Transformations

12 Key excerpts on "Transformations"

• 

  eBook - ePub

  Linear Algebra

  An Inquiry-Based Approach

  • Jeff Suzuki(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  3

  Transformations

  3.1 Geometric Transformations

  An important concept in mathematics is that of a transformation, where we take some object and alter it to produce a new object. The most familiar type of transformation is a geometric transformation. We can look at these Transformations in two ways:

  • Geometrically: We reflect a point across an axis, or rotate it about a center of rotation, or translate it some distance horizontally and vertically,
  • Algebraically: We take the coordinates of the point

    ( x , y )

    and alter them to produce a new set of coordinates for the point

    (

    x ′

    ,

    y ′

    )

    .

  In either case, we say that the new point is the image of the original under the transformation. Additionally, we might say the original point is the preimage of the transformed point.

  One way to view the transformation is as a set of functions, where the new coordinates

  (

  x ′

  ,

  y ′

  )

  are functions of the old coordinates,

  x ′

  = f ( x , y )

  y ′

  = g ( x , y )

  where

  f ( x , y )

  and

  g ( x , y )

  are some formulas that involve x and y.

  Example 3.1. Consider a translation that shifts every point 3 units to the right. Express the new coordinates

  (

  x ′

  ,

  y ′

  )

  in terms of the old coordinates

  ( x , y )

  .

  Solution. Let’s draw a picture showing how a generic point with coordinates

  ( x , y )

  is affected by this translation:

  Since the point has been shifted 3 units to the right, the new x-coordinate must be 3 more than the original:

  x ′

  = x + 3

  But since the point has not been shifted up or down, the new y-coordinate is the same as the original:

  y ′

  = y

  Activity 3.1: Geometric Transformations

  A3.1.1 For each transformation given, find formulas that compute the new coordinates

  (

  x ′

  ,

  y ′

  )

  from the original coordinate

  ( x , y )

  .

  1. M

     x

     , the reflection of a point across the x

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  Mathematics for Elementary School Teachers

  • Tom Bassarear, Meg Moss(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Figure 10.22 SUMMARY 10.1 We have examined four fundamental ways in which we can transform a geometric figure: translation, reflection, rotation, and glide reflection. We have uncovered connections among these Transformations. For example, performing two specific reflections is equivalent to performing a specific rotation. We have discovered that there are many similarities between numbers and shapes: They have important subsets, we do operations on them, we can make tables for the operations, we can decompose numbers and shapes, and there are important properties that help us better understand our operations on numbers and shapes. These investigations of geometric Transformations of two-dimensional figures in a plane can be extended. What if we examined Transformations of three-dimensional objects? These investigations occupy the attention of many mathematicians and have applications in other fields. For example, the way in which atoms are packed helps to determine the properties of a compound. 1. Look back at Figures 10.1, 10.2, and 10.3. What transforma- tions do you see in each one? 2. Below are a figure and a translation vector. Determine the translation. Explain at least two ways to describe the transla- tion other than providing the vector. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 x y L M I S E P 3. Below are a figure and a translation vector. Without using pen- cil or pen, determine whether the image overlaps the original figure or not. Explain your reasoning. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 x y Exercises 10.1 Section 10.1 Congruence Transformations 583 4. Find the image of the kite reflected across line r . 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 x y r 5. The figure below shows quadrilateral ABCD and a line of reflection. Determine the coordinates of    A , B ,C , and  D using only reasoning (that is, without folding). Explain your reasoning. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 x y r A D C B 6.

