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chapter 1 | The Necessity of Perspective<br />
chapter 1 | The Necessity of Perspective<br />
chapter 2 | Collinearity and Angle<br />
chapter 2 | Collinearity and Angle<br />
chapter 2 | Collinearity and Angle<br />
chapter 2 | Collinearity and Angle<br />
chapter 2 | Collinearity and Angle<br />
chapter 2 | Collinearity and Angle<br />
In a given plane, the perceived length of an object depends on the pair of light rays emanating from the two points of the...
chapter 2 | Collinearity and Angle<br />Points that are collinear with the eye will appear to share the same position, and...
The apparent dimension of an object does not depend directly on what its actual dimension is, nor at what distance from th...
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3 | The Linear and Planar Scales<br />
chapter 3.1 | Size<br />
chapter 3.1 | Size<br />
	There’s a car whose breadth is 2 meters, and you hold a scale about 10 cm from your eye. <br />	100 m away,<br />	1 m awa...
chapter 3.1 | Size<br />l’∝ df<br />
chapter 3.1 | Size<br />
chapter 3.1 | Size<br />5.1 mm, 1 m away<br />
chapter 3.1 | Size<br />n∆l,  n∆l’<br />
chapter 3.1 | Size<br />
chapter 3.1 | Size<br />
chapter 3.2 | Position<br />
chapter 3.2 | Position<br />In ΔOP’N’, tanθ = x’/df⇒x’ =dftanθ.<br />
chapter 3.2 | Position<br />
chapter 3.2 | Position<br />
chapter 3.2 | Position<br />θ’ = θ.<br />
?<br />chapter 3.2 | Position<br />(r’, θ’)  ≡<br />But the vanishing point is not always <br />at the centre, along our l...
chapter 3.2 | Position<br />Use this with functions f(x,y,z,t) to project objects at a single go.<br />E = E(t)<br />
chapter 3.2 | Position<br />
chapter 3.2 | Position<br />
chapter 3.3 | Motion<br />For non-planar displacements, project end-points, then join.<br />
chapter 3.3 | Motion<br />
chapter 3.3 | Motion<br />
chapter 3.3 | Motion<br />
Review of Part 1<br />
Review of Part 1<br />
chapter 3.4.1 | Applications: The Cube<br />A cube of side-length 50cm<br />dx = 20cm<br />dy = 30cm<br />dz = 1m<br />The...
chapter 3.4.1 | Applications: The Cube<br />
chapter 3.4.1 | Applications: The Cube<br />
?<br />chapter 3.4.1 | Applications: The Cube<br />But is it correct to just join the projected corners?<br />
chapter 3.4.1 | Applications: The Straight Line<br />x’ {m (ez- z1) – n (ey- y1)} – df m (ex - x1) = y’ {l (ez- z1) – n (e...
chapter 3.4.1 | Applications: The Straight Line<br />,<br />
chapter 3.4.1 | Applications: The Straight Line<br />(x’, y’) ≡ <br />(x’, y’, df)<br />= (x’ n/df, y’ n/df, n) = (l, m, n...
chapter 3.4.1 | Applications: The Straight Line<br />
chapter 3.4.1 | Applications: The Straight Line<br />l1l2+m1m2+n1n2 = l1l3+m1m3+n1n3 = l2l3+m2m3+n2n3 = 0.<br />x1x2 + y1y...
chapter 3.4.1 | Applications: The Straight Line<br />v1 (x1, y1)<br />v2 (x2, y2)<br />v3 (x3, y3)<br />v1.v2 = v1.v3 ⇒ v1...
