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? 

Glide Reflections Glide Reflections

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Contents
Table of contents

    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.

    A glide reflection is the combination of two transformation methods; translation and reflection, to map a point P to P".

    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.

    Glide reflections, Footstep, StudySmarterFig. 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.

    Glide reflections, glide reflection patterns, StudySmarterFig. 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. \[\text{Positive x,y-axis translation}: (x,y) \rightarrow (x+h,y+k)\] \[\text{Negative x,y-axis translation}: (x,y) \rightarrow (x-h,y-k)\] \[\text{Positive x-axis and Negative y-axis translation}: (x,y) \rightarrow (x+h,y-k)\] \[\text{Negative x-axis and Positive y-axis translation}: (x,y) \rightarrow (x-h,y+k)\]

    All the points on the coordinate system are shifted in the same number of units and directions.

    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.

    Glide reflection, translation example, StudySmarterFig. 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.

    Glide reflection, glide reflection example, StudySmarterFig. 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.

    Given a figure to be \(P(-2,-3), Q(-3,-1), R(-5,-4)\),

    a. Translate \((x,y)\rightarrow (x+8,y)\).

    b. Reflect in the x-axis.

    Solution:

    Let us plot the pre-image.

    Glide reflections, Triangle, StudySmarterFig. 5. Pre-image of triangle PQR.

    Now our figure will be translated into \(8\) units right to give us \(\bigtriangleup P'Q'R'\).

    Finding \(\bigtriangleup P'R'Q'\) means that we will add \(8\) units to the x-axis of each point.

    \[P(-2,-3) \rightarrow P'(-2+8,-3)\]

    \[P'(6,-3)\]

    \[Q(-3,-1) \rightarrow Q'(-3+8,-1)\]

    \[Q'(5,-1)\]

    \[R(-5,-4) \rightarrow R'(-5+8,-4)\]

    \[R'(3,-4)\]

    The translated figure has coordinates \(P'(6,-3), Q'(5,-1), R'(3,-4)\).

    We will now want to reflect the translated figure to give us \(\bigtriangleup P''Q''R''\).

    Reflection over the x-axis means \((x,y)\rightarrow (x,-y)\).

    \[P'(6,-3) \rightarrow P''(6,-(-3))\]

    \[P''(6,3)\]

    \[Q'(5,-1) \rightarrow Q''(5,-(-1))\]

    \[Q''(5,1)\]

    \[R'(3,-4) \rightarrow R''(3,-(-4))\]

    \[R''(3,4)\]

    The coordinates for the glided and reflected figure now are \(P''(6,3), Q''(5,1), R''(3,4)\).

    Glide reflections, Triangle, StudySmarterFig. 6. Image of triangle PQR.

    We will look at another example.

    Given triangle to be \(A(3,1), B(6,-2), C(4,5)\),

    a. Translate \((x,y)\rightarrow (x,y-2)\).

    b. Reflect in the y-axis.

    Solution:

    Plot triangle \(\bigtriangleup ABC\).

    Glide reflections, Triangle, StudySmarterFig. 7. Pre-Image triangle ABC.

    We will translate the traiangle \(\bigtriangleup ABC\) and name it \(\bigtriangleup A'B'C'\). So, we will subtract \(2\) units from the y-axis of each point.

    \[A(3,1) \rightarrow A'(3,1-2)\]

    \[A'(3,-1)\]

    \[B(6,-2) \rightarrow B'(6,-2-2)\]

    \[B'(6,-4)\]

    \[C(4,5) \rightarrow C'(4,5-2)\]

    \[C'(4,3)\]

    The translated figure has coordinates \(A'(3,-1), B'(6,-4), C'(4,3)\).

    We will now want to reflect the translated figure to give us \(\bigtriangleup A''B''C''\).

    Reflection over the y-axis means \((x,y)\rightarrow (-x,y)\).

    \[A'(3,-1) \rightarrow A''(-3,-1)\]

    \[B'(6,-4) \rightarrow B''(-6,-4)\]

    \[C'(4,3) \rightarrow C''(-4,3)\]

    Glide reflections, Triangle, StudySmarterFig. 8. Image of triangle ABC.

    Given a triangle with coordinates to be \(A(-5,3), B(-2,4), C(-1,0)\),

    1. Translate \((x,y) \rightarrow (x+4,y)\)
    2. Reflect over line \(y=x\).

    Solution:

    First plot triangle \(\bigtriangleup ABC\).

    Glide reflections, Triangle, StudySmarterFig. 9. Pre-image of triangle ABC.

    Finding \(\bigtriangleup A'B'C'\) means that we will add \(4\) units to the x component of each point.

    \[(x,y)\rightarrow (x+4,y)\]

    \[A(-5,3) \rightarrow A'(-5+4,3)\]

    \[A'(-1,3)\]

    \[B(-2,4) \rightarrow B'(-2+4,4)\]

    \[B'(2,4)\]

    \[C(-1,0) \rightarrow C'(-1+4,0)\]

    \[C'(3,0)\]

    The translated figure has coordinates \(A'(-1,3), B'(2,4), C(3,0)\). We will now reflect it over line \(y=x\). Reflecting over line \(y=x\) means that \((x,y) \rightarrow (y,x)\).

    \[A'(-1,3) \rightarrow A''(3,-1)\]

    \[B'(2,4) \rightarrow B''(4,2)\]

    \[C'(3,0) \rightarrow C''(0,3)\]

    Glide reflections, Trianggle, StudySmarterFig. 10. Image of triangle ABC.

    Glide Reflections - Key takeaways

    • Glide reflection is the combination of two transformation methods; translation and reflection, to map a point \(P\) to \(P''\).
    • 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.
    • It does not matter whether translation or reflection is done first, all that matters is that the line of reflection is parallel to the translation.
    Frequently Asked Questions about Glide Reflections

    What is a glide reflection?

    Glide reflection is the combination of two transformation methods; translation and reflection, to map a point P to P''. 

    What is an example of a glide reflection? 

    A figure A(3, 1), B(6, -2), C(4, 5) is glided 2 units down and reflected over the y-axis to give figure A''(-3,-1), B''(-6,-4), C''(-4,3).

    What two transformations make up a glide reflection? 

    The two transformations that make up a glide reflection are translations and reflections.

    What is a glide reflection equation? 

    Glide reflections comprise of the translation and reflection rules;

    1) Translation rule

    • Right h units (x, y) → (x + h, y).
    • Left h units (x, y) → (x – h, y).
    • Up k units (x, y) → (x, y + k).
    • Down k units (x, y) → (x, y – k).

    2) A line to reflect over

    • Reflection in x-axis: (x, y) → (x, -y)
    • Reflection in y-axis: (x, y) → (-x, y)
    • Reflection in y = x: (x, y) → (y, x)
    • Reflection in y = -x: (x, y) → (-y, -x).

    What is a glide reflection in geometry?

    A glide reflection is a symmetry operation which is a composition of reflection and translation transformations.

    Test your knowledge with multiple choice flashcards

    Translating positively on the x-axis would shift the image to the image left whilst translating negatively on the x-axis would shift the image to the right.

    Which process needs to be performed first?

    The size and shape of the pre-image are the same as the images.

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