Translations

3 Key excerpts on "Translations"

• 

  eBook - ePub

  Linear Methods

  A General Education Course

  • David Hecker, Stephen Andrilli(Authors)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  • Similarity: A transformation of the plane that preserves angles. Scalings, rotations, reflections, and Translations are all similarities.

  • Transformation of the plane: A function which moves each point of the plane to some (image) point in the plane. Examples include contractions, dilations, rotations, reflections, and Translations.

  • Translation: A transformation of the plane in which every point is moved the same distance in the same direction as a given vector t. The distance moved is the length of t. For any vector v in the plane, the image of v under the translation is v + t.

  Exercises for Section 7.1

  1.

  In each part, if possible, find a 2 × 2 matrix that represents the given geometric transformation. Also, in each case, specify whether the transformation preserves angles and whether it preserves distances.

  (a)

  A scaling that doubles the lengths of all vectors, but keeps them in the same direction.

  (b)

  A scaling that multiplies the length of vectors by

  1 3

  , but switches them to the opposite direction.

  (c)

  A reflection about the line

  y =

  1 2

  x

  .

  (d)

  A counterclockwise rotation about the origin through an angle of 75°. Round the entries of your matrix to three places after the decimal point.

  (e)

  A clockwise rotation about the origin through an angle of 140°. Round the entries of your matrix to three places after the decimal point.

  (f)

  A transformation for which every point is moved 4 units to the left and 7 units up.

  2.

  In each part, if possible, find a 2 × 2 matrix that represents the given geometric transformation. Also, in each case, specify whether the transformation preserves angles and whether it preserves distances.

  (a)

  A scaling that halves the lengths of all vectors, but keeps them in the same direction.

  (b

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  eBook - ePub

  Space, Time, Matter

  • Hermann Weyl(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  h.

  We succeed in introducing logical order into the structure of geometry only if we first narrow down the general conception of congruent transformation to that of translation, and use this as an axiomatic foundation (§§2 and 3). By doing this, however, we arrive at a geometry of translation alone, viz. affine geometry within the limits of which the general conception of congruence has later to be re-introduced (§ 4). Since intuition has now furnished us with the necessary basis we shall in the next paragraph enter into the region of deductive mathematics.

  § 2. The Foundations of Affine Geometry

  For the present we shall use the term vector to denote a translation or a displacement a in the space. Later we shall have occasion to attach a wider meaning to it. The statement that the displacement a transfers the point P to the point Q (“transforms” P into Q) may also be expressed by saying that Q is the end-point of the vector a whose starting-point is at P. If P and Q are any two points then there is one and only one displacement a which transfers P to Q. We shall call it the vector defined by P and Q, and indicate it by .

  The translation c which arises through two successive Translations a and b is called the sum of a and b, i.e. c = a + b. The definition of summation gives us : (1) the meaning of multiplication (repetition) and of the division of a vector by an integer; (2) the purport of the operation which transforms the vector a into its inverse - a; (3) the meaning of the nil-vector o, viz. “identity,” which leaves all points fixed, i.e. a + o = a and a + (- a) = o. It also tells us what is conveyed by the symbols , in which m and n are any two natural numbers (integers) and λ denotes the fraction . By taking account of the postulate of continuity this also gives us the significance of λa, when λ is any

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  eBook - ePub

  The Ellipse

  A Historical and Mathematical Journey

  • Arthur Mazer(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Figure 4.4.

  Figure 4.4

  Equations in translated coordinates.

  Let points in original coordinates be denoted by (x , y ) and points in translated coordinates be denoted by As indicated in Figure 4.4 , when Similarly, when In general, we have the following equalities:

  (4.2)

  Substituting the equalities from equation (4.2) into the equation for the curve, equation (4.1) , results in the equation for the curve in translated coordinates,

  Example 4.1

  Express the circle x 2 + y 2 – 9 = 0 in a coordinate system centered at (1, 2).

  Solution

  Translation of the coordinates has a complementary operation, translating the geometric object. Translating the coordinates by a specified distance in a given direction yields the equivalent result of translating the object by the same distance in the opposite direction. In the example, the coordinates are moved to the point (1,2). This has the equivalent effect of moving the circle to a new location centered at a new point in the opposite direction (–1, –2). Accordingly, to translate a figure from a given position to a new position, subtract the values that determine the shift from the values of x and y (Figure 4.5 ). The formula for translating the object by the quantity (a, b) while maintaining the same coordinate axes is f (x – a , y – b ) =0.

  Figure 4.5

  Translation of center from (0, 0) to (a, b)

  Example 4.2

  Shift the circle with radius 3 centered at the origin to a circle with radius 3 centered at (–1,–2).

  Solution

  Remarks

  • This problem demonstrates the power of unifying geometric and algebraic concepts in a single framework. The geometric figure is constructed as an algebraic object. The geometry of translation is then expressed algebraically. Finally, geometric translation is algebraically introduced into the equations, yielding a new algebraic expression. All along none of the geometric intuition is lost; indeed the geometry guides the algebra.

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