chapter 3.4.1 | Applications: The Straight Line<br />v1<br />v2<br />v3<br />
chapter 3.4.2 | Applications: The Wall<br />z = mx + d, y = ±h.<br />
chapter 3.4.2 | Applications: The Wall<br />d = 2m<br />h = 1.5m<br />df = 10cm<br />θ = 60°<br />
chapter 3.4.3 | Applications: The Staircase<br />20 steps<br />
chapter 3.4.3 | Applications: The Staircase<br />10 steps above eye level<br />7 steps below<br />
chapter 3.4.3 | Applications: The Staircase<br />Till the 9th step below<br />
chapter 3.4.3 | Applications: The Staircase<br />The 10th step<br />
chapter 3.4.3 | Applications: The Staircase<br />
chapter 3.4.4 | Applications: The Circle<br />r = R, z= do<br />
chapter 3.4.4 | Applications: The Circle<br />x2 + y2 + (z-d)2 = r2, z = mx + d<br />
chapter 3.4.4 | Applications: The Circle<br />
chapter 3.4.4 | Applications: The Circle<br />
chapter 3.4.5 | Applications: The Sphere<br />x2 + y2 + (z-d)2 = r2<br />The perceived size of an object depends on the pa...
chapter 3.4.5 | Applications: The Sphere<br />
chapter 3.4.5 | Applications: The Sphere<br />
P’: (x’, y’)<br />chapter 3.5 | Working Backwards: From Image to Object<br />
chapter 3.5 | Working Backwards: From Image to Object<br />r’ = R <br />
chapter 3.4.5 | Binocular Vision<br />
?<br />chapter 3.4.5 | Binocular Vision<br />(x, y, z) ≡ <br />But does your brain know this formula?<br />
chapter 3.4.5 | Binocular Vision<br />
chapter 4 | The Circular and Spherical Scales<br />
chapter 4 | The Circular and Spherical Scales<br />
chapter 4.1 | Position<br />x’ =rfθ<br />
chapter 4.1 | Position<br />
chapter 4.2 | Applications: The Wall<br />z = d, y = ±h<br />But this graph is WRONG!<br />
chapter 4.2 | Applications: The Wall<br />
chapter 4.2 | Applications: The Wall<br />
chapter 5 | The Cylindrical Scale<br />
chapter 5.1 | Position<br />
chapter 5.2 | Applications: The Wall<br />z = d, y = ±h<br />
chapter 5.2 | Applications: The Wall<br />
chapter 6| The Semicircular and Hemispherical Scales<br />
chapter 6.1 | Position<br />
chapter 6.2 | Applications: The Wall<br />z = d, y = ±h<br />
chapter 6.2 | Applications: The Wall<br />
chapter 7| More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
chapter 7 | More on Binocular Vision<br />
Microsoft© Word TM and PowerPoint TM<br />MathCast<br />Copyright © 2004-2007 Tom Chakam<br />   www.mathcast.sf.net<br />...
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Perspective: the maths of seeing

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This is the two-part seminar I gave in college on my book that is being published. Hope you like it, although a lot of the information was delivered verbally and is not contained in the slides.

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Transcript of "Perspective: the maths of seeing"

  1. 1.
  2. 2. chapter 1 | The Necessity of Perspective<br />
  3. 3. chapter 1 | The Necessity of Perspective<br />
  4. 4. chapter 2 | Collinearity and Angle<br />
  5. 5. chapter 2 | Collinearity and Angle<br />
  6. 6. chapter 2 | Collinearity and Angle<br />
  7. 7. chapter 2 | Collinearity and Angle<br />
  8. 8. chapter 2 | Collinearity and Angle<br />
  9. 9. chapter 2 | Collinearity and Angle<br />
  10. 10. In a given plane, the perceived length of an object depends on the pair of light rays emanating from the two points of the object in that plane that subtend the greatest angle at the eye. <br />chapter 2 | Collinearity and Angle<br />
