Essential Mathematics: Transformations

Unit 6: Geometric transformations

84 flashcards · Mathematics · by lakesare

1. What's a unit square?

   In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).

2. What transformations are affine?

   Affine transformations are functions that map parallel lines to parallel lines. Roughly speaking, these are functions that preserve parallelism; that is, they map parallel lines to parallel lines. An affine transformation is any transformation that: preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation)preserves ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation)

3. What transformations are isometries?

   It turns out that every isometry is one of four basic types: a translation, a rotation, a reflection, or a glide-reflection. A proof of this fact is beyond the scope of this module.

4. What transformations are isometries?

   f() is an isometry g() is NOT an isometry

5. Name 4 isometries. Every isometry is one of four basic types.

   translation p units horizontally, q units vertically rotation through a fixed angle about a fixed point reflection in a line L - maps P to P' so that L is a perpendicular bisector of PP'. glide-reflection in a line L - reflection in L followed by a translation parallel to L (or vice versa)

6. What isometric transformations reverse the orientation of the figure? One characteristic of a reflection that distinguishes it from a rotation or a translation is that it reverses the orientation of figures. To see what this means, consider what happens to a triangle with vertices at A, B and C when it is reflected in a line H, as shown in Figure 19. Under the reflection, the vertices A, B, C are mapped to the vertices A! , B! , C! of the image triangle on the right. The vertices A, B, C of the…

   Reflections and glide-reflections. translation - orientation stays the same rotation - orientation stays the same reflection - REVERSE glide-reflection - REVERSE Whereas reflections reverse the orientation of figures, rotations and translations preserve orientation. Actually, there is one other type of isometry that reverses orientation, namely the glide-reflections.

7. Prove that a composite of two isometric transformations is an isometric transformation.

   Suppose f and g are isometric transformations. We need to prove that PQ = g ◦ f (P) g ◦ f (Q). Since f is an isometric transformationDistance between P and Q = distance between f(P) and f(Q). Since g is an isometric transformationDistance between f(P) and f(Q) = distance between g(f(P)) and g(f(Q)). Overall, therefore, the distance between P and Q = distance between (g ◦ f)(P) and (g ◦ f)(Q). It follows that g ◦ f is an isometry.

8. [Composing reflections]

9. What's the result of composing 2 rotations about the same point?

   Their composite is the rotation through the angle θ + φ.

10. What's the result of composing 2 rotations about different points?

    Their composite is the rotation through the angle θ + φ, but the center of rotation is neither of these points. It may interest you to know that, even if the rotations through θ and φ are about different points, P and Q say, then their composite is still a rotation through the angle θ + φ, but the centre of rotation is neither P nor Q. The only exception to this is if θ + φ is a multiple of 2π, in which case the composite is a translation.

11. What's the result of composing 2 translations?

12. Which of the isometric transformations are commutative? translation rotation reflection glide-reflection

    translation rotation - only in 2 dimensions, about the same point reflection glide-reflection All two-dimensional rotations about the same point commute.Rotations about different points, in general, do not commute. Rotations in three-dimensional space differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not commutative, so the order in which rotations are applied is important even about the same point.

13. What's the result of composing 2 reflections in line l and m (that CROSS)?

    Their composite is the rotation through TWICE the angle from l to m.

14. What's the result of composing 2 reflections in line l and m (that are PARALLEL)?

    Their composite is the translation through TWICE the distance from l to m.

15. What are 2 possible results of composing 2 reflections in line l and m?

    If lines l and m cross - their composite is the rotation through TWICE the angle from l to m. If lines l and m are parallel - their composite is the translation through TWICE the distance from l to m.

16. Why does it make sense that 2 possible results of composing 2 reflections is either rotation, or translation?

    The first reflection reverses orientation and the second reflection reverses it back again, so the composite is an isometry that preserves orientation. Hence it must be either a translation or a rotation. Which of these it is depends on the lines of reflection. If they intersect, then the composite fixes the point of intersection, and is therefore a rotation. If they don't intersect, then the composite is a translation.

17. [Mathematics] What's a fixed point of a function?

    Fixed point is such a point that f(point) = point. A fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

18. What isometry is this? (a) Try to describe this process as a composite of two isometries. (b) Does this composite preserve orientation? (c) Does your composite leave any points fixed? Consider the process of superimposing the footprint of one step onto the footprint of the next.

    It's glide-rotation. It can't be a rotation or a translation because it reverses orientation, and it can't be a reflection because there are no fixed points. It follows that it must be a type of isometry that we have not yet considered. It is known as a glide-reflection.

19. What's reflection (in a line L)?

    Maps P to P' so that L is a perpendicular bisector of PP'.

20. What's glide-reflection (in a line L)?

    Reflection in L, followed by a translation parallel to L (or vice versa).