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  Transformational Plane Geometry

  • Ronald N. Umble, Zhigang Han(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 4 Translations, Rotations, and Reflections In grades 9–12 all students should apply Transformations and use symmetry to...understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function nota-tion, and matrices . Principles and Standards for School Mathematics National Council of Teachers of Mathematics (2000) Studying geometric Transformations provides opportunities for learners to describe patterns, discover basic features of isometries, make generalizations, and develop spatial competencies . H. Bahadir Yanick, Math. Educator Anadolu University In this chapter we consider three important families of isometries and investigate some of their properties. We motivate each section with an ex-ploratory activity. The instructions for these activities are generic and can be performed using any software package that supports geometric constructions. The Geometer’s Sketchpad commands required by these activities appear in the appendix at the end of the chapter. 4.1 Translations A translation of the plane is a transformation that slides the plane a finite distance in some direction. Exploratory activity 1, which follows below, uses the vector notation in the following definition: 45 46 Transformational Plane Geometry Definition 97 A vector v = a b is a quantity with norm (or magnitude) k v k := √ a 2 + b 2 and direction Θ defined by the equations k v k cos Θ = a and k v k sin Θ = b . The values a and b are called the x -component and y -component of v , respectively. The vector 0 = 0 0 , called the zero vector , has magnitude 0 and arbitrary direction. Given vectors v = a b and w = c d , the dot product is defined to be v · w := ac + bd . Thus v · v = a 2 + b 2 = k v k 2 . If P = x y and Q = x 0 y 0 are points, the vector PQ = x 0 -x y 0 -y is called the position vector from P to Q, and P and Q are called the initial and terminal points of PQ .

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  An Integrated Introduction to Computer Graphics and Geometric Modeling

  • Ronald Goldman(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 4 Af fi ne Transformations for Satan himself is transformed into an angel of light. – 2 Corinthians 11:14 4.1 Transformations Transformations are the lifeblood of geometry. Euclidean geometry is based on rigid motions — translation and rotation — Transformations that preserve distances and angles. Congruent triangles are triangles where corresponding lengths and angles match. Transformations generate geometry. The turtle uses translation (FORWARD), rotation (TURN), and uniform scaling (RESIZE) to generate curves by moving about on the plane. We can also apply translation (SHIFT), rotation (SPIN), and uniform scaling (SCALE) to build new shapes from previously de fi ned turtle programs. These three Transformations — translation, rotation, and uniform scaling — are called conformal Transformations . Conformal Transformations preserve angles, but not distances. Similar triangles are triangles where corresponding angles agree, but the lengths of corresponding sides are scaled. The ability to scale is what allows the turtle to generate self-similar fractals like the Sierpinski gasket. In Computer Graphics, Transformations are employed to position, orient, and scale objects as well as to model shape. Much of elementary Computational Geometry and Computer Graphics is based upon an understanding of the effects of different fundamental Transformations. The Transformations that appear most often in two-dimensional Computer Graphics are the af fi ne Transformations. Af fi ne Transformations are composites of four basic types of Transformations: translation, rotation, scaling (uniform and nonuniform), and shear. Af fi ne Transformations do not necessarily preserve either distances or angles, but af fi ne Transformations map straight lines to straight lines and af fi ne Transformations preserve ratios of distances along straight lines (see Figure 4.1). For example, af fi ne Transformations map midpoints to midpoints.

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Figure 16.1 Notice that when we fold the kite over BD, we are actually forming a one-to-one correspondence between the points of the kite. For example, points A and C cor- respond to each other, points along segments AB and CB correspond, and points along segments AD and CD correspond. In this chapter we will investigate correspon- dences between points of the plane. A transformation is a one-to-one correspondence between points in the plane such that each point P is associated with a unique point ′ P , called the image of P. Transformations that preserve the size and shape of geometric figures are called isometries (iso means “same” and metry means “measure”) or rigid motions. In the remainder of this subsection, we’ll study the various types of isometries. Translations Consider the following transformation that acts like a “slide.” NCTM Standard All students should recognize and apply slides, flips, and turns. Children’s Literature www.wiley.com/college/musser See “A Cloak for the Dreamer” by Aileen Friedman. Reflection from Research Learning about Transformations in a computer-based environment significantly increases students‘ two-dimensional visualization (Dixon, 1995). Common Core – Grade 8 Verify experimentally the proper- ties of translations that lines are taken to lines, and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are take to parallel lines. In each of the following three tilings, describe the type of motion required to move Figure 1 to Figure 2 which will move points A to ′ A , B to ′ B , C to ′ C , and D to ′ D . A 1 B C D A9 2 D9 C9 B9 A 1 B C D A9 2 B9 C9 D9 A 1 B C D A9 2 B9 C9 D9 Transformations 824 Chapter 16 Geometry Using Transformations The sliding motion of Example 16.1 can be described by specifying the distance and direction of the slide. The arrow from A to ′ A in Figure 16.3 conveys this infor- mation.

