Isometry

11 Key excerpts on "Isometry"

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  Geometry: The Line and the Circle

  • Maureen T. Carroll, Elyn Rykken(Authors)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Definition 15.3. An Isometry , or rigid motion , is a mapping (or transformation), ? , of a plane onto itself that preserves distances, that is, given any two points ? and ? in the plane, we have ?(?, ?) = ?(?(?), ?(?)). Notice that the word “onto” in the definition specifies that an Isometry of a plane must be onto , or surjective . This means that the range of the map ? must be all points of the plane, that is, for any point ? on the plane, there is another point ? so that ?(?) = ? . [As demonstrated at the beginning of this section, Euclidean analytic geometry can be a useful tool for the reader to build examples and counterexamples to test the properties of functions. For example, take a moment to convince yourself that, in the Euclidean plane where points are given by ordered pairs, ?(?) = ? ( (𝑎 1 , 𝑎 2 ) ) = (2𝑎 1 , 2𝑎 2 ) = ? is an onto map but not an Isometry since it does not preserve distance. As another example, show that ?(?) = ? ( (𝑎 1 , 𝑎 2 ) ) = (1 + 𝑎 1 , 2 + 𝑎 2 ) = ? is an Isometry.] Since an Isometry preserves distances, it must be one-to-one , or injective ; that is, given an Isometry, ? , and points ? and ? , if ?(?) = ?(?) , then ? = ? . We leave the 376 Chapter 15 Isometries proof of this fact as an exercise for the reader. [As an example, show that ? as given above is a one-to-one function.] Functions that are both injective and surjective are called bijective , and every bijective function has an inverse. Thus, every Isometry is a bijection, and consequently, has an inverse. We leave the fact that the set of isometries of a plane forms a group under composition as an exercise. The identity element in this group is the identity transformation, and the unique inverse of an Isometry ? is denoted by ? −1 . Isometries do more than just preserve distances, they also preserve our two main characters, the line and the circle, as well as angles and triangles. We start by showing any Isometry is line-preserving.

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  Geometry

  • David A. Brannan, Matthew F. Esplen, Jeremy J. Gray(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  We were able to do this because reflections leave lengths and angles unchanged. Of course, reflections are not the only transformations that preserve lengths and angles: other examples include rotations and translations. In general, any transformation that preserves lengths and angles can be used to tackle problems which involve these properties. In fact, we need worry only about leaving distances unchanged, since any transformation from R 2 onto R 2 that This is because, once we know the lengths of the sides of a triangle, the angles are uniquely determined. changes angles must also change lengths. Transformations that leave distances unchanged are called isometries. Definition An Isometry of R 2 is a function which maps R 2 onto R 2 and preserves distances. In fact, every Isometry has one of the following forms: The identity Isometry can be regarded as a rotation through an angle that is a multiple of 2 π . a translation along a line in R 2 ; a reflection in a line in R 2 ; a rotation about a point in R 2 ; a composite of translations, reflections and rotations in R 2 . The composite of any two isometries is also an Isometry, and so it is easy to verify that the set S ( R 2 ) of isometries of R 2 forms a group under composition of functions. These observations can be used to build up the transformations we need in order to prove Euclidean results. Example 2 Prove that if ABC and DEF are two triangles such that AB = DE , AC = DF and ∠ BAC = ∠ EDF , then BC = EF , ∠ ABC = ∠ DEF and ∠ ACB = ∠ DFE . B A C E D F Solution It is sufficient to show that there is an Isometry which maps ABC onto DEF . We construct this Isometry in stages, starting with the translation 64 2: Affine Geometry which maps A to D . This translation maps ABC onto DB C , where B and Of course, A and D may already coincide, in which case we omit the translation stage.

