Mathematics
Glide Reflections
Glide reflections are a type of transformation in which a figure is first reflected over a line and then translated parallel to that line. This transformation is a combination of a reflection and a translation, and it preserves orientation. Glide reflections are commonly used in geometry and crystallography.
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eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 14.2 Reflections 541 14.2 Reflections Understand and use the concept of a reflection in a line. Understand and use the definition of a glide reflection. Understand and use the definition of a frieze pattern. Reflections Standards Grades 3–5 Geometry Students should draw and identify lines and angles, and classify shapes by properties of their lines and angles. Grades 6–8 Geometry Students should understand congruence and similarity using physical models, transparencies, or geometry software. Definition of Reflection A reflection is a transformation in which a Flip Line of reflection figure flips in a line called the line of reflection. A reflection creates a mirror image of the original figure. Flip Line of reflection The original figure and its image are congruent. EXAMPLE 1 Identifying Reflections Determine whether the blue figure is a reflection of the red figure. a. b. SOLUTION a. The red parallelogram flips to form the blue parallelogram. So, the blue parallelogram is a reflection of the red parallelogram. b. The red arrow cannot be flipped in the line to form the blue arrow. When you flip the red arrow in the line, it will point to the left. So, the blue arrow is not a reflection of the red arrow. It is a translation. EXAMPLE 2 Identifying Reflections in Typesetting Which of the letters of the alphabet are the same when reflected in a horizontal line? SOLUTION For the font shown above, the letters B, C, D, E, H, I, K, O, and X are the same. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
eBook - PDF- Michael Hvidsten(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
By the previous theorem we have that r 3 ◦ r 2 ◦ r 1 is a glide reflection. Finally, suppose that all three meet at a single point P . Then, by Lemma 5.14 we have that r 3 ◦ r 2 ◦ r 1 is a reflection about some line through the common intersection point. � 5.6.1 Glide Reflection Symmetry Definition 5.15. A figure is said to have glide symmetry if the figure is preserved under a glide reflection. Where does glide symmetry ap-pear in nature? You may be sur-prised to discover that your feet are creators of glide symmetric patterns! For example, if you walk in a straight line on a sandy beach, your footprints will create a pattern that is invariant under glide reflection (footsteps created by Preston Nichols). 222 � Exploring Geometry Many plants also exhibit glide symmetry in the alternating struc-ture of leaves or branches on a stem. Exercise 5.6.1. Find examples of two objects in nature that exhibit symme-tries of a glide reflection (other than the ones we have given). Draw sketches of these and illustrate the glide reflection for each object on your sketch. Exercise 5.6.2. Finish the proof of Theorem 5.18; that is, show in the proof that G ( P ) = P � . [Hint: Let P �� be the reflection of P across ← QS . Show that → = Δ QSP and use this to show the result.] Δ P � P �� Q ∼ Exercise 5.6.3. Show that the only invariant line under a glide reflection T AB ◦ r l (with −−→ = (0 � , 0) ) is the line of reflection l . [Hint: If m is invariant, AB then it is also invariant under the glide reflection squared.] Exercise 5.6.4. Show that if a glide reflection has a fixed point, then it is a pure reflection—it is composed of a reflection and a translation by the vector v = (0 , 0) . [Hint: Use a coordinate argument.] Exercise 5.6.5. Show that the composition of a glide reflection with itself is a translation and find the translation vector in terms of the original glide reflection. A group of symmetries is a set of Euclidean isometries that have the following properties: 1.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Probably the most famous example of glide reflection—is a diagram of footprints on the sand (Figure 10.15). How would we transform a left foot print to put it in the position of a right footprint? Does it matter which order we do the transformations in? Translation line Line of reflection Figure 10.15 c. The pentomino was translated in a diagonal direction. d. The pentomino was rotated 180 degrees. 180° We could first translate a left foot print to the right and then reflect it across the line of reflection; then it would be in the location of the right foot print. We could also do these in the 578 CHAPTER 10 Geometry as Transforming Shapes Let me give you an example of a translation followed by a reflection that is not a glide reflection. Figure 10.17 shows a flag being translated to position 2 and then being reflected across a vertical line that is not parallel to the translation vector. The composite transformation of the flag is not a glide reflection, because the translation vector and the reflection line are not parallel. Because they are not parallel, we would not get the same image if we reflected first across the line and then translated up and to the right. opposite order, and it would be the same. What is the relationship between the translation line and the line of reflection? They are parallel, which always has to be the case with a glide reflection. Figure 10.16 shows a flag being translated to position 2 and then being reflected across a horizontal line (which is parallel to the translation vector). The transformation of the flag from position 1 to position 3 is a glide reflection. Look at Figure 10.18. Do you see the Glide Reflections? How would you explain them? In the top row, if we took any “flower” and translated it to the right, and then reflected it across the horizontal line of center, then it would line up with the next “flower.” This is also true for the third row and the bottom row.