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  Geometry Revisited

  • H. S. M. Coxeter, S. L. Greitzer(Authors)
  • 1967(Publication Date)
  • American Mathematical Society

    (Publisher)

  C H A P T E R 4 Transformations By faith E n d was translated that he should not see death; and was not found, because God had translated him: for before his translation he had this testimony, that he pleased God. Hcbmus, 11 :5 In a remark at the end of Section 1.6, we obtained the right angle between F D and OB (Figure 1.6A) by rotating the perpendicular lines H D and CB through equal angles a about D and B, respectively. In the preamble to Theorem 1.71, we observed that the two similar triangles ABC and A'B'C' have the same centroid and that, since their orthocenters are H and 0, A H = 20A'. Finally, in the remark after Theorem 1.81, we used a half-turn to interchange the orthocenters of the two congruent triangles A'B'C' and KLM. The rotation, dilatation, and half-turn are three instances of a transjmation which (for our present purposes) means a mapping of the whole plane onto itself so that every point P has a unique image PI, and every point Q' has a unique prototype Q. This idea of a mapping figures prominently in most branches of mathematics; for instance, when we write y = f(x) we are mapping the set of values of x on the set of corresponding values Euclidean geometry is only one of many geometries, each having its own primitive concepts, axioms, and theorems. Felix Klein, in his inaugural address at Erlangen in 1872, proposed the classification of geometries according to the groups of Transformations that can be applied without changing these concepts, axioms, and theorems. In particular, Euclidean geometry is characterized by the group of simi- larities; these are angle-preserving Transformations. An important special case of a similarity is an isometry. This is a Zength-preserving transforma- tion such as a rotation or, in particular, a half-turn. Isometries are at the bottom of the familiar idea of congruence: two figures are congruent if and only if one can be transformed into the other by an isometry. 80 of y.

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  Mathematical Tools for Applied Multivariate Analysis

  • Paul E. Green(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  In the case of basis vector changes, these kinds of Transformations require the use of matrix inverses; hence, the basic ideas of matrix inversion are introduced at this point. The next major section of the chapter is devoted to the geometric representation of various types of matrix Transformations, such as rotations, stretches, central dilations and reflections. The geometric character of combinations of various matrix Transformations is also illustrated so that the reader can see how simple geometric changes, when taken in combination, lead to complex representations. The remainder of the chapter focuses on the solution of simultaneous equations and the central roles that matrix inversion and matrix rank play in this activity. In particular, we discuss the solution of linear equations in multivariate analysis and, in the process, tie in the present topic with material presented in earlier chapters on determinants and the pivotal method for solving sets of linear equations. 127 128 4. LINEAR Transformations FROM A GEOMETRIC VIEWPOINT 4.2 SIMULTANEOUS EQUATIONS AND MATRIX Transformations The concept of a function or mapping is fundamental to all mathematics. By a mapping we mean an operation by which elements of one set of mathematical entities are transformed into elements of another. In scalar algebra we recall that functions like the following are often employed: y=f(x) = bx y=f(x) = e x y=f(x) = ax b ; y=f(x) = ab x For example, for a specific value of x, and a value for the parameter b, we can find a value of y from the linear equation y = bx. The possible values that x can assume are called the domain of the function. The possible values that y can assume are called the range of the function. In scalar algebra our interest centers on the description of rules (i.e., the functions) by which pairs of numbers are related. In vector algebra we are interested in the rules by which pairs of vectors or points are related.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  EXAMPLE 1 Identifying Transformations in Art What types of Transformations are shown in the tessellation? Explain your reasoning. SOLUTION There are translations and rotations. Rotation Translation There are no reflections or dilations in this tessellation. For the tessellation shown in Example 1, you can convince your students that there are no dilations because all of the lizards are congruent. To convince your students that there are no reflections, make a copy of the print, cut out one of the lizards, and reflect it. The reflected image does not occur in the print. Reflection Classroom Tip Copyright 2014 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. 532 Chapter 14 Transformations Translations Definition of Translation A translation is a transformation in which a figure slides but does not turn. Every point of the figure moves the same distance and in the same direction. Slide Slide The original figure and its image are congruent. EXAMPLE 2 Identifying Translations Determine whether the blue figure is a translation of the red figure. a. b. SOLUTION a. The red trapezoid slides to form the blue trapezoid. So, the blue trapezoid is a translation of the red trapezoid. b. The red 3 turns to form the blue 3. So, the blue 3 is not a translation of the red 3. EXAMPLE 3 Identifying Translations in Real Life You rearrange the icons on your smartphone. Do the icons represent examples of translations? Explain your reasoning. Original Rearranged SOLUTION. Except for the message icon ( ) , all of the icons slide without changing their size or shape.