  11. 11. chapter 2 | Collinearity and Angle<br />Points that are collinear with the eye will appear to share the same position, and hence, appear to be the same point. <br />
  12. 12. The apparent dimension of an object does not depend directly on what its actual dimension is, nor at what distance from the eye it is placed. It depends directly on one, and only one thing: the angle subtended by it at the eye. The larger this angle compared to the angles of other objects, the larger it will look compared to them.<br />chapter 2 | Collinearity and Angle<br />
  13. 13. chapter 3 | The Linear and Planar Scales<br />
  14. 14. chapter 3 | The Linear and Planar Scales<br />
  15. 15. chapter 3 | The Linear and Planar Scales<br />
  16. 16. chapter 3 | The Linear and Planar Scales<br />
  17. 17. chapter 3 | The Linear and Planar Scales<br />
  18. 18. chapter 3 | The Linear and Planar Scales<br />
  19. 19. chapter 3 | The Linear and Planar Scales<br />
  20. 20. chapter 3 | The Linear and Planar Scales<br />
  21. 21. chapter 3 | The Linear and Planar Scales<br />
  22. 22. chapter 3 | The Linear and Planar Scales<br />
  23. 23. chapter 3.1 | Size<br />
  24. 24. chapter 3.1 | Size<br />
  25. 25. There’s a car whose breadth is 2 meters, and you hold a scale about 10 cm from your eye. <br /> 100 m away,<br /> 1 m away,<br />10,000 times!<br />chapter 3.1 | Size<br />
  26. 26. chapter 3.1 | Size<br />l’∝ df<br />
  27. 27. chapter 3.1 | Size<br />
  28. 28. chapter 3.1 | Size<br />5.1 mm, 1 m away<br />
  29. 29. chapter 3.1 | Size<br />n∆l, n∆l’<br />
  30. 30. chapter 3.1 | Size<br />
  31. 31. chapter 3.1 | Size<br />
  32. 32. chapter 3.2 | Position<br />
  33. 33. chapter 3.2 | Position<br />In ΔOP’N’, tanθ = x’/df⇒x’ =dftanθ.<br />
  34. 34. chapter 3.2 | Position<br />
  35. 35. chapter 3.2 | Position<br />
  36. 36. chapter 3.2 | Position<br />θ’ = θ.<br />
  37. 37. ?<br />chapter 3.2 | Position<br />(r’, θ’) ≡<br />But the vanishing point is not always <br />at the centre, along our line of sight.<br />
  38. 38. chapter 3.2 | Position<br />Use this with functions f(x,y,z,t) to project objects at a single go.<br />E = E(t)<br />
  39. 39. chapter 3.2 | Position<br />
  40. 40. chapter 3.2 | Position<br />
  41. 41. chapter 3.3 | Motion<br />For non-planar displacements, project end-points, then join.<br />
  42. 42. chapter 3.3 | Motion<br />
  43. 43. chapter 3.3 | Motion<br />
  44. 44. chapter 3.3 | Motion<br />
  45. 45. Review of Part 1<br />
  46. 46. Review of Part 1<br />
  47. 47. chapter 3.4.1 | Applications: The Cube<br />A cube of side-length 50cm<br />dx = 20cm<br />dy = 30cm<br />dz = 1m<br />The reference frame is at a distance df = 10cm, and graduated in meters.<br />
  48. 48. chapter 3.4.1 | Applications: The Cube<br />
  49. 49. chapter 3.4.1 | Applications: The Cube<br />
  50. 50. ?<br />chapter 3.4.1 | Applications: The Cube<br />But is it correct to just join the projected corners?<br />
  51. 51. chapter 3.4.1 | Applications: The Straight Line<br />x’ {m (ez- z1) – n (ey- y1)} – df m (ex - x1) = y’ {l (ez- z1) – n (ex - x1)} - df l (ey- y1).<br />
  52. 52. chapter 3.4.1 | Applications: The Straight Line<br />,<br />
  53. 53. chapter 3.4.1 | Applications: The Straight Line<br />(x’, y’) ≡ <br />(x’, y’, df)<br />= (x’ n/df, y’ n/df, n) = (l, m, n) <br />
  54. 54. chapter 3.4.1 | Applications: The Straight Line<br />
  55. 55. chapter 3.4.1 | Applications: The Straight Line<br />l1l2+m1m2+n1n2 = l1l3+m1m3+n1n3 = l2l3+m2m3+n2n3 = 0.<br />x1x2 + y1y2 = x1x3 + y1y3 = x2x3 + y2y3 = -df2.<br />
