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“Rosetón 1” Construction: Using angle and transforms to generate hyperbola-relat

“Rosetón 1” Construction: Using angle and transforms to generate hyperbola-relat
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2024-02-23 02:45
AngleVectorTransformationGeometric ConstructionLocus
In collections Mosaicos - Superficies - Rosetones
This GeoGebra example demonstrates the “Rosetón 1” geometric configuration: starting from angle constraints and a network of key points, it applies symmetry and rotation to build structured constructions, which then lead to a hyperbola-related curve/trajectory and clear angle relationships. The main mathematical ideas include geometric transformations (Mirror, Rotate), angle construction and angle bisectors (Angle, LineBisector), vector-based direction and orthogonal constructions (Vector, OrthogonalVector, OrthogonalLine), and intersection-based geometry (Line, Intersect). Circles and arcs (Circle, CircleArc) and polygonal scaffolding (Polygon) provide the geometric framework, while CurveCartesian supports rendering the final curve in Cartesian form. From the construction summary, the key workflow is: create direction vectors, derive orthogonal lines, enforce angle/bisector constraints, compute intersections, form circular auxiliary elements, then apply Mirror/Rotate to obtain the rosette-like arrangement, finishing with CurveCartesian to display the resulting curve behavior. Suitable for secondary to early university students, especially for lessons on geometric construction, transformation geometry, and curve visualization.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. OrthogonalVector(u) → v
  3. Polygon(A, B, 4) → poly1, n, r, t, a, N, O
  4. PointIn(poly1) → Puntopoly1
  5. Centroid(poly1) → C
  6. PointIn(poly1) → D
  7. Circle(C, D) → c
  8. LineBisector(A, B) → f
  9. OrthogonalLine(C, f) → g
  10. LineBisector(F, G) → h
  11. Intersect(f, h) → L
  12. Mirror(M, f) → M'
  13. LineBisector(M, G) → i
  14. Intersect(eq1, i) → P
  15. Polygon(M, F, M') → t1, m', m, f_1
  16. Rotate(M, 45°, C) → M'_{1}
  17. Rotate(F, 45°, C) → F'
  18. Rotate(M', 45°, C) → M''
  19. Polygon(M'_{1}, F', M'') → t1', m'', m'_{1}, f'
  20. Rotate(M'_{1}, 45°, C) → Q
  21. Rotate(F', 45°, C) → F''
  22. Rotate(M'', 45°, C) → M'''
  23. Polygon(Q, F'', M''') → t1'', m''', q, f''
  24. Mirror(M, g) → M'_{2}
  25. Mirror(F, g) → F'_{1}
  26. Mirror(M', g) → M''_{1}
  27. Polygon(M'_{2}, F'_{1}, M''_{1}) → t1'_{1}, m''_{1}, m'_{2}, f'_{1}
  28. Mirror(Q, C) → Q'
  29. Mirror(F'', C) → F'''
  30. Mirror(M''', C) → R
  31. Polygon(Q', F''', R) → t1''', r_1, q', f'''
  32. Rotate(Q', 45°, C) → Q''
  33. Rotate(F''', 45°, C) → S
  34. Rotate(R, 45°, C) → R'
  35. Polygon(Q'', S, R') → t2, r', q'', s
  36. Rotate(Q', -45°, C) → Q''_{1}
  37. Rotate(F''', -45°, C) → T
  38. Rotate(R, -45°, C) → R'_{1}
  39. Polygon(Q''_{1}, T, R'_{1}) → t3, r'_{1}, q''_{1}, t_1
  40. Mirror(Q''_{1}, f) → U
  41. Mirror(T, f) → T'
  42. Mirror(R'_{1}, f) → V
  43. Polygon(U, T', V) → t3', v_1, u_1, t'
  44. Rotate(E, 8°, C) → E'
  45. Rotate(E', 30°, C) → E''
  46. Rotate(E'', 15°, C) → E'''
  47. Rotate(E''', 30°, C) → K
  48. LineBisector(E'', E') → j
  49. Intersect(j, g_1) → W
  50. CircleArc(C, E', E'') → e_1
  51. Polygon(E'', W, E') → t4, l, b, w
  52. Rotate(E'', 45°, C) → E'''_{1}
  53. Rotate(W, 45°, C) → W'
  54. Rotate(E', 45°, C) → E''_{1}
  55. Polygon(E'''_{1}, W', E''_{1}) → t4', h_1, e'''_{1}, w'
  56. Mirror(E', f) → E''_{2}
  57. Mirror(E'', f) → E'''_{2}
  58. Segment(E''_{2}, E'''_{2}) → w'_{1}
  59. Mirror(E'', f) → E'''_{3}
  60. Mirror(W, f) → W'_{1}
