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Polar Coordinates