  56. 56. chapter 3.4.1 | Applications: The Straight Line<br />v1 (x1, y1)<br />v2 (x2, y2)<br />v3 (x3, y3)<br />v1.v2 = v1.v3 ⇒ v1.(v2-v3) = 0.<br />
  57. 57. chapter 3.4.1 | Applications: The Straight Line<br />v1<br />v2<br />v3<br />
  58. 58. chapter 3.4.2 | Applications: The Wall<br />z = mx + d, y = ±h.<br />
  59. 59. chapter 3.4.2 | Applications: The Wall<br />d = 2m<br />h = 1.5m<br />df = 10cm<br />θ = 60°<br />
  60. 60. chapter 3.4.3 | Applications: The Staircase<br />20 steps<br />
  61. 61. chapter 3.4.3 | Applications: The Staircase<br />10 steps above eye level<br />7 steps below<br />
  62. 62. chapter 3.4.3 | Applications: The Staircase<br />Till the 9th step below<br />
  63. 63. chapter 3.4.3 | Applications: The Staircase<br />The 10th step<br />
  64. 64. chapter 3.4.3 | Applications: The Staircase<br />
  65. 65. chapter 3.4.4 | Applications: The Circle<br />r = R, z= do<br />
  66. 66. chapter 3.4.4 | Applications: The Circle<br />x2 + y2 + (z-d)2 = r2, z = mx + d<br />
  67. 67. chapter 3.4.4 | Applications: The Circle<br />
  68. 68. chapter 3.4.4 | Applications: The Circle<br />
  69. 69. chapter 3.4.5 | Applications: The Sphere<br />x2 + y2 + (z-d)2 = r2<br />The perceived size of an object depends on the pairs of light rays emanating from it that subtend the greatest angle at the eye.<br />
  70. 70. chapter 3.4.5 | Applications: The Sphere<br />
  71. 71. chapter 3.4.5 | Applications: The Sphere<br />
  72. 72. P’: (x’, y’)<br />chapter 3.5 | Working Backwards: From Image to Object<br />
  73. 73. chapter 3.5 | Working Backwards: From Image to Object<br />r’ = R <br />
  74. 74. chapter 3.4.5 | Binocular Vision<br />
  75. 75. ?<br />chapter 3.4.5 | Binocular Vision<br />(x, y, z) ≡ <br />But does your brain know this formula?<br />
  76. 76. chapter 3.4.5 | Binocular Vision<br />
  77. 77. chapter 4 | The Circular and Spherical Scales<br />
  78. 78. chapter 4 | The Circular and Spherical Scales<br />
  79. 79. chapter 4.1 | Position<br />x’ =rfθ<br />
  80. 80. chapter 4.1 | Position<br />
  81. 81. chapter 4.2 | Applications: The Wall<br />z = d, y = ±h<br />But this graph is WRONG!<br />
  82. 82. chapter 4.2 | Applications: The Wall<br />
  83. 83. chapter 4.2 | Applications: The Wall<br />
  84. 84. chapter 5 | The Cylindrical Scale<br />
  85. 85. chapter 5.1 | Position<br />
  86. 86. chapter 5.2 | Applications: The Wall<br />z = d, y = ±h<br />
  87. 87. chapter 5.2 | Applications: The Wall<br />
  88. 88. chapter 6| The Semicircular and Hemispherical Scales<br />
  89. 89. chapter 6.1 | Position<br />
  90. 90. chapter 6.2 | Applications: The Wall<br />z = d, y = ±h<br />
  91. 91. chapter 6.2 | Applications: The Wall<br />
  92. 92. chapter 7| More on Binocular Vision<br />
  93. 93. chapter 7 | More on Binocular Vision<br />
  94. 94. chapter 7 | More on Binocular Vision<br />
  95. 95. chapter 7 | More on Binocular Vision<br />
  96. 96. chapter 7 | More on Binocular Vision<br />
  97. 97. chapter 7 | More on Binocular Vision<br />
  98. 98. chapter 7 | More on Binocular Vision<br />
  99. 99. chapter 7 | More on Binocular Vision<br />
  100. 100. Microsoft© Word TM and PowerPoint TM<br />MathCast<br />Copyright © 2004-2007 Tom Chakam<br /> www.mathcast.sf.net<br />MathGV<br />Graph<br /> www.padowan.dk<br />MetaCreations Bryce 4<br />Adobe© Photoshop TM<br />Credits<br />

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