Rotations

8 Key excerpts on "Rotations"

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  Computer Graphics

  Theory and Practice

  • Jonas Gomes, Luiz Velho, Mario Costa Sousa(Authors)
  • 2012(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  4.2 Introduction to Rotations in Space The study of Rotations in three dimensions is more complex than in two, and will occupy us for the rest of the chapter. The material necessarily involves a lot of mathematics; readers 78 4. The Space of Rotations tempted to skip the calculations should nonetheless strive to follow the text, to grasp the key ideas. This will allow them to understand the details later on, when needed in practice. Here again, we define a rotation to be a linear transformation of R 3 that is orthogonal (hence distance-preserving) and positive. Rotations form a subgroup of the group of all linear transformations of R 3 —why? We denote the group of Rotations of R 3 by SO(3). A rotation R is of course determined by what it does to the standard basis {e 1 , e 2 , e 3 } of R 3 . Suppose R takes the standard basis to the basis {b 1 , b 2 , b 3 }, which is orthonormal by definition. The matrix of R has as its columns the expressions of b 1 , b 2 , b 3 in the standard basis. Unlike the case of R 2 , it is not enough to know b 1 , or the first column of the matrix: there are many Rotations that take e 1 to b 1 . Indeed, given one such rotation, we can then apply any rotation about the axis determined by b 1 : the composition is still a rotation taking e 1 to b 1 . However, if both b 1 and b 2 are known, b 3 is determined. This is the cross product familiar from physics: b 3 = b 1 × b 2 . (Similarly, b 1 = b 2 × b 3 and b 2 = b 3 × b 1 .) So one way to understand the space SO(3) is to study the space of possibilities for b 1 , and the space of possibilities for b 2 once a choice of b 1 is made. The space of choices for b 1 is just the 2D sphere S 2 , the set of vectors of length 1: S 2 = {(x, y, z ) ∈ R 3 : x 2 + y 2 + z 2 = 1}. (Compare with the 2D case, where the space of choices of b 1 was a circle.) Once b 1 is chosen, the choices for b 2 narrow down to a circle, since b 2 must be perpendicular to b 1 .

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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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  Backgrounds Of Arithmetic And Geometry: An Introduction

  An Introduction

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

    (Publisher)

  In order to make R o T explicit, we shall notice that (/? o T)' [ = T' o /?' and we have thus returned to the initial problem. We shall find out that R o T is a rotation R va , where V = T' ] (U), point U being the one described above. We find out that: RoT = ToR** U = V** T 1. Geometrical Transformations 155 Fig.27 Fig.26 In order to make RoR' explicit let us suppose R = R 0 „ and R' = R 0 , d .. If 0 = O,, then RoR' = R 0i ^,, according to Theorem 6.2. Let us suppose 0 * O' and let s 2 be a ray of origin O' situated on <* 2 = (0O*). We determine s' so that R'(s)) = s 2 and let d x be the support straight line of the bisector of the angle formed bys 2 and s'. We also consider the support straight line <*, of the bisector of the angle formed by [0; O') and s = R([0; O').. We also consider the symmetries 5, related to the straight lines d l (for i = 1, 2, 3). It results R' = S 2 o 5,, R = S, o 5, and so rto'l' = S 3 o 5 , . /?o/J ' is a translation ifand oriy if 4 1 4 , thatis £ +£' =0;inthis case, the translation applies 0 in 0 = /f(0). If <3+<3' ; *0 , let be t / G ^ D ^ / / ? o/? ' is a rotation of centre U and angle a + « ' . Remark. In figure 27 there also appears the centre V of the rotation R' o R, symmetric with U related to ( 0 0 ' ) ; the planned conditions make impossible the equality Ro'' = R' OR. One calls rotation related to an axis d an isometry R of E } which has the straight line d or all the points of the space as set of fixed points. Theorem 6.6. 77i

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  Matrices and Transformations

  • Anthony J. Pettofrezzo(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  H of linear homogeneous transformations.

  Figure 3-15

  The set of ordered products of the rigid motion transformations of the plane may be represented by a matrix of the form where

  (3-20)

  Furthermore,

  (3-21)

  and

  (3-22)

  Two theorems which state that a general linear transformation of the plane is a rigid motion transformation if and only if the matrix representing the transformation satisfies conditions (3-21) and (3-22) will be proved now.