eBook - ePub- Michael Hvidsten(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Such isometries will turn out to be equivalent to either a reflection or a glide reflection. Definition 5.14. An isometry that is made up of a reflection and a translation parallel to the line of the reflection is called a glide reflection. A glide reflection is essentially a flip across a line and then a glide (or translate) along that line. If A B → is a vector with T AB translation by this vector and if l is a line parallel to A B → with r l reflection across l, then the glide reflection defined by these isometries is G l, A B = T A B ∘ r l Our first theorem about Glide Reflections says that it doesn’t matter if you glide and then reflect or reflect and then glide. You always end up at the same place. Theorem 5.17. Let l be a line and A B → a vector parallel to l. G l,AB = T AB ○ r l ○ T AB G l, A B − 1 = T B A ∘ r l Figure 5.15 Proof: For the first statement of the theorem, let P be a point not on l. Let P ′ = r l (P) and P ″ = T AB (P) (Figure 5.15). Let G = T AB (r l (P)) and H = r l (T AB (P)). We know that P P ′ ¯ is perpendicular to l. We also know that APP ″ B and AP ′ GB will be parallelograms, by Theorem 5.9. Thus, ∠ P ′ PP ″ and ∠ PP ′ G are right angles, as P P ″ ¯ and P ′ G ¯ are both parallel to l and P P ′ ¯ crosses l at right angles. The angles at P ″ and G in quadrilateral PP ″ GP ′ are also right angles, as translation preserves angles. Thus, PP ″ GP ′ is a rectangle. A similar argument will show that PP ″ HP ′ is also a rectangle and thus G = H, or T AB (r l (P)) = r l (TAB (P)). If P lies on l, then since translation of T AB (P) will still lie on l, we have that r l (TAB (P)) = T AB (P) = T AB (r l (P)). For the second statement we reference one of the earlier exercises of the chapter, which said that if a function h was the composition of f and g (h = f ○ g),. then h −1 = g −1 ○ f −1
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
872 Chapter 16 Geometry Using Transformations Glide reflection determined by directed line segment AB and glide axis l (or T AB followed by M AB ) 829 Clockwise orientation 830 Counterclockwise orientation 830 Translation symmetry 830 Rotation symmetry 830 Reflection symmetry 830 Glide reflection symmetry 832 Escher-type patterns 832 Size transformation S O k , with center O and scale factor k 834 Magnification 835 Dilation 835 Dilitation 835 Similitude 835 Exercises 1. Draw directed line segment AB and P anywhere. Show how to find the image of P determined by T AB . 2. Draw a directed angle of measure a and vertex O and P any- where. Show how to find the image of P determined by R O a , . 3. Draw a line l , Q on l , and P anywhere. Show how to find the images of Q and P determined by M l . 4. Draw a line l , a directed line segment AB parallel to l , and P anywhere. Show how to find the image of P determined by the glide reflection l and AB. 5. Sketch a pattern that has translation symmetry. What must be true about such patterns? 6. Sketch a pattern that has a rotation symmetry of less than 180°. 7. Sketch a pattern that has reflection symmetry. 8. Sketch a pattern that has glide reflection symmetry. What must be true about such patterns? 9. Draw nABC and point O anywhere. Show how to find the image of nABC determined by S O,2 . 10. Using the idea of a similitude, show that any two equilat- eral triangles are similar. Exercises 1. Name the four types of isometries. 2. Given that an isometry maps line segments to line segments and preserves angle measure, show that the isometry maps any triangle to a congruent triangle. 3. Given that an isometry maps lines to lines and preserves angle measure, show that the isometry maps parallel lines to parallel lines. 