultaneously. \[\text{Positive x,y-axis translation}: (x,y) \rightarrow (x+h,y+k)\] \[\text{Negative x,y-axis translation}: (x,y) \rightarrow (x-h,y-k)\] \[\text{Positive x-axis and Negative y-axis translation}: (x,y) \rightarrow (x+h,y-k)\] \[\text{Negative x-axis and Positive y-axis translation}: (x,y) \rightarrow (x-h,y+k)\]

What are the rules for reflection?

• Reflection over the \(x-axis\) : \((x,y)\) images \((x,-y)\).
• Reflection over the \(y-axis\) -axis : \((x,y)\) images \((-x,y)\).
• Reflection over \(y=x\) : \((x,y)\) images \((y,x)\).
• Reflection over \(y=-x\) : \((x,y)\) images \((-y,-x)\).

Glide reflection symmetry with example

Let us understand how to perform glide reflection symmetry with an example. Suppose we have a pre-image of \(\bigtriangleup XYZ\) with point coordinates \(X(-4,-1), Y(-6,-4), Z(-1,-3)\). This triangle \(\bigtriangleup XYZ\) is positively translated to the right with \(10\) units by forming \(\bigtriangleup X'Y'Z'\). So, this pre-image is shifted to the right, we will add \(10\) units to the x-axis of all the coordinate points.

\[X(-4,-1) \rightarrow X'(-4+10,-1)=X'(6,-1)\]

\[Y(-6,-4) \rightarrow Y'(-6+10,-4)=Y'(4,-4)\]

\[Z(-1,-3) \rightarrow Z'(-1+10,-3)=Z'(9,-3)\]

We can see this translation in the below figure.

Fig. 3. Translation of triangle XYZ.

Now the translated image \(\bigtriangleup X'Y'Z'\) is made reflected over x-axis. That is, we will take the negation of the y-axis coordinates for all the points.

\[X'(6,-1) \rightarrow X''(6,-(-1))=X''(6,1)\]

\[Y'(4,-4) \rightarrow Y''(4,-(-4))=Y''(4,4)\]

\[Z'(9,-3) \rightarrow Z''(9,-(-3))=Z''(9,3)\]

We can see the reflected image \(\bigtriangleup X''Y''Z''\) in the below figure this \(\bigtriangleup X''Y''Z''\) is the resulting image of \(\bigtriangleup XYZ\) by glide reflection.

Fig. 4. Glide reflection of triangle XYZ.

Glide reflection example

As established earlier, glide reflections involve translations and reflections on the same figure. It really does not matter which is done first; reflecting or translation, what matters is that the line of reflection is parallel to the translation. Let us look at a few examples below.

We will look at another example.