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  College Geometry with GeoGebra

  • Barbara E. Reynolds, William E. Fenton(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  C H A P T E R E I G H T Transformational Geometry F rom your work in previous courses, you are familiar with functions whose inputs and outputs are real numbers. In this chapter, you will be working with functions that use points of the plane as inputs and outputs. If a set of points forms a geometric figure in the plane, the functions we will study in this chapter will transform that figure in various ways. A figure may simply be moved from one place to another, or it may be deformed in some way—perhaps stretched or twisted. Of particular interest are Transformations of the plane that do not alter the distances between points. These special functions, called isometries, are the main topic of this chapter. We introduce the four basic types of isometry and prove that every isometry is one of these basic types. We show how isometries can be used to study congruence of geometric figures. As functions, isometries can be combined by composition. This leads to notions of closure, identity, inverse—and eventually to the important concept of group. 183 8.1 A C T I V I T I E S Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate. 1. Consider the function f 1 (x, y) = (x, 1 y ). This function takes points in the coordinate plane as inputs and returns points in the plane as outputs. Let us examine the effect this function has on its input. Carry out the following steps: • Create an xy-coordinate system. • Construct a circle of radius 2 centered at the origin. • Construct a point P on this circle. • Use the Input bar to plot the new point P ′ = (x(P), 1/y(P)). • Drag the original point P around the circle and watch what happens to the new point P ′ . It may be helpful to set Trace On for P ′ as you drag P. To turn the trace on, right-click on point P (in the Graphics pane) or on its description (in the Algebra pane) and select the Trace On option.

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  SuperFractals

  • Michael Fielding Barnsley(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  96 Transformations of points, sets, pictures and measures Figure 2.3 This picture is invariant under a familiar type of transformation on R 2 , a reflection. This invariance partly defines the picture. Figure 2.4 This picture shows a mathematically perfect reflection. But photographs of real swans on real water are not exactly invariant under reflection. 2.2 Transformations of pictures 97 Figure 2.5 Both pictures here are invariant under the rotation transformation R 36 ◦ . The left-hand picture also represents an invariant set. There are many instances of sets and pictures that are invariant under trans- formations. In graphic design and art the Transformations under which a picture is invariant may be referred to as its symmetries. Wallpaper pictures, pictures of flowers and architectural motifs may be invariant under translational and/or rotational Transformations. As a more complicated example, Figure 2.6 illustrates a set S ⊂ R 2 that is invariant under the M¨ obius transformation (Section 2.6) M =  M ρ ◦ R θ ◦  M ρ , where  M ρ : C → C is defined by  M ρ (z ) = ρ z 1 + (ρ − 1)z (2.2.1) for values of ρ > 1. This transformation obeys  M ρ (0) = 0 and  M ρ (1) = 1. The visible part of the invariant set is S ∩ {(x , y ) ∈ R 2 : −2 ≤ x , y ≤ 2}. R θ denotes a rotation through angle θ about the origin. An even more complicated example of an invariant picture is illustrated in Figure 2.7. In this case the transformation f :  → , where  = {(x , y ) ∈ R 2 : 0 ≤ x , y < 1}, is defined by f (x , y ) =  ( 1 2 x , 2 y − 1 ) when 1 2 ≤ y < 1, ( 1 2 x + 1 2 , 2 y ) when 0 ≤ y < 1 2 . (2.2.2) Elaborate sets and pictures that are invariant under simple Transformations, those whose formulas may be described explicitly in a succinct manner involving less than say sixteen free parameters, can be produced in various ways.