21. What transformation does rotation(reflection(x, y)) result in?

    In fact it can be shown that, if you choose a reflection at random and compose it with either a rotation or a translation, then the chances are that the composite will be a glide-reflection.Only in certain special cases, where the line of reflection passes through the centre of rotation, or where the line of reflection is at a right angle to the direction of translation, will the composite turn out to be a reflection. Reflection in the y-axis. Glide-reflection.

22. [Composing a reflection with a glide-reflection] What types of isometries would you expect to obtain when you compose a reflection with a glide-reflection?

    Reflections and glide-reflections both reverse orientation, and so their composite will preserve orientation. It follows that the composite will be either a translation or a rotation.

23. Prove that isometry is a one-to-one transformation.

    Because if 2 points are mapped to 1 point - nonzero distance would map to 0 distance!

24. Why is it true that every isometry has an inverse?

    Because every isometry is one-to-one. Any one-to-one (& onto) function has an inverse, and every isometry is one-to-one.

25. [Finding the inverses of some transformations] Find the inverse of the transformation k. k(x, y) = (x + 3y, −y)

    k-1(x, y) = (x + 3y, −y) const k = (x, y) => { const newX = x + 3y const newY = −y return (newX, newY)} const k -1 = (newX, newY) => { const x = newX + 3newY const y = −newY return (x, y)}

26. Draw the Venn diagram of linear transformations and isometries.

    Some isometries are linear transformations, while others are not. Also there are many examples of linear transformations that are not isometries.

27. What kind of transformation is linear?

    One whose rule can be represented by f(x, y) = (ax + by, cx + dy).

28. [Finding the matrices of some linear transformations] Find the matrix of this linear transformation: g(x, y) = (2x, y) (scaling in the x-direction by the factor 2)

29. [Finding the matrices of some linear transformations] Find the matrix of this linear transformation: k(x, y) = (x + 2y, y) (shear)

30. [Finding the images of points under a linear transformation]

31. Why does linear transformation leave the origin fixed?

    Because linear if f(x, y) = (ax + by, cx + dy), then f(0, 0) = (0, 0).

32. How can we determine whether linear transformation is one-to-one?

    Linear transformation is one-to-one if determinant is any NUMBER but zero. Mnemonic: determinant is 1 - transformation is 1-to-1.

33. Is this linear transformation one-to-one?

34. Is this linear transformation one-to-one?

35. Explain what it means for the transformation to: preserve linearity preserve parallelism

    preserve linearity - if AB is a line - then f(A)f(B) is a line (that is, all the points in-between A and B map appropriately, and not do some random shit). In fact, you can use the matrix of the transformation to find the image of any given line under a one-to-one linear transformation.preserve parallelism - if m and n are parallel lines, then f(m) and f(n) are also parallel lines. Transformation preserves linearity - it means that if AB is a line - then f(A)f(B) is a line? That is, all the poi…

36. What's meant by the transformation of the line?

37. [image of the unit square under a linear transformation] Let f be the linear transformation represented by the matrix A. Illustrate the effect that f has on the unit square.

    Finding the image of the unit square under a linear transformation is fairly straightforward.The origin (0, 0) is mapped to itself, andthe points (1, 0) and (0, 1) are mapped to points whose position vectors are respectively the first and second columns of the matrix of the transformation.So you just need to work out the image of the remaining vertex (1, 1). The image of the unit square is a parallelogram. This is because the parallel sides of the unit square are mapped to parallel sides. So it…

38. [Finding images of intersection points on the unit grid] What's the image of (0, 3) (2, -1)? The grid on the right consists of two families of parallel lines, one in the direction of the image i' of i under f, and the other in the direction of the image j' of j under f.These lines partition the plane into parallelograms, each congruent to the parallelogram that is the image of the unit square under f.The individual squares or parallelograms into which a grid partitions the plane are known as gr…

    (0, 3) - A (2, -1) - D

39. How to prove that [one-to-one linear] transformation scales the area by |ad - bc|?

40. [One-to-one linear] transformation scales the areas of all figures by the factor that it scales the unit square. Why?

    If F is a figure that surrounds N grid squares, then the image of F will be a figure that surrounds the N corresponding parallelograms. The factor by which the area of F is scaled is therefore equal to the factor by which f scales the area of the unit square S. But the area of S is 1, so this factor is equal to the area of the parallelogram f(S).

41. We can draw a grid (of parallelograms each congruent to the transformed unit square) because 1-to-1 linear transformation preserves lines and parallelism. Name 2 interesting consequences of this mapping-to-grid.

    We can determine: whether orientation is preserved or reversed, and by what factor the area of all figures changed By looking at the orientation & area of the unit square.

