Similarity Transformations

10 Key excerpts on "Similarity Transformations"

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  eBook - PDF

  Transformational Plane Geometry

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

    (Publisher)

  Chapter 8 Similarity Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, trans-lations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Common Core State Standards for Mathematics CCSS.MATH.CONTENT.8.G.A.4 In this chapter we consider transformations that magnify or stretch the plane. Such transformations are called “similarities” or “size transformations.” One uses a similarity to relate two similar triangles in much the same way one uses an isometry to relate two congruent triangles. The “stretch” similarities, which linearly expand the plane radially outward from some fixed point, are especially important because they provide an essential component of every non-isometric similarity. Indeed, we shall prove that every similarity is one of the following four distinct similarity types: an isometry, a stretch, a stretch rotation, or a stretch reflection. 8.1 Plane Similarities Recall Definition 93: A similarity of ratio r > 0 is a transformation α : R 2 → R 2 such that if P and Q are points, P 0 = α ( P ) , and Q 0 = α ( Q ) , then P 0 Q 0 = rPQ. Note that if r = 1 , then P 0 Q 0 = PQ and α is an isometry. Furthermore, if α has distinct fixed points P and Q, then P 0 = P, Q 0 = Q, and P 0 Q 0 = P Q, the ratio of similarity r = 1 and α is an isometry. Thus α is the identity or a reflection by Theorem 170. Of course, if α has three non-collinear fixed points, then α = ι by Theorem 135. This proves: Proposition 254 A similarity of ratio 1 is an isometry; a similarity with two or more distinct fixed points is a reflection or the identity; a similarity with three non-collinear fixed points is the identity. 169 170 Transformational Plane Geometry Similarities are bijective by Proposition 95. In fact: Proposition 255 The set S of all similarities is a group with respect to com-positions.

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  Mathematics for Elementary School Teachers

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

    (Publisher)

  567 10 SECTION 10.1 Congruence Transformations SECTION 10.2 Symmetry and Tessellations SECTION 10.3 Similarity Geometry as Transforming Shapes In Chapter 9 we focused on understanding measurement of shapes, and in this chapter, we are going to look at what happens when we transform shapes. This will help us better understand the patterns that emerge when shapes are put together, as in quilts, floor patterns, art, and other situations. Look at the pictures in Figures 10.1 to 10.3. What do you see? What did the artist who created these have to think about? Note your ideas before reading on. Figure 10.1 Azat1976/Shutterstock.com Figure 10.2 Margrit Hirsch/Shutterstock.com Figure 10.3 OKing/Shutterstock.com 568 CHAPTER 10 Geometry as Transforming Shapes n n Transformations Before we get into the mathematical aspect of transformations, what do you think when you hear this word? The interesting examples offered by my students include the transformation of a frog into a prince in fairy tales and the transformation of a caterpillar to a butterfly. You are in the process of transforming from a student to a teacher. Virtually all examples of trans- formation give a sense of movement and change. In mathematics, there are many kinds of transformations. Explorations Manual 10.1, 10.2, 10.3 What do you think? l Where have you heard the words translation, reflection, and rotation in everyday life? l How might you describe the translation, reflection, or rotation of a three-dimensional figure in space? l How are the operations translation, reflection, and rotation like the operations addition, subtraction, multiplication, and division? Congruence Transformations SECTION 10.1 INVESTIGATION 10.1a How Do Things Move? A. Take an object such as your cell phone, a tangram triangle, an index card, and so on, and explore all the different ways you can move that object. Describe each unique type of movement and give it a name that makes sense to you.

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  Geometry and Its Applications

  • Walter Meyer(Author)
  • 2022(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  7 Transformation Geometry III: Similarity, Inversion and Projections

  DOI: 10.1201/9780429198328-7

  Until now, we have studied isometries extensively, but they are not the only geometric transformations which are theoretically interesting and useful. In this chapter, we introduce other types of transformations and their applications in computer graphics, cartography, the theory of mechanical linkages, and computer vision.