   

  Theorem 3-3 Let

  represent a general linear transformation of the plane under which distance is a scalar invariant. Then

  and

  a 11 a 12 + a 21 a 22 = 0.

  Proof : Consider the points O : (0, 0, 1) and P : (1, 0, 1). Under the transformation represented by T , the image points of O and P are O′: (a 13 , a 23 , 1) and P ’: (a 11 + a 13 , a21 , + a 23 , 1), respectively. Since under the transformation represented by T ,

  In a similar manner, choosing O : (0, 0, 1) and P : (0, 1, 1), it can be shown that

  Now, consider the points O : (0, 0, 1) and P : (1, 1, 1). Under the transformation represented by T , the image points of O and P are O′: (a 13 , a 23 , 1) and P ’: (a 11 + a 12 + a 13 , a 21 + a 22 + a 23 , 1), respectively. Again, since under the transformation represented by T ,

  Theorem 3-4 Let

  represent a general linear transformation of the plane such that , , and a 11 a 12 + a 21 a 22 = 0. Then distance is a scalar invariant under the transformation represented by T.

  Proof : Consider any two points P 1 : (x 1 , y 1 , 1) and P 2 : (x 2 , y 2 , 1) on the plane. Under the transformation represented by T, the image points of P 1 and P 2 are (a 11 x 1 + a 12 y 1 + a 13 , a 21 x 1 + a 22 y 1 . + a 23 , 1) and (a 11 x 2 + a 12 y 2 + a 13 , a 21 x 2 + a 22 y 2 + a 23

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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)

  Are any of the ships translations of each other? SOLUTION There are two ships that are 2 units long, two that are 3 units long, two that are 4 units long, and one that is 6 units long. The only ships that you can slide, without turning, to coincide with another ship are the 2-unit ships. So, the 2-unit ships are translations of each other. A translation cannot be flipped, enlarged, or rotated. Notice that the triangle below is not a translation of the original triangle in Example 4 because it is not a slide of the original triangle. 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. 534 Chapter 14 Transformations Rotations Definition of Rotation Turn Angle of rotation Center of rotation A rotation is a transformation in which a figure turns about a point called the center of rotation. The number of degrees a figure rotates is the angle of rotation. The original figure and its image are congruent. EXAMPLE 6 Identifying Rotations Determine whether the blue figure is a rotation of the red figure. a. b. SOLUTION a. The red figure rotates, or turns, 90º to 90° form the blue figure. So, the blue figure is a rotation of the red figure. b. The red trapezoid slides to form the blue trapezoid. So, the blue trapezoid is not a translation of the red trapezoid. EXAMPLE 7 Finding the Center of Rotation The red triangle rotates to form the blue triangle. Find the center of rotation. SOLUTION 1. Identify a pair of corresponding points (one point on each triangle).

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  An Introduction to Groups and their Matrices for Science Students

  • Robert Kolenkow(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 ROTATION 7.1 Introduction This chapter treats rotation to lay the groundwork for Chapter 8 on angular momen- tum, the subject that historically led physicists to recognize the power of group theory in quantum mechanics. 7.2 Two Ways of Looking at Rotation The next two sections show that rotating a vector counterclockwise (positive angle) in a fixed coordinate system gives the same result as keeping the vector fixed and rotating the coordinate system clockwise (negative angle). 7.2.1 Rotated Vector, Fixed Axes In a 2-dimensional Cartesian coordi- nate system, vector A from the origin is written A x O i C A y O j with respect to the Cartesian unit vectors.  is the angle between A and O i as shown, and let A be the magnitude of A. Suppose that A is rotated counter- clockwise about the origin around the z-axis by an additional angle  , result- ing in a new vector A 0 that points to a new point .x 0 ; y 0 /. This picture of rota- tion is called the alibi picture because vector A can argue that it was some- where else at the time: 168 7.2 Two Ways of Looking at Rotation 169 A D A x O i C A y O j D A cos  O i C A sin  O j A 0 D A 0 x O i C A 0 y O j D A cos . C / O i C A sin . C / O j D A.cos  cos   sin  sin / O i C A.sin  cos  C cos  sin / O j D A cos   cos  O i C sin  O j  C A sin    sin  O i C cos  O j  D A x  cos  O i C sin  O j  C A y   sin  O i C cos  O j  D .A x cos   A y sin / O i C .A x sin  C A y cos / O j or in matrix form with A 0 z D A z ,  A 0 x A 0 y A 0 z  D  cos   sin  0 sin  cos  0 0 0 1  A x A y A z  : (7.1) A and A 0 both define the same length from the origin, so a vector transforma- tion is identified as a rotation if it causes no change in the vector’s magnitude. Another identifier of rotation is that the matrix in Eq. (7.1) has determinant cos 2  C sin 2  D 1. Rotation Notation A convenient notation for counterclockwise rotation by angle ˛ about axis n is R.˛; n/.