42. [Finding properties of a linear transformation]

43. [Finding properties of some linear transformations] For the transformations represented by each of the following matrices, find the factor by which the transformation scales areas, and whether it preserves orientation.

44. [Linear transformations] What are the matrices for scaling VS dilation linear transformations?

45. [Linear transformations] Imagine line L. What's a shear about a line L? Note, however, that only shears about lines through the origin are linear transformations (since a linear transformation fixes the origin).

    Shear about a line L displaces every point P: in the direction parallel to L by an amount proportional to its distance from L.

46. [Linear transformations] What's the matrix of a horizontal shear VS the matrix of a vertical shear? Only shears about lines through the origin are linear transformations (since a linear transformation fixes the origin).

47. [Linear transformations] What's the shear factor for horizontal shear for vertical shear? Only shears about lines through the origin are linear transformations (since a linear transformation fixes the origin).

    for horizontal shear: k. for vertical shear: −k. We defined the shear factor to be positive when points are mapped in a clockwise direction! Horizontal shear leaves the horizontal grid lines looking the same, but it tilts all the vertical grid lines about the points where they intersect the x-axis. They are tilted clockwise if k > 0 and anticlockwise if k < 0. Points on the x-axis remain fixed. The shear factor is equal to k.Similarly, a vertical shear leaves the vertical grid lines looking the…

48. Which of the isometries are linear transformations? Isometries (transformations of the plane that keep the distances between pairs of points unchanged): translations, rotations, reflections, and glide-reflections. Isometries are transformations of the plane that keep the distances between pairs of points unchanged. Some isometries are linear transformations, while others are not. Also there are many examples of linear transformations that are not isometries.

    rotations - is a linear transformation (but only about an origin) reflections - is a linear transformation (but only in a line through the origin) Now any isometry that moves the origin cannot be a linear transformation, but what about an isometry that fixes the origin? You’ll see that such an isometry is indeed a linear transformation, and that it’s either a rotation about the origin or a reflection in a line through the origin.

49. Why does this grid game work for linear transformations? So by using grids you can visualise where the image of each intersection point on the unit grid is located. In fact, you can locate the image of any point in the plane in the same way.

    Because (abcd)(1, 0) = (ac), and keeping x and y - we can represent the transformed point using new i' and j'.

50. Name 2 types of affine transformations.

    linear - can be represented by the abcd matrix. isometry - keeps the distance between 2 points. Translation, rotation, reflection, glide-reflection.

51. [Linear transformations] Draw why this matrix works for rotations.

52. [Linear transformations] Draw why this matrix works for reflections.

53. [Linear transformations] How would you find the matrix that represents the reflection in the line y = x √3.

    Find the angle α Use the matrix

54. [Linear transformations] Consider linear transformations f(x) = Ax for which det(A) = 0. A zero determinant suggests that all areas are reduced to zero under the transformation, and this in turn suggests that the transformation collapses the plane in some way. Name 2 ways in which det(A) = 0 matrix can 'collapse the plane'. Every isometry that fixes the origin is a linear transformation.

    zero matrix - collapsing the plane into one point at the origin. nonzero matrix (with zero determinant) - collapsing the plane into the line through the origin.

55. [Linear transformations] Name 3 criteria you can use to determine whether the transformation is a flattening.

    1st column of A is a multiple of the 2nd column of A. det(A) = 0. f is not one-to-one.

56. [Finding the image set of a flattening of the plane] Show that f is a flattening.

    There are 3 ways to show that A is a flattening: 1st column of A is a multiple of the 2nd column of A. det(A) = 0 f is not one-to-one.

57. [Finding the image set of a flattening of the plane] Find the image set of f.

58. Identify the type of linear transformation.

    The transformation is a horizontal shear with shear factor 5. The bottom left entry of the matrix is zero and there are 1s down the main diagonal, so the transformation is a horizontal shear.

59. Identify the type of linear transformation.

    Rotation about the origin. Since both entries off the main diagonal are non-zero, this matrix cannot be a (k, l)-scaling, nor can it be a horizontal or vertical shear. Perhaps it’s a rotation or a reflection.

60. Identify the type of linear transformation.

    The matrix is non-zero but it has determinant 2 × 6 − 3 × 4 = 0, so the transformation is a flattening onto a line through the origin. Here again, the matrix cannot be a (k, l)-scaling, or a horizontal or vertical shear, but the determinant is zero.

61. Identify the type of linear transformation.

    This is a diagonal matrix so the transformation is a (5, 2)-scaling.

62. [Finding the matrix of a composite linear transformation] Find the matrix of the composite transformation g ◦ f. f is represented by A, g is represented by B.