  Prerequisites: Sections 1 and 2 rely on some facts about similar triangles (material covered in Section 4 of Chapter 2 ).

  In Section 1 , the composition of transformations (covered in Section 2 of Chapter 4 ) comes up but can be avoided. In any case, only the definition is needed.

  In Sections 3 , 4 , 5 , we undertake our most extensive study of three-dimensional geometry, using vector equations of lines and planes (covered in Sections 1 and 2 of Chapter 5 ), as well as the axioms pertaining to three dimensions (Section 3 in Chapter 1 ).

  Section 1. Central Similarity and Other Similarity Transformations

  We are surrounded by pictures which are not life-size: huge faces on billboards, tiny ones on our drivers' licenses, etc. The process of getting an image blown up or squeezed down can be thought of in terms of the geometric transformation called central similarity.

  Definition: Let C be any fixed point in the plane and r any positive number. The transformation called1 central similarity S

  C,r

  is defined as follows:

  1. The image of C is C itself.
  2. For any point other point P, the image P'—also denoted S
     C,r
     (P)—is the point with the following two properties:
     1. CP' = r CP.
     2. P and P' are on the same side of C on the line
        C P

        ↔
        .

  The point C is called the centre of the transformation and r is called the ratio of similarity.

  It is also possible to define central similarity for a negative value of r: First, take the absolute value of r and proceed as above. Finally, apply rotation of 180° to the resulting point. Figure 7.1 shows central similarities with r = 2 and −2. Unless we say otherwise, we deal exclusively with central similarities with r

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  No longer available |Learn more

  First Course in Euclidean Geometry, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  This criterion means that if a triangle is copied to preserve the shape, then the copy is to scale. ________________________ WORLD TECHNOLOGIES ________________________ • SSS : If the ratio of corresponding sides of two triangles does not depend on the sides chosen, then the triangles are similar. This means that if any triangle copied to scale is also copied in shape. • SAS : if two sides are taken in a triangle, that are proportional to two corres-ponding sides in another triangle, and the angles included between these sides have the same measure, then the triangles are similar. This means that to enlarge a triangle, it is sufficient to copy one angle, and scale just the two sides that form the angle. Similar curves Several other types curves are similar, with all examples of that type being similar to each other. These include: • Parabola • Catenary • Graphs of the logarithm function for different bases • Logarithmic spiral Similarity in Euclidean space One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r , so that for any two points x and y we have where d ( x , y ) is the Euclidean distance from x to y . Two sets are called similar if one is the image of the other under such a similarity. A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry. Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E(n) is a subgroup of the affine group. Viewing the complex plane as a 2-dimensional space over the reals, the 2D Similarity Transformations expressed in terms of the complex plane are f ( z ) = az + b and , and all affine transformations are of the form ( a , b , and c complex).

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  Geometry Transformed

  Euclidean Plane Geometry Based on Rigid Motions

  • James R. King(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 8 Dilations and Similarity In this chapter, we will begin the study of similar figures. Similarity is a relationship among figures much like congruence, except that instead of “same size, same shape,” it formalizes an idea of “enlarged or reduced size, same shape”. As with congruence, certain transformations will be used to define this relation; these transformations are called similitudes. Definition 8.1. A similitude with scale factor ? > 0 is a transformation that scales distances by the factor ? . To be specific, if ? and ? are any points and ? is a similitude, then ‖?(?)?(?)‖ = ?‖??‖ . If ? is a similitude, ? −1 is a similitude with scale factor 1/? . To see ‖? −1 (?)? −1 (?)‖ = (1/?)‖??‖ for all ? and ? , let ? = ? −1 (?) and ? = ? −1 (?) . Then this equation becomes ‖??‖ = (1/?)‖?(?)?(?)‖ , which is true for all ? and ? . By the Dilation Axiom, any dilation 𝒟 ?,? is a similitude with scale factor ? . From its definition, a rigid motion is a similitude with scale factor 1 . Conversely, a similitude with scale factor ? = 1 is an isometry and so by Theorem 4.10 is a rigid motion. If ? and ? are similitudes with scale factors ? and ? , then ?? is a similitude with scale factor ?? , since ‖??(?)??(?)‖ = ?‖?(?)?(?)‖ = ??‖??‖ for any points ? and ? . Similitudes preserve angle measure. This is not part of the definition, but we can prove this as a theorem. Theorem 8.2 (Similitudes and Angles) . Every similitude preserves angle measure. Con-sequently, since angle measures of 0 and 180 are preserved, a similitude maps lines to lines, segments to segments, and rays to rays. Proof. Let ? be a similitude with scale factor ? . For any point ? , 𝒟 ?,1/? is a similitude with scale factor 1/? . Then, as observed above for ?? , the product ? = 𝒟 ?,1/? ? is a similitude with scale factor ?(1/?) = 1 , so it is an isometry and hence is a rigid motion by Theorem 4.10.