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  Spatial Cognition

  The Structure and Development of Mental Representations of Spatial Relations

  • D. R. Olson, E. Bialystok(Authors)
  • 2014(Publication Date)
  • Psychology Press

    (Publisher)

  r, the transformational relation between corresponding constituents:

  r (arm a, arm a′).

  That is, r is the spatial predicate relating the corresponding constituents of object A and A′. However, to see specifically how r is represented it is necessary to look in detail at the type, direction, and extent of the Rotations that make up r.

  Just as there are three dimensions in terms of which features of objects can be specified: top/bottom, front/back, left/right; so too there are three axes around which Rotations can occur. Furthermore, Rotations can occur in one of the two directions for a certain distance. Any rotational transformation, r, must then, be specified in terms of values for each of these 3 variables. The axes are the familiar Cartesian x-axis, y-axis, and z-axis, but the axes are assigned, we suggest, on the basis of ego space with top, front and sides. The x-axis involves Rotations in depth around a Horizontal axis; rotation can be either top towards or top away from viewer; and it can go any distance. The y-axis rotation involves depth rotation around a Vertical axis again in one of two directions, either right towards (counterclockwise) or right away from (clockwise) the viewer, and again any distance. The z-axis involves Rotations in a plane around a frontal axis either top to the right (clockwise) or top to the left (counterclockwise) for a particular distance. These Rotations we may call lateral axis Rotations, vertical axis rotation and frontal axis rotation respectively. Any rotation keeps the values in one dimension invariant while allowing the other two to vary. These characteristics may be summarized as shown in Table 8.1 .

  TABLE 8.1

  It may be recalled from Chapter 4

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  3D Game Engine Design

  A Practical Approach to Real-Time Computer Graphics

  • David Eberly(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  C h a p t e r 17 Rotations R otation of vectors is a common operation in computer graphics. Most pro-grammers are comfortable with rotation matrices. A few are uncomfortable with the quaternions, an alternate representation for rotation matrices, because of their inherent mathematical nature. This chapter summarizes both topics, but in the end the emphasis is on the comparison of rotation matrices and quaternions regard-ing memory usage and computational speed. 17.1 Rotation Matrices A 2D rotation of the vector (x , y) to the vector (x , y ) by an angle θ is represented by the matrix x y = cos θ − sin θ sin θ cos θ x y = R 2 x y (17.1) where the last equality defines the rotation matrix R 2 . The rotation direction is coun-terclockwise in the xy -plane when θ > 0. We saw such a rotation in Figure 2.3. Using the rotation matrix in Equation (17.1) as motivation, a 3D rotation of the vector (x , y , z) to the vector (x , y , z ) by an angle θ about the z -axis is represented by the matrix ⎡ ⎣ x y z ⎤ ⎦ = ⎡ ⎣ cos θ − sin θ 0 sin θ cos θ 0 0 0 1 ⎤ ⎦ ⎡ ⎣ x y z ⎤ ⎦ = R 3 ⎡ ⎣ x y z ⎤ ⎦ (17.2) 759 760 Chapter 17 Rotations where the last equality defines the rotation matrix R 3 . The idea is to rotate the 2D component of the vectors in the xy -plane using the rotation matrix of Equation (17.1). 17.1.1 Axis/Angle to Matrix We may use the matrix of Equation (17.2) to create a matrix representing a general ro-tation. Let the rotation axis have direction W , a unit-length vector. Let θ be the angle of rotation. Choose any pair of unit-length vectors U and V so that the set { U , V , W } is a right-handed orthonormal set. That is, the vectors are mutually perpendicular and W = U × W . A vector P will be rotated to a vector Q . We may represent P as P = x U + y V + z W where x = U . P , y = V . P , and z = W . P . Similarly, we may represent Q as Q = x U + y V + z W where x = U .

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