4. Describe how isometries are used to prove that two tri- angles are congruent.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 588 CHAPTER 10 Geometry as Transforming Shapes Unless otherwise noted, all content on this page is © Cengage Learning are virtually limitless. However, as soon as we replace design by pattern, the designers are limited by the mathematical laws that underlie the formation of patterns. Let us see what this means with one kind of pattern. Now that we have the idea of translation symmetry, we can give a definition of pattern : a figure with translation symmetry. Do you see the connection between this definition and the earlier discussion of characteristics of patterns: having a unit that is repeated in an organized manner? We will begin by looking at the mathematically simplest kinds of patterns, which are often called border or strip patterns because the motif (unit) is repeated in only one direc-tion. With your understanding of translations, reflections, rotations, and Glide Reflections, determine the translation, reflection, rotation, and glide reflection symmetry for each strip pattern below. A. What kind(s) of symmetry does the following strip pattern have? Think before reading on. . . . DISCUSSION This pattern has translation symmetry because we can translate this pattern so that we can place it on top of itself. The translation vector is parallel to the bottom of the page, and the length of the vector is equal to the distance between any two shapes. One way to verify this symmetry is to trace the flags on a blank sheet of paper or on an overhead transparency.
eBook - PDF- Robert Bix(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
The axes of the reflections in g are lines l s and // for all integers s and t, where the lines l s are spaced by x x and the lines // are spaced by x. The trivial Glide Reflections in g are determined by the lines l s and // and Theorem 8.8. The axes of the nontrivial Glide Reflections in g are lines m s and m[ for all integers s and t, where each line m s is parallel to T and lies midway between l s and l s+1 , and each line m' t is parallel to z t and lies midway between // and . The nontrivial glide T' Z z o o z z z ° z z o z o z FIGURE 12.8. FIGURE 12.9. 228 II Transformation Geometry FIGURE 12.10. reflections in g are determined by the lines m s and m' t and Theorem 8.9. The 2-centers of g are the points where the lines l s and // intersect and the points where the lines m s and m' t intersect for all pairs of integers s and /. This type of group is named cmm. Conversely, for any perpendicular translations r and x x , the symmetries of Figure 12.11 are the isometries in the last paragraph, as Figure 12.12 shows. Thus, these isometries form a symmetry group of type cmm (by Theorems 7.16 and 8.6). FIGURE 12.11. FIGURE 12.12. Section 12. Wallpaper Groups with 180° or 120° Rotations 229 MP M_I NIG T' ; j ; { j j ; I i III I i ? i I T i O 0 o 0 o o o o o o o o o o o o o o o o o s * 3 V FIGURE 12.13. Case 5: g contains both reflections and nontrivial Glide Reflections, but it does not contain a reflection and a nontrivial glide reflection that have parallel axes. There are perpendicular translations r and r' such that the translations in g and the identity map are the maps r l r' J for all integers i and j (Figure 12.13). The axes of the reflections in g are lines l s spaced by x' for all integers s. The trivial Glide Reflections in g are determined by the lines 4 and Theorem 8.8. The axes of the nontrivial Glide Reflections in g are lines m t spaced by x for all integers t. The nontrivial Glide Reflections in g are determined by the lines m t and Theorem 8.9.
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