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  Exploring Geometry

  • Michael Hvidsten(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Thus, we will define transformations as follows: Transformational Geometry � 189 Definition 5.2. A function f on the plane is a transformation of the plane if f is a one-to-one function that is also onto the plane. An Isometry is then a length-preserving transformation. One impor-tant property of any transformation is that it is invertible . Definition 5.3. Let f, g be functions on a set S . We say that g is the inverse of f if f ( g ( s )) = s and g ( f ( s )) = s for all s in S . That is, the composition of g and f ( f and g ) is the identity function on S . We denote the inverse by f − 1 . It is left as an exercise to show that a function that is one-to-one and onto must have a unique inverse. Thus, all transformations have unique inverses. A nice way to classify transformations (isometries) is by the nature of their fixed points. Definition 5.4. Let f be a transformation. P is a fixed point of f if f ( P ) = P . How many fixed points can an Isometry have? Theorem 5.2. If points A, B are fixed by an Isometry f , then the line through A, B is also fixed by f . Proof: We know that f will map the line ←→ f ( A ) f ( B ). AB to the line ←−−−−−→ Since A, B are fixed points, then ←→ AB gets mapped back to itself. Suppose that P is between A and B . Then, since f preserves be-tweenness, we know that f ( P ) will be between A and B . Also AP = f ( A ) f ( P ) = Af ( P ) This implies that P = f ( P ). A similar argument can be used in the case where P lies elsewhere on ←→ AB . � 190 � Exploring Geometry Definition 5.5. The Isometry that fixes all points in the plane will be called the identity and will be denoted as id . Theorem 5.3. An Isometry f having three non-collinear fixed points must be the identity. A B C P Q R Figure 5.2 Proof: Let A, B, C be the three non-collinear fixed points. From the previous theorem we know that f will fix lines ←→ AC , and ← BC → . AB , ←→ Let P be a point not on one of these lines. Let Q be a point between A, B (Figure 5.2).

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  Geometry Revisited

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

    (Publisher)

  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. TRANSLATION 81 4.1 Translation Apart from the identity, which leaves all points just where they were before, the most familiar transformation is the translation, which preserves the distance between any two points and the direction of the line through them. A' 8' Figure 4.iA Figure 4.1B If A'B' is the translated image of a line segment AB, then either A , B, A', B' lie on a line, as in Figure 4.1A, or A A'B'B is a parallelo- gram, as in Figure 4.1B. (In the former case we naturally speak of a degmer& parallelogram A A'B'B. ) Thus the translation is determined by the directed segment AA', or equally well determined by infinitely many other segments, such as BB', having the same distance and di- rection. Another name for a translation is a vector, and we use the nota- tion z' = 5'. In particular, the identity may be regarded as a translation through no distance, or as the zero vector. D C Figure 4.1C The fact that a translation preserves the shape and size of any figure is used in the proofs of various theorems on area. For example (see Figure 4.1C), in deriving the usual formula for the area of a parallelo- gram ABCD with an acute angle at A , we cut off a right-angled triangle AHD and stick it on again after translating it to the position BH'C, thus obtaining a rectangle HH'CD. Figure 4.1D illustrates the problem of inscribing, in a given circle, a rectangle with two opposite sides equal and parallel to a given line seg- ment a. This can be solved by translating the circle along either of the two equal and opposite vectors represented by a.

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  Geometry and Symmetry

  • L. Christine Kinsey, Teresa E. Moore, Efstratios Prassidis(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  a. Construct the point about which the object is rotated. 270 • 8. ISOMETRIES b. Construct the point about which the object is rotated. c. Construct the line about which the object is reflected. d. Construct the line about which the object is reflected. e. Construct the direction and distance of translation. 8.1. TRANSFORMATION GEOMETRY • 271 f. Construct the line of reflection and the distance of translation for this glide reflection. As mentioned above, one example of a function that preserves geometric properties is rotation: let us rotate the plane, including the triangle ABC shown, about the origin by 45 ◦ to get a new triangle A  B  C  : A C B B C A Functions such as the rotation above are called isometries (from the Greek words for “same measure”), or rigid motions. Definition 8.1. An Isometry is a function T defined on the plane so that for any pair of points P and Q, the distance from P to Q is the same as the distance from T(P) to T(Q). I.e., PQ = T(P)T(Q). Theorem 8.2. If T is an Isometry, then T is one-to-one and T takes lines to lines. Proof: To show that a function is one-to-one, we must show that if T (A) = T (B) for points A and B, then A = B. If T (A) = T (B), then T (A)T (B) = 0. Since T is an Isometry, T (A)T (B) = 0 = AB, so A = B. We next show that T takes lines to lines. Let A, B, and C be three points on a line and assume that B lies between A and C, so AC = AB + BC. Since T is an Isometry, T (A)T (C) = T (A)T (B) + T (B)T (C). If T (A), T (B) and T (C) were not collinear, then the triangle inequality (Euclid’s Proposition I.20) would imply that T (A)T (C) < T (A)T (B) + T (B)T (C). Therefore, T (B) must lie on the line determined by T (A) and T (C) and furthermore, must lie between those two points. Thus, T takes all of the points on the line AB to points on the line T (A)T (B). 272 • 8. ISOMETRIES  Exercise 8.1. Let ABC be a triangle and T be an Isometry of the plane, with A  = T(A), B  = T (B), and C  = T (C).