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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)

  Describe how the original object and the image appear to be related. Is there a single transformation that would map the original object to its image? If yes, what is it? Analyzing Student Thinking 26. Racquel claims that under an isometry a triangle will have the same area as its image. Is she correct? Explain. 27. Monica asks you this question about isometries and parallel lines: “If a figure has parallel sides, will they still be parallel after the transformation?” How should you respond? Explain. 28. Thomas wants to know: “In a size transformation, would the corresponding line segments of a figure and its image be parallel?” How would you respond? Explain. 29. Kristie asks, “If there are two congruent shapes anywhere on my paper, will I always be able to find an isometry that maps one onto the other, even if one is the flip image of the other, like two mittens where the right-hand mitten is horizontal and the left-hand mitten is vertical?” How would you respond? Explain. 30. Eli can’t see why a size transformation preserves orienta- tion, but a similitude does not. How could you help him understand this concept? 31. Under a similitude, all pairs of corresponding sides between a triangle and its image are in the same propor- tion. Petra asks why this is not listed as a property for isometries. How should you respond? 32. Hector says that there are four isometries. Candace disagrees and says that there are really only three. Which student is correct or are they both correct? Explain. Section 16.3 Geometric Problem Solving Using Transformations 863 Using Transformations to Solve Problems The use of transformations provides an alternative approach to geometry and gives us additional problem-solving techniques. Our first example is a transformational proof of a familiar property of isosceles triangles. You and your friends are shooting pool on the pool table as shown when you are confronted with three collinear balls.

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  Methods of Geometry

  • James T. Smith(Author)
  • 2011(Publication Date)
  • Wiley-Interscience

    (Publisher)

  Notice that OVr* O'V; and these triangles aren't similar. The rest of this chapter is a detailed, fairly complete study of those geometric transformations that do preserve distance—called isometries—and those that preserve similarity but not necessarily distance—called similarities. All isometries and similarities are linear, but not vice-versa. Detailed study of linear transformations that aren't isometries hes beyond the scope of this book." 6.2 Isometries Concepts Definition Physical motions and time are not geometric concepts Translations Rotations Reflections The isometry group Noncommutativity Invariance Rigidity theorem Uniqueness theorem Chapter 15 of Martin 1982 is an introduction to that theory in two dimensions. Virtually all the results in Methods of geometry about isometries and similarities have both two- and three-dimensional analogues in the theory of general linear transformations. But those are often cast in projective geometric language, and hard to find in the literature. You could start a search with the research paper Ellers 1979 and its references. 6.2 ISOMETRIES 241 This section introduces plane isometries and some related concepts. It presents common examples, but gives few details. Those come later, in sections 6.3 to 6.5 and 6.7, which study in depth four types of isometries. This section and the next are arranged to provide a quick route to the classification theorem, which shows that every isometry is one of those four types. The rest of chapter 6 is concerned with points in a single plane . It's always the same plane, so it's never again mentioned explicitly. For example, in the rest of this chapter, point means point in .

