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No-Pivot Turn-Flip: Rotation and Reflection Combined

No-Pivot Turn-Flip: Rotation and Reflection Combined
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2023-10-24 22:03
PolygonAngleVectorTransformationGeometric Construction
In collections Congruence Transformations
This GeoGebra example demonstrates what happens when you combine a rotation with a reflection—producing a “turn-flip” style transformation on a planar figure. It focuses on the geometric relationship between the original polygon and its transformed image, emphasizing Euclidean transformation composition. Key ideas include rigid motions, the reflection across a line (Mirror), rotation about a center (Rotate), and how corresponding points (e.g., A' , B' , C' …) are obtained and tracked under the composite map. Construction-wise, the model uses Point, Polygon, Segment, and Line to build the preimage geometry; Angle is used to control or display the rotation angle; Rotate and Mirror implement the two fundamental transformations, with a line of reflection defined by a Line object. Additional commands such as Translate, Vector, and the overall transform-object workflow help assemble the construction chain and visualize the final image after the combined rotation-reflection effect. Suitable for middle to early university geometry, interactive learning, and teaching transformation properties via direct manipulation.

GeoGebra Commands & Parameters

  1. Point(Circle[A, 1.2851459061133876]) → B
  2. Polygon(A, B, C, D, E, F) → preimage, a, b, c, d, e, f
  3. Vector(G, H) → Vector
  4. Line(I, J) → lineOfReflection
  5. Translate(A, Vector) → A'_{5}
  6. Translate(B, Vector) → B'_{5}
  7. Translate(C, Vector) → C'_{5}
  8. Translate(D, Vector) → D'_{5}
  9. Translate(E, Vector) → E'_{5}
  10. Translate(F, Vector) → F'_{5}
  11. Polygon(A'_{5}, B'_{5}, C'_{5}, D'_{5}, E'_{5}, F'_{5}) → translated, a', b', c', d', e', f'
  12. Mirror(A, lineOfReflection) → A'_{1}
  13. Mirror(B, lineOfReflection) → B'_{1}
  14. Mirror(C, lineOfReflection) → C'_{1}
  15. Mirror(D, lineOfReflection) → D'_{1}
  16. Mirror(E, lineOfReflection) → E'_{1}
  17. Mirror(F, lineOfReflection) → F'_{1}
  18. Polygon(A'_{1}, B'_{1}, C'_{1}, D'_{1}, E'_{1}, F'_{1}) → reflected, a'_{1}, b'_{1}, c'_{1}, d'_{1}, e'_{1}, f'_{1}
  19. Rotate(A, α, Center) → A'_{2}
  20. Rotate(B, α, Center) → B'_{2}
  21. Rotate(C, α, Center) → C'_{2}
  22. Rotate(D, α, Center) → D'_{2}
  23. Rotate(E, α, Center) → E'_{2}
  24. Rotate(F, α, Center) → F'_{2}
  25. Polygon(A'_{2}, B'_{2}, C'_{2}, D'_{2}, E'_{2}, F'_{2}) → rotated, a'_{2}, b'_{2}, c'_{2}, d'_{2}, e'_{2}, f'_{2}
  26. Line(K, lineOfReflection) → h
  27. Point(h) → L
  28. Vector(K, L) → Vector2
  29. Mirror(A, lineOfReflection) → A'_{3}
  30. Mirror(B, lineOfReflection) → B'_{3}
  31. Mirror(C, lineOfReflection) → C'_{3}
  32. Mirror(D, lineOfReflection) → D'_{3}
  33. Mirror(E, lineOfReflection) → E'_{3}
  34. Mirror(F, lineOfReflection) → F'_{3}
  35. Polygon(A'_{3}, B'_{3}, C'_{3}, D'_{3}, E'_{3}, F'_{3}) → intermediary, a'_{3}, b'_{3}, c'_{3}, d'_{3}, e'_{3}, f'_{3}
  36. Translate(A'_{3}, Vector2) → M
  37. Translate(B'_{3}, Vector2) → N
  38. Translate(C'_{3}, Vector2) → O
  39. Translate(D'_{3}, Vector2) → P
  40. Translate(E'_{3}, Vector2) → Q
  41. Translate(F'_{3}, Vector2) → R
  42. Polygon(M, N, O, P, Q, R) → image_{1}, m, n, o, p, q, r
  43. Line(S, T) → g
  44. Rotate(preimageList, angleAnimator, rotationCenter) → RFanimRotList
  45. Rotate(A, rotationAngle) → A'_{4}
  46. Rotate(C, rotationAngle) → C'_{4}
  47. Rotate(D, rotationAngle) → D'_{4}
  48. Rotate(E, rotationAngle) → E'_{4}
  49. Rotate(F, rotationAngle) → F'_{4}
  50. Rotate(A, rotationAngle, rotationCenter) → A'_{6}
  51. Rotate(C, rotationAngle, rotationCenter) → C'_{6}
  52. Rotate(D, rotationAngle, rotationCenter) → D'_{6}
  53. Rotate(E, rotationAngle, rotationCenter) → E'_{6}
  54. Rotate(F, rotationAngle, rotationCenter) → F'_{6}
  55. Mirror(A'_{6}, g) → W
  56. Mirror(C'_{6}, g) → F_1
  57. Mirror(D'_{6}, g) → G_1
  58. Mirror(E'_{6}, g) → H_1
  59. Mirror(F'_{6}, g) → I_1
  60. Rotate(preimageList, rotationAngle, rotationCenter) → RFintermList
  61. Mirror(RFintermList, g) → RFfinalList
  62. Mirror(preimageList, g) → FRintermList
  63. Rotate(FRintermList, rotationAngle, rotationCenter) → FRfinalList
  64. Rotate(FRintermList, angleAnimator, rotationCenter) → FRanimRotList
  65. Polygon(FRanimFlipList) → FRanimFlip
  66. Polygon(FRanimRotList) → FRanimRot
  67. Polygon(FRintermList) → FRinterm
  68. Polygon(RFanimRotList) → RFanimRot
  69. Polygon(RFintermList) → RFinterm
  70. Polygon(RFanimFlipList) → RFanimFlip
  71. Polygon(RFfinalList) → RFfinal
  72. Polygon(FRfinalList) → image
  73. Segment(rotationCenter, FRanimRotList(2)) → FRrotationSegment
  74. Segment(rotationCenter, RFanimRotList(2)) → RFrotSegment
  75. Line(rotationCenter, V) → i
  76. Rotate(V, rotationAngle / 2, rotationCenter) → V'
  77. Angle(V, rotationCenter, V') → β
  78. Line(rotationCenter, V') → line2

AI Drawing Prompt Reference

Draw a hexagon along with multiple transformed images, including the original polygon, a line of reflection, a rotation center, and an adjustable angle. Create a hexagon as the base figure, draw a line as the axis of reflection, and specify a rotation center with an angle value. Sequentially show: the translated polygon, the reflected polygon, the rotated polygon, and the final image obtained by reflecting then translating. Use the adjustable angle to control rotation, demonstrating the combined rotation and reflection transformation and how corresponding points change.
The prompt above was compiled from the original drawing commands and is for reference only.