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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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  A Mathematical Bridge

  An Intuitive Journey in Higher Mathematics

  • Stephen Hewson(Author)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  θ y x 1 0 1 0 M = M M a c d b a b d c a c b d 1 1 Area = det( M ) 1 1 M c+d a+b a+b c+d Fig. 4.14 Determinant as a scale factor for volume in 2D. 4.6 Symmetry Length preserving maps, or isometries, can be thought of as symmetries of R n equipped with some scalar product, with which the distances are measured. These are maps for which the distance between two points before and after the transformation are the same. Surfaces described within a particular vector space will possibly inherit some of this symmetry. In general, an object is said to be symmetric, or invariant, under the action of some transformation if it is mapped exactly into itself by the transformation in question. Clearly, the sphere is invariant under any reflection through or rotation about its centre. On the other hand, the circular hyperboloid x 2 + y 2 − z 2 = 1 only has one continuous rotational symmetry, about its fixed axis, although it is still symmetric under reflections in the x − y,x − z and y − z planes. A nice way of thinking 266 A Mathematical Bridge about the symmetry structure of this hyperboloid is that it is exactly the same as that of a circle in the x − y plane, with one additional symmetry corresponding to a reflection in the z = 0 plane. The sphere and the hyperboloid have an infinite number of symmetries, since rotations can be about any angle between 0 and 2 π . Some shapes have only a finite number of symmetries. Let us consider the example of a rigid square in the z = 0 plane. How many isometries preserve its structure? We can think of the square as being defined by its four vertices. For a unit square the distance between adjacent vertices is 1 whereas the distance between opposite vertices is √ 2. If the square is to be transformed exactly onto itself by a length preserving map then opposite points must be mapped onto opposite points (Fig. 4.15). 1 3 4 2 4 4 1 1 2 2 3 3 All lengths preserved Lengths not preserved -- opposite pairs mixed Fig.

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  Methods of Geometry

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

    (Publisher)

  Some axiomatizations of geometry do exactly that. But this idea has more practical consequences, too. You can now easily extend the notion of congruence: Two figures 9 and 3F' are 260 PLANE ISOMETRIES AND SIMILARITIES called congruent—in symbols, & = —if &' =

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  Backgrounds Of Arithmetic And Geometry: An Introduction

  An Introduction

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

    (Publisher)

  Let 9(YYl, be the power set of YYl. It is easy to find out that we can conceive f as transformatior from 9(YYl) to 9(tl) ; f is called the globalized of T. Usually, the symbol whicl appeared in the notation is omitted; we shall also do so when this omission does noi generate confusions. §2. Isometries One calls distance on the set YYl a map d: YYl 2 —> R satisfying the conditions: l.d(A,B) = 0e>A=B, 2d(A,B) = d(B,A); 3.VA.B.C eYYl : d(A, B) +d(B,C)Z d(A,Q. One calls metric space a couple (YYl, d), where d is a distance for YYl . Within the metric space (YYl, d) one calls sphere of centre O and ray r the geometrical locus ol points M satisfying d (0,M) = r. In the particular case when YYl is an Euclidean plane, instead of the word sphere that of circle is used. We call Isometry of (YYl, d) a surjective map T YYl -> YYl which preserves the distance, that is : /A,B&YYl d(T(A),T(B)) = d(A,B). Theorem 2.1. Any Isometry is injective; the isometries of a metric space make up a group related to the compounding operation. Remark. The condition of surjectiveness included here into the definition ol isometries simplifies essentially the proof .d(A,B) = 0A=B; 2d(A,B) = d(B,A); 3.VA.B.C eYYl : d(A, B) +d(B,C)Z d(A,Q. /A,B&YYl d(T(A),T(B)) = d(A,B). 3. VA,B, C e m d(A,B) + d(B,C)Z d(A,Q. j d : m 2 -> R : Geometrical Transformations 141 Theorem 2.2. An Isometry T of an Euclidean plane preserves the ternary relations of to be between and to be co-linear: the image through T of a straight line, a ray, a segment, a semi-plane, a circle, an arc of a circle is a set characterized by the same name. Proof. The condition A-B-C, equivalent with B G | AC , is expressed here by d(A, B)+d(B, C) = d(A, C) mdd(A, B)-d(B, C)-d(A, C) * 0 It results that the Isometry T preserve the relation to be between and 7(|ytC|) = | T(A)T(C).