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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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  College Geometry

  A Unified Development

  • David C. Kay(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  If k = 1, the mapping preserves distances, and is an isometry . Transformations in Modern Geometry 439 It is easy to see that a similitude maps distinct points to distinct points, thus is one-to-one, and maps any triangle to one that is similar to it by the SSS similarity criterion. Thus, a similitude is angle preserving. It will eventually be shown that given any two similar triangles, a unique simili-tude exists which maps the first triangle onto the second. An analogous result is true for isometries. This shows that similitudes and isometries exist in abundance. By the result of Problem 16 above, the most general similitude has the coordinate form (1) x ax by x y bx ay y a b ʹ ʹ = -+ = + + uni23A7 uni23A8 uni23AA uni23A9 uni23AA + > = ± e e e 0 0 0 2 2 1 , where a and b are constants. (The mapping is direct iff ε = 1.) To find the dilation factor, consider A (0, 0) and B (1, 0) and their images A ′ ( x 0 , y 0 ) and B ′ ( a + x 0 , b + y 0 ). Then A ′ B ′ = kAB = k · 1 implies k = A ′ B ′ , or, by the distance formula, (2) k a b = + 2 2 If k = 1 then (1) is an isometry, so the coordinate form of an isometry is given by (1) , with a 2 + b 2 = 1. Example.1 Consider the transformation f given by ʹ = -+ ʹ = + -uni23A7 uni23A8 uni23A9 x x y y x y 3 4 5 4 3 2 (a) Find the coordinates of the images A ′ , B ′ , and C ′ for the points A (0, 1), B (0, 2), and C (1, 1) and sketch the graph showing Δ ABC and its image Δ A ′ B ′ C ′ . (b) Find the dilation factor k , and show that the lengths of the sides of Δ A ′ B ′ C ′ are k times those of Δ ABC , thus showing that the triangles are similar. Solution (a) Substituting the coordinates of A , B , and C into the equations for f , we find A ′ = (1, 1), B ′ = (−3, 4), and C ′ = (4, 5). The graph is shown in Figure 8.15. (b) From (2) we find k = 3 4 2 2 + = 25 = 5. The dis-tance formula produces A ′ B ′ = ( ) ( ) 1 3 1 4 2 2 + + -= 16 9 + = 5 = 5 AB . Similar calculations show that B ′ C ′ = 5· BC and A ′ C ′ = 5· AC .

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  Handbook of Computer Vision Algorithms in Image Algebra

  • Joseph N. Wilson, Gerhard X. Ritter(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  Geometric transformations change the spatial relationships among the objects in an image or change the geometry of an object in an image. Examples of geometric transformations are affine and perspective transforms. These types of transforms are commonly used in image registration and rectification. For example objects in magnetic resonance imagery (MRI) are displaced because of the warping effects of the field and raster-scanned satellite images of the earth exhibit the phenomenon that adjacent scan lines are offset slightly with respect to one another because the earth rotates as successive lines of an image are recorded. Functions between spatial domains provide the underlying foundation for realizing naturally induced operations for spatial manipulation of image data. In particular, if f : Y + X and a E IfX, then we define the induced image a o f E IfY by Thus, the operation defined by Eq. 1 1.1.1 transforms an If-valued image defined over the space X into an F-valued image defined over the space Y. Figure 11.1.1 provides a visual interpretation of this composition operation. Figure 1 1.1.1. The spatial transform a o f The basic transformations of the plane give rise to the basic geometric image transformations. Basic planar transformations are translations, rotations, reflections, con- tractions, and expansions. In the image domain, these correspond to shifting an image, rotating an image, reflecting an image across a straight line, and shrinking or magnifying an image. In this section we restrict our attention to the basic operations of image reflec- tion and magnification. These operations are particularly easy if integral grid points map to integral grid points and lines of reflections are vertical or horizontal; i.e., of form y = k or X = k , where k is an integer. In case of magnification, we have X C Y C Z2, and f : U -+ X.

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