  61. Mirror(E', f) → E''_{3}
  62. Polygon(E'''_{3}, W'_{1}, E''_{3}) → t4'_{1}, e''_{3}, e'''_{3}, i_1
  63. Mirror(E'''_{1}, f) → A_1
  64. Rotate(e_1, 45°, C) → k_1
  65. Mirror(k_1, f) → p_1
  66. Mirror(E'''_{1}, f) → B_1
  67. Mirror(W', f) → W''
  68. Mirror(E''_{1}, f) → C_1
  69. Polygon(B_1, W'', C_1) → t4'', j_1, b_1, l_1
  70. Mirror(e_1, f) → q_1
  71. Mirror(e_1, g) → s_1
  72. Mirror(s_1, f) → c_2
  73. Mirror(k_1, g) → d_2
  74. Mirror(d_2, f) → e_2
  75. Mirror(E'''_{1}, g) → D_1
  76. Mirror(W', g) → W''_{1}
  77. Mirror(E''_{1}, g) → o
  78. Polygon(D_1, W''_{1}, o) → t4''_{1}, m_1, n_1, w''_{1}
  79. Mirror(E'''_{1}, C) → F_1
  80. Mirror(W', C) → W''_{2}
  81. Mirror(E''_{1}, C) → G_1
  82. Polygon(F_1, W''_{2}, G_1) → t4''_{2}, a_1, f_2, w''_{2}
  83. Mirror(E'''_{3}, g) → H_1
  84. Mirror(W'_{1}, g) → I_1
  85. Mirror(E''_{3}, g) → J_1
  86. Polygon(H_1, I_1, J_1) → t5, g_2, h_2, i_2
  87. Mirror(E'', g) → E'''_{4}
  88. Mirror(W, g) → W'_{2}
  89. Mirror(E', g) → E''_{4}
  90. Polygon(E'''_{4}, W'_{2}, E''_{4}) → t4'_{2}, e''_{4}, e'''_{4}, w'_{2}
  91. Rotate(K_1, 22.5°, C) → L_1
  92. Rotate(L_1, 90°, C) → M_1
  93. Line(M'', M) → j_2
  94. Angle(D, C, H) → α
  95. CurveCartesian(((1 - t)^(3) * N_1) + (((3 * t) * (1 - t)^(2)) * O_1) + (((3 * (1 - t)) * t^(2)) * P_1) + (t^(3) * M_1), t, 0, 1) → o_1
  96. Mirror(o_1, j) → w_1
  97. Mirror(o_1, f) → a_2
  98. Mirror(w_1, f) → b_2
  99. Rotate(o_1, 45°, C) → k_2
  100. Rotate(w_1, 45°, C) → l_2
  101. Mirror(k_2, f) → m_2
  102. Mirror(l_2, f) → n_2
  103. Mirror(l_2, g) → p_2
  104. Mirror(k_2, g) → q_2
  105. Mirror(q_2, f) → r_2
  106. Mirror(p_2, f) → s_2
  107. Mirror(w_1, g) → t_2
  108. Mirror(o_1, g) → u_2
  109. Mirror(t_2, f) → v_2
  110. Mirror(u_2, f) → w_2
  111. Polygon(M, G, M') → t6, f_3, m_3, g_3
  112. Rotate(M, 45°, C) → M'_{3}
  113. Rotate(G, 45°, C) → G'
  114. Rotate(M', 45°, C) → M''_{2}
  115. Polygon(M'_{3}, G', M''_{2}) → t6', m''_{2}, m'_{3}, g'
  116. Rotate(M, 90°, C) → M'_{4}
  117. Rotate(G, 90°, C) → G'_{1}
  118. Rotate(M', 90°, C) → M''_{3}
  119. Polygon(M'_{4}, G'_{1}, M''_{3}) → t6'_{1}, m''_{3}, m'_{4}, g'_{1}
  120. Mirror(M, g) → M'_{5}
  121. Mirror(G, g) → G'_{2}
  122. Mirror(M', g) → M''_{4}
  123. Polygon(M'_{5}, G'_{2}, M''_{4}) → t6'_{2}, m''_{4}, m'_{5}, g'_{2}
  124. Rotate(M, 135°, C) → M'_{6}
  125. Rotate(G, 135°, C) → G'_{3}
  126. Rotate(M', 135°, C) → M''_{5}
  127. Polygon(M'_{6}, G'_{3}, M''_{5}) → t6'_{3}, m''_{5}, m'_{6}, g'_{3}
  128. Rotate(M, -45°, C) → M'_{7}
  129. Rotate(G, -45°, C) → G'_{4}
  130. Rotate(M', -45°, C) → M''_{6}
  131. Polygon(M'_{7}, G'_{4}, M''_{6}) → t6'_{4}, m''_{6}, m'_{7}, g'_{4}
  132. Rotate(M, -90°, C) → M'_{8}
  133. Rotate(G, -90°, C) → G'_{5}
  134. Rotate(M', -90°, C) → M''_{7}
  135. Polygon(M'_{8}, G'_{5}, M''_{7}) → t6'_{5}, m''_{7}, m'_{8}, g'_{5}
  136. Rotate(M, -135°, C) → M'_{9}
  137. Rotate(G, -135°, C) → G'_{6}
  138. Rotate(M', -135°, C) → M''_{8}
  139. Polygon(M'_{9}, G'_{6}, M''_{8}) → t6'_{6}, m''_{8}, m'_{9}, g'_{6}

AI Drawing Prompt Reference

Describe a rosette-shaped figure generated by symmetry and rotation transformations, containing a central point, multiple symmetric points, line segments, and a circle, with a hyperbola trajectory displayed. The figure is constructed by creating direction vectors, orthogonal vectors, angle bisectors, and circles, then applying mirror and rotation operations to generate points and connect them, forming a rosette arrangement. The curve part shows the characteristics of a hyperbola trajectory.
The prompt above was compiled from the original drawing commands and is for reference only.