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  College Geometry

  A Unified Development

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

    (Publisher)

  15. Referring to Problem 14, there exists a unique affine trans-formation f which maps Δ ABC to Δ DEF . Using the coordi-nate form for an affine transformation, use the information given to find the coefficients and thus verify that f is a similitude. 8 .4. . Line.Reflections:.Building.Blocks. for.Isometries.and.Similitudes It is perhaps surprising that all isometries in the plane are simply prod-ucts of reflections in lines—the simplest type of Isometry. Line reflections have a multitude of applications in mathematics and physics, as well as in the real world. It is fitting that a section be devoted to this important area of geometry. For convenience, we repeat the definition from a previ-ous section where reflections were originally introduced (in particular, see Section 3.5 ). Definition A reflection in the line l is the mapping which takes each point P in P to the unique point P ′ also in P such that l is the perpen-dicular bisector of the segment PP ′ . (If P ∈ l , we take P ′ = P ; thus l is a line of fixed points for the reflection.) It is evident that one may construct the reflection P ′ for each point P not on l by dropping the perpendicular PM to point M on l , then doubling the segment PM to segment PP ′ (segment-doubling theorem). It should be noticed that reflections are special cases of affine reflections. It is almost obvious that reflections preserve distance, and consequently, reflections are isometries. In Figure 8.21 is shown two cases, (1) when A and B lie on the same side of l , and (2) when they do not. Thus, we see that in case (1) ◊ ABNM ≅ ◊ A ′ B ′ NM (by SASAS), so that AB = A ′ B ′ by CPCPC, and in case (2) AB ′ = A ′ B by case (1) so that trapezoid AA ′ BB ′ is isosceles, with congruent diagonals, AB and A ′ B ′ . We include one property of reflections that has practical applications.

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  Euclidean Geometry and Transformations

  • Clayton W. Dodge(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  19 appear in high school geometry texts (with different proofs). The high school student who understands the workings of isometries should find the methods of transformations quite logical and understandable.

  The procedure we are using might best be called transform-solve-transform, for when we are given a problem, we transform it into a new problem through the use of isometries, solve the new problem, then transform the solution back to the original problem. In Theorem 18.4 , for example, the given problem is transformed into one in which the two triangles share a common side, then the problem is solved (the theorem is proved) for this case, and finally this solution is applied to the given problem of proving any two triangles that satisfy the SSS condition congruent.

  Remember that we have assumed the SAS condition for congruence as a postulate, so that other congruence conditions must be proved.

  18.2 Theorem Vertical angles are congruent.

  Figure 18.1

  Let AB and CD meet at P as in Fig. 18.1 . Then σ

  P

  ( APC) = BPD and σ

  P

  ( APD) = BPC, since lines through P are fixed under the map σ

  P

  .

  18.3 Theorem If a point is equidistant from two points, then it lies on the perpendicular bisector of the segment between the two points.

  Let A be equidistant from P and Q as in Fig. 18.3 . Let m be the bisector of angle P A Q. Then σ

  m

  maps line AP into line A Q, and since AP ≅ A Q, σ

  m

  (P) = Q. Hence m is the perpendicular bisector of PQ.

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