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  Mathematics of Shape Description

  A Morphological Approach to Image Processing and Computer Graphics

  • Koichiro Deguchi, Pijush K. Ghosh(Authors)
  • 2009(Publication Date)
  • Wiley

    (Publisher)

  In the mathematical literature, congruent Transformations are also called isometries . A geometric transformation α is an isometry if | p q | = | pq | for all points p and q in the plane, where p = α ( p ) and q = α ( q ); by the term | p q | = | pq | , we mean that the distance from p to q is equal to the distance from p to q . This means that “isometry” is a geometric transformation that “preserves distance.” The term comes from the Greek words “ isos ” (equal) and “ metron ” (measure). The set of all isometries on the plane is generally denoted by the symbol I . The set I contains the identity transformation, all translations, all rotations, all reflections, and all glide reflections on the plane (Figure 3.2). If there is an isometry that transforms one figure into another figure, we say that these two figures are congruent. This is why isometry is also known as congruent transformation. It is easy to see that ( I , ◦ ) is an algebraic system, since the composition of any two isometries is another isometry. In our subsequent discussions, we shall come across many more algebraic systems with internal composition laws. 3.1.2 Algebraic Systems with External Composition Laws Example 3.4. Let R + be the set of all positive real numbers and let N be the set of all natural numbers. Consider the mapping of N × R + that assigns to every couple ( n, r ) the positive real number r n , where n ∈ N and r ∈ R + . This is an external composition law for R + , commonly known as exponentiation . Example 3.5. Let C be the set of all complex numbers of the form a + ib , where a, b ∈ R , the set of real numbers. Consider the mapping of R × C into C and assign to every couple ( r, c ) 82 Mathematics of Shape Description 1 2 3 4 m–2 m–1 m tick tick tick tick tic k tick tick Figure 3.3 A clock algebra the complex number c = ra + irb , where a + ib = c ∈ C .

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  Backgrounds Of Arithmetic And Geometry: An Introduction

  An Introduction

  • Dan Branzei, Radu Miron(Authors)
  • 1995(Publication Date)
  • World Scientific

    (Publisher)

  CHAPTER VII GEOMETRICAL Transformations §1. Generalities Given the sets 171, IX, we call (geometrical) transformation of 171 into fl a map y ; ffi -» fi-(jj e presence of the qualificative in bracket is justified only when certain geometrical structures are indicated or implied on the two given sets and T satisfies certain conditions referring to these structures. When there is one more transformation V : m ' -W , in the hypothesis that It C 171', we can consider the transformation 7 o T : 17 1 -H ' called composition of Transformations T and T defined for any M G 171 by (7' o T)(M ) = T' {T{M . ). The partial operation of compounding the (geometrical) Transformations is associative, that is Transformations T o ( V o T) and and coincide (The coincidencfi of two Transformations returns to the coincidence of domains, ranges and correspondences.) For the map T considered above, in the supplementary hypothesis It C 171, composition To T more often noted with T 2 can be defined. The more general symbol T is defined for any non-null natural number n, recoursively, by 7' = T and 7 *1 = j o T. If n is the smallest non-null natural number for which V = T holds it is said that transformation T is involutive of n order; when the order is not specified we shall take n = 2.

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