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Ellipse Parametric Curve with Rotation Transformation

Ellipse Parametric Curve with Rotation Transformation
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2023-03-22 02:34
CircleTrigonometryTransformationParametric EquationGeometric Construction
In collections Realidades de GeoGebra
This GeoGebra example demonstrates drawing a parametric ellipse curve that rotates with changing angle. The core mathematical concepts include parametric equations, geometric properties of ellipses, trigonometric functions in coordinate transformations, and rotation/translation transformations. Construction key points: Use CurveCartesian command to define the parametric ellipse, employ Sequence and Element to generate serialized curve point sets, draw connecting lines with Segment commands, and implement dynamic rotation transformation through the rotation angle zAngRot. Suitable for high school and above students learning parametric equations, coordinate geometry, and transformation concepts, also useful for teachers demonstrating relationships between parametric curves and geometric transformations.

GeoGebra Commands & Parameters

  1. Ellipse((-6.62, 5), (-5.38, 5), 0.65) → zElipse1
  2. Point(zElipse1) → XY
  3. CurveCartesian(((3 * cos(t)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(p / 30) * t) + 5, t, (-34 * p), 0) → h0
  4. Sequence(h0(((-i) * p / 3)), i, 0, 98) → L
  5. Sequence(Circle[h0(((-i) * p / 3)) + (0.19, 0.1), h0(((-i) * p / 3))], i, 0, 98) → L1
  6. Element(L1, 2) → p01
  7. Element(L1, 3) → p02
  8. Element(L1, 5) → p03
  9. Element(L1, 7) → p04
  10. Element(L1, 11) → p05
  11. Element(L1, 13) → p06
  12. Element(L1, 17) → p07
  13. Element(L1, 19) → p08
  14. Element(L1, 23) → p09
  15. Element(L1, 29) → p10
  16. Element(L1, 31) → p11
  17. Element(L1, 37) → p12
  18. Element(L1, 41) → p13
  19. Element(L1, 43) → p14
  20. Element(L1, 47) → p15
  21. Element(L1, 53) → p16
  22. Element(L1, 59) → p17
  23. Element(L1, 61) → p18
  24. Element(L1, 67) → p19
  25. Element(L1, 71) → p20
  26. Element(L1, 73) → p21
  27. Element(L1, 79) → p22
  28. Element(L1, 83) → p23
  29. Element(L1, 89) → p24
  30. Element(L1, 97) → p25
  31. Element(L1, 10) → m502
  32. Element(L1, 15) → m503
  33. Element(L1, 20) → m504
  34. Element(L1, 25) → m505
  35. Element(L1, 30) → m506
  36. Element(L1, 35) → m507
  37. Element(L1, 40) → m508
  38. Element(L1, 45) → m509
  39. Element(L1, 50) → m510
  40. Element(L1, 55) → m511
  41. Element(L1, 60) → m512
  42. Element(L1, 65) → m513
  43. Element(L1, 70) → m514
  44. Element(L1, 75) → m515
  45. Element(L1, 80) → m516
  46. Element(L1, 85) → m517
  47. Element(L1, 90) → m518
  48. Element(L1, 95) → m519
  49. Element(L1, 14) → m702
  50. Element(L1, 21) → m703
  51. Element(L1, 28) → m704
  52. Element(L1, 35) → m705
  53. Element(L1, 42) → m706
  54. Element(L1, 49) → m707
  55. Element(L1, 56) → m708
  56. Element(L1, 63) → m709
  57. Element(L1, 70) → m710
  58. Element(L1, 77) → m711
  59. Element(L1, 84) → m712
  60. Element(L1, 91) → m713
  61. Element(L1, 98) → m714
  62. Element(L1, 22) → m112
  63. Element(L1, 33) → m113
  64. Element(L1, 44) → m114
  65. Element(L1, 55) → m115
  66. Element(L1, 66) → m116
  67. Element(L1, 77) → m117
  68. Element(L1, 88) → m118
  69. Element(L1, 99) → m119
  70. Segment(h0(((-p)) / 3) + (0.2, -0.12), h0((-97 * p / 3)) + (0.2, -1)) → s2a
  71. Segment(h0((-p)) + (0.2, 0.1), h0((-93 * p / 3)) + (0.2, -1)) → s2b
  72. Segment(h0((-5 * p / 3)) + (0.15, 0.1), h0((-95 * p / 3)) + (0.15, -1)) → s2c
  73. Segment(h0((-2 * p / 3)) + (0.2, -0.12), h0((-98 * p / 3)) + (0.2, -1)) → s3a
  74. Segment(h0((-5 * p / 3)) + (0.2, 0.1), h0((-95 * p / 3)) + (0.2, -1)) → s3b
  75. CurveCartesian(0.2 + ((3 * cos(t + p)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(27.7°) * t) + 10.7, t, (-11 * p), (-11 * p / 3) - 0.1) → h5
  76. CurveCartesian(0.2 + ((3 * cos(t)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(36.4°) * t) + 13.7, t, (-17 * p / 2), (-4 * p) - 0.1) → h7
  77. CurveCartesian(0.2 + ((3 * cos(t + p)) * sin(zAngRot)) + ((3 * sin(t)) * cos(zAngRot)), (tan(49°) * t) + 17.2, t, (-33 * p / 5), (-11 * p / 3) - 0.1) → h11
  78. Ellipse((-3, 5.5), (3.4, 5.5), 3.21) → base

AI Drawing Prompt Reference

Draw a rotating parametric curve ellipse to demonstrate parametric equations, trigonometric functions, and rotation transformations. Start by plotting an ellipse, then create a curve based on a parametric equation that rotates as the angle changes. Use sequences to generate points on the curve and connect them with line segments to form a dynamically rotating ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.