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Cycloid

Cycloid
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2015-04-06 14:40
Coordinate GeometryCalculusDerivativeIntegralParametric Equation
This example demonstrates the cycloid, the path traced by a point on the rim of a circular wheel rolling along a straight line without slippage. It involves core math concepts like parametric equations, coordinate geometry, derivatives, integrals, and limits. Construction highlights include commands like Circle, Segment, Line, OrthogonalLine, Intersect, Mirror, Point, and Midpoint, as per the GeoGebra summary. Suitable for high school students and math enthusiasts exploring curve constructions and related theories.

GeoGebra Commands & Parameters

  1. Circle(C, r) → c
  2. Segment(P, C) → a
  3. Line(P, C) → b
  4. OrthogonalLine(C, b) → d
  5. Intersect(d, b) → A
  6. Intersect(c, d) → B, D
  7. Intersect(c, b) → E, F
  8. Line(D, C) → e
  9. Line(F, C) → f
  10. Line(C, B) → g
  11. Segment(C, D) → h
  12. Segment(C, F) → i
  13. Segment(C, B) → j
  14. Mirror(c, yAxis) → c'
  15. Mirror(P, yAxis) → P'
  16. Point(xAxis) → G
  17. OrthogonalLine(G, xAxis) → k
  18. Mirror(c', k) → c''
  19. Mirror(P', k) → P''
  20. Mirror(C, yAxis) → C'
  21. Mirror(B, yAxis) → B'
  22. Segment(C', B') → j'
  23. Mirror(C, yAxis) → C'_1
  24. Mirror(D, yAxis) → D'
  25. Segment(C'_1, D') → h'
  26. Mirror(C, yAxis) → C'_2
  27. Mirror(B, yAxis) → B'_1
  28. Segment(C'_2, B'_1) → j'_1
  29. Mirror(C', k) → C''
  30. Mirror(C', k) → C''_1
  31. Mirror(B', k) → B''
  32. Segment(C''_1, B'') → j''
  33. Mirror(C'_1, k) → H
  34. Mirror(D', k) → D''
  35. Segment(H, D'') → h''
  36. Mirror(P, yAxis) → P'_1
  37. Mirror(C, yAxis) → C'_3
  38. Segment(P'_1, C'_3) → a'
  39. Mirror(P'_1, k) → I
  40. Mirror(C'_3, k) → J
  41. Segment(I, J) → a''
  42. Mirror(C, yAxis) → C'_4
  43. Mirror(F, yAxis) → F'
  44. Segment(C'_4, F') → i'
  45. Mirror(C'_4, k) → K
  46. Mirror(F', k) → F''
  47. Segment(K, F'') → i''
  48. Segment(C, C'') → l
  49. Midpoint(C'', C) → L
  50. OrthogonalLine(L, l) → m
  51. OrthogonalLine(C, l) → n
  52. OrthogonalLine(C'', l) → p
  53. Point(m) → M
  54. Line(C, M) → q
  55. Line(C'', M) → s
  56. OrthogonalLine(M, m) → a_1
  57. Intersect(a_1, p) → N
  58. Intersect(n, a_1) → O
  59. Line(N, L) → b_1
  60. Line(O, L) → c_1
  61. Line(N, s) → d_1
  62. Line(C, c_1) → e_1
  63. Midpoint(N, M) → Q
  64. Midpoint(M, O) → R
  65. Intersect(a_1, e_1) → S
  66. Midpoint(O, S) → T
  67. Segment(R, T) → f_1
  68. Midpoint(R, O) → U
  69. Midpoint(O, T) → V
  70. Segment(U, V) → g_1
  71. Segment(U, V) → h_1
  72. Segment(V, U) → i_1
  73. Intersect(b_1, s) → W
  74. Midpoint(N, W) → Z
  75. Circle(N, Z) → k_1
  76. Intersect(k_1, b_1) → A_1, B_1
  77. Segment(B_1, Z) → j_1
  78. Segment(N, C'') → l_1
  79. Segment(O, C) → m_1
  80. Segment(U, V) → n_1
  81. Segment(V, U) → p_1
  82. Segment(O, V) → q_1
  83. Segment(T, R) → r_1
  84. Segment(T, U) → s_1
  85. Segment(V, U) → t_1
  86. Intersect(k_1, a_1) → C_1, D_1
  87. Segment(N, C_1) → a_2
  88. Intersect(q, c_1) → E_1
  89. Segment(E_1, C) → b_2
  90. Segment(W, C'') → c_2
  91. Segment(W, C'') → d_2
  92. Midpoint(M, E_1) → F_1
  93. Midpoint(M, W) → G_1
  94. Line(F_1, G_1) → e_2
  95. Intersect(n, e_2) → H_1
  96. Intersect(p, e_2) → I_1
  97. Segment(H_1, I_1) → f_2
  98. Segment(F_1, E_1) → g_2
  99. Segment(W, G_1) → h_2
  100. Segment(F_1, E_1) → i_2
  101. Segment(F_1, E_1) → j_2
  102. Circle(C, r / 3) → Circ
  103. Intersect(Circ, n, 1) → J_1
  104. Intersect(n, m_1) → K_1
  105. Intersect(n, m_1) → L_1
  106. Intersect(n, m_1) → M_1
  107. Intersect(n, m_1) → N_1
  108. Intersect(Circ, n, 2) → O_1
  109. Segment(O_1, C'') → k_2
  110. Segment(J_1, C'') → l_2
  111. Intersect(d_1, e_2) → P_1
  112. Segment(I_1, P_1) → m_2
  113. Segment(I_1, P_1) → n_2
  114. Segment(I_1, P_1) → p_2
  115. Segment(I_1, P_1) → q_2
  116. Intersect(d_1, xAxis) → Q_1
  117. Intersect(s, xAxis) → R_1
  118. Line(Q_1, R_1) → r_2
  119. Intersect(f_2, m_1) → S_1

AI Drawing Prompt Reference

Draw the path traced by a point on the rim of a circular wheel rolling along a straight line without slippage, known as a cycloid. Start by sketching a circle and the line connecting the circle's center to the straight line. Then find the intersection points of the circle with this line and draw the perpendicular line from the circle's center. Next, locate the intersection points of the circle with the perpendicular line and draw lines from the center to these points. Finally, reflect the circle vertically and connect the reflected key points to form the cycloid.
The prompt above was compiled from the original drawing commands and is for reference only.