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Mirror-and-rotate construction of piecewise parametric curves

Mirror-and-rotate construction of piecewise parametric curves
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2021-04-17 03:46
FunctionTransformationParametric EquationGeometric Construction
In collections GeoGebra and Laser Cutting
This example demonstrates how to construct and combine curves using piecewise-function ideas and parameter-geometry building blocks, highlighting the visual effect of symmetry obtained via mirroring and translation/rotation-style transformations. Key concepts include piecewise functions, parabola-related geometry, parametric curves/equations, and the systematic use of geometric transformations to generate matching curve segments. In the construction, GeoGebra commands such as Point, Segment, OrthogonalLine, Intersect, Midpoint, Mirror, Circle, Vector, and Translate are used, along with many dependent geometric objects (e.g., key points A–N, auxiliary lines/functions like f, g, h and their indexed variants, and mirrored/translated counterparts such as F', E', l'). Suitable for advanced secondary students and introductory university learners studying geometric construction, coordinate geometry, and transformation-driven curve modeling.

GeoGebra Commands & Parameters

  1. Point(g_3) → A
  2. Point(g_3) → B
  3. Point(xAxis) → C
  4. Point(xAxis) → D
  5. Segment(A, C) → h
  6. Segment(C, D) → i
  7. Segment(D, B) → j
  8. OrthogonalLine(E, i) → k
  9. Intersect(g_3, k, (-9.7, 4.000081982889263)) → G
  10. Midpoint(G, E) → F
  11. Segment(F, E) → l
  12. Mirror(F, yAxis) → F'
  13. Mirror(E, yAxis) → E'
  14. Segment(F', E') → l'
  15. Circle(H, 4.5) → c
  16. Intersect(c, xAxis, 1) → J
  17. OrthogonalLine(J, xAxis) → m
  18. OrthogonalLine(I, m) → n
  19. Intersect(m, n) → K
  20. Segment(I, H) → p
  21. Segment(H, J) → q
  22. Segment(K, J) → r
  23. Segment(K, I) → s
  24. Midpoint(I, H) → M
  25. Segment(L, N) → g_1
  26. Segment(N, M) → h_1
  27. Segment(K, L) → k_1
  28. Segment(K, J) → t
  29. Segment(J, H) → a
  30. Circle(H, 0.3) → d
  31. Segment(M, H) → b
  32. Intersect(d, b, 1) → O
  33. OrthogonalLine(O, b) → e
  34. Intersect(e, t) → P
  35. Midpoint(P, O) → Q
  36. Segment(Q, O) → f_1
  37. Circle(J, 0.3) → c_1
  38. Intersect(c_1, a, 1) → S
  39. Intersect(a, xAxis) → T
  40. OrthogonalLine(S, a) → i_1
  41. Circle(S, l) → d_1
  42. Intersect(d_1, i_1, 2) → R
  43. Segment(S, R) → j_1
  44. OrthogonalLine(Ponto(EixoY), yAxis) → l_1
  45. Mirror(L, l_1) → L'
  46. Mirror(N, l_1) → N'
  47. Segment(L', N') → m_1
  48. Mirror(N, l_1) → N'_1
  49. Mirror(M, l_1) → M'
  50. Segment(N'_1, M') → n_1
  51. Mirror(K, l_1) → K'
  52. Mirror(L, l_1) → L'_1
  53. Segment(K', L'_1) → p_1
  54. Mirror(K, l_1) → K'_1
  55. Mirror(J, l_1) → J'
  56. Segment(K'_1, J') → t'
  57. Mirror(J, l_1) → J'_1
  58. Mirror(H, l_1) → H'
  59. Segment(J'_1, H') → a'
  60. Mirror(M, l_1) → M'_1
  61. Mirror(H, l_1) → H'_1
  62. Segment(M'_1, H'_1) → b'
  63. Mirror(Q, l_1) → Q'
  64. Mirror(O, l_1) → O'
  65. Segment(Q', O') → q_1
  66. Mirror(S, l_1) → S'
  67. Mirror(R, l_1) → R'
  68. Segment(S', R') → r_1
  69. Mirror(A, l_1) → A'
  70. Mirror(C, l_1) → C'
  71. Segment(A', C') → h'
  72. Mirror(C, l_1) → C'_1
  73. Mirror(D, l_1) → D'
  74. Segment(C'_1, D') → i'
  75. Mirror(D, l_1) → D'_1
  76. Mirror(B, l_1) → B'
  77. Segment(D'_1, B') → j'
  78. Mirror(F, l_1) → F'_1
  79. Mirror(E, l_1) → E'_1
  80. Segment(F'_1, E'_1) → l'_1
  81. Mirror(F', l_1) → F''
  82. Mirror(E', l_1) → E''
  83. Segment(F'', E'') → l''
  84. Intersect(s_1, h, (-10, 0.6967855704262355)) → U
  85. Intersect(h', i') → V
  86. Vector(U, V) → u
  87. Translate(s_1, u) → t_1
  88. Segment(A', B') → a_1
  89. Circle(B', 0.4) → e_1
  90. Intersect(e_1, j', 1) → W
  91. OrthogonalLine(W, j') → b_1
  92. Intersect(b_1, h') → Z
  93. Segment(Z, W) → g_2
  94. Segment(C', Z) → h_2
  95. Segment(D'_1, W) → i_2
  96. Segment(A_1, B_1) → j_2
  97. Circle(A_1, 4.22) → c_2
  98. OrthogonalLine(A_1, j_2) → k_2
  99. Intersect(c_2, k_2, 1) → C_1
  100. OrthogonalLine(C_1, k_2) → l_2
  101. OrthogonalLine(B_1, j_2) → m_2
  102. Intersect(l_2, m_2) → D_1
  103. Segment(A_1, C_1) → n_2
  104. Segment(B_1, D_1) → p_2
  105. Segment(C_1, D_1) → q_2
  106. Circle(A_1, 0.3) → d_2
  107. Intersect(d_2, j_2, 1) → E_1
  108. Circle(E_1, q_1 - 0.28) → e_2
  109. Intersect(e_2, k, 1) → F_1
  110. Segment(E_1, F_1) → r_2
  111. Mirror(E_1, yAxis) → G_1
  112. Mirror(F_1, yAxis) → H_1
  113. Segment(G_1, H_1) → s_2
  114. Intersect(g_3, yAxis, (0, 5)) → I_1
  115. Circle(I_1, 0.5) → t_2
  116. Intersect(t_2, yAxis) → J_1, K_1
  117. Circle(J_1, 0.15) → c_3

AI Drawing Prompt Reference

Draw a coordinate plane with a horizontal axis as reference. Construct a few key points on a given parameter-defined curve and connect them into short segments, then erect an auxiliary line through a chosen point that is perpendicular to one of those segments; find their intersection with the parameter curve and take a midpoint to form a segment intended for symmetry. Mirror that segment across the vertical axis and connect the mirrored endpoints to create matching curve/segment parts. Also draw a circle centered at the main point with radius 4.5, intersect it with the horizontal axis to get a secondary point, then draw perpendiculars to form further intersection points. Use smaller circles and perpendicular lines to generate additional intersection points, join them with segments and midpoints, and finally mirror the later constructed segment set across the vertical axis to produce a clear piecewise, symmetry-based composite curve.
The prompt above was compiled from the original drawing commands and is for reference only.