63. [Finding the matrix of a composite linear transformation] What composite transformation is illustrated by the two-stage diagram below, g◦f or f◦g?

    g◦f. The two-stage diagram shows the effect of first doing f, the horizontal shear, then doing g.

64. How would you go about algebraically showing that reflection in θ and φ is equivalent to rotation 2 * (φ - θ)?

    A - matrix that reflects around θ B - matrix that reflects around φ We want to prove that AB rotates around 2 * (φ - θ). Use the expression for a reflection in a line through the origin to write down the matrices of f and g. Use the angle sum trigonometric identities to simplify each element of the matrix. Compare this with the matrix of a rotation about the origin.

65. [Composing a reflection and a rotation] How would you go about proving that g◦f is a reflection (in a line l through the origin)?

    This matrix represents the reflection in a line H through the origin.

66. How to quickly find an inverse of the 2x2 matrix A-1?

    To get the inverse of A: swap diagonal elements, change the sign of the off-diagonal elements and divide by the determinant.

67. [Finding the inverse of a linear transformation] Show that f is invertible.

    Check whether the determinant of A is non-zero. If it is, then f−1 exists.

68. [Finding the inverse of a linear transformation] Find f-1.

69. [Finding the inverse of a linear transformation] How would you find the point (x, y) such that f(x, y) = (2, 1)?

70. [Finding the image of the unit circle under an invertible linear transformation] How would you find equation of the image f(C) of the unit circle C?

    Get A-1 What point does f map to (x, y)? Find with A-1(x, y). So our f(C) is the equation of C (a circle), except with (3x - 2y, x - y) in place of x and y.

71. [Finding the image of the unit circle under an invertible linear transformation] How would you calculate the area enclosed by f(C) of the unit circle C?

    Area enclosed by f(C) = area of the C scaled by det(A).

72. [Finding the area of an ellipse. Excellent proof!] Show that the area of the ellipse is π a b.

73. [Finding the matrix of a shear about the line y = x] How to use matrix inverses to find the matrix of a shear that is neither a horizontal nor a vertical shear. (Similar techniques can be used to find the matrix of a scaling that is neither horizontal nor vertical.)

    Use the conjugation of isometries. [Isometry] Apply a rotation g about the origin so that the line y = x ends up along the x-axis. [Transform] Apply a horizontal shear h with shear factor 3. [Isometry inverse] Apply the inverse rotation g−1, so that the line along the x-axis is sent back to the line y = x.

74. [Finding the matrix of a shear about the line y = −x] Find the matrix of the shear about the line y = −x with shear factor 2.

    This idea of composing a transformation with an isometry and its inverse in order to perform the same action at a different location of the plane is known as conjugation.

75. What's an affine transformation?

    Transformation in the form of f(x) = Ax + a Affine transformation = linear transformation + translation.

76. [Finding an affine transformation] Find the affine transformation f that maps the points (0, 0), (1, 0) and (0, 1) to the points (1, 4), (7, 4) and (3, 5), respectively. An affine transformation as the composite of a linear transformation and a translation.

77. Why does the set of affine transformations contain the set of all linear transformations?

    Because each linear transformation f(x) = Ax can be written as the affine transformation f(x) = Ax + 0.

78. When is linear transformation a flattening VS affine transformation a flattening? An affine transformation as the composite of a linear transformation and a translation.

79. [Inverse of an affine transformation] Why is this the rule for the inverse of the affine transformation?

80. [Finding the inverse of an affine transformation] Find the inverse of the following affine transformation.

81. [Finding the area of a triangle] Find the area of the triangle T with vertices at (2, 5), (3, −7), (1, −3).

    10.

82. How would you prove that composite of 2 affine transformations is an affine transformation?

    Just find the composite of 2 matrices f ◦ g = B(Ax + a) + b

83. [Recognising an affine transformation that is an isometry] How would you go about describing the following transformation geometrically? (Provided you know it's an isometry!) (it's the composite of the reflection in the line y = x followed by the translation through −2i + 2j)

    Find equation for points that f(x, y) = (x, y). Then see what fits: translations - no fixed points rotations - fix 1 point reflections - fix ∞ points along the line. glide-reflections - no fixed points Translations are easy to spot because they have the form f(x) = x + a; that is, the matrix that multiplies x is the identity matrix (which is therefore usually omitted). If an affine transformation has no fixed points and the matrix is not the identity matrix, then it must be a glide-reflection.…

84. What's conjugation of isometries?

    gtg−1 - when we compose transformation t with a transformation and its inverse in order to perform t at a different location. This idea of composing a transformation with an isometry and its inverse in order to perform the same action at a different location of the plane is known as conjugation. Use the conjugation of isometries. [Isometry] Apply a rotation g about the origin so that the line y = x ends up along the x-axis. [Transform] Apply a horizontal shear h with shear factor 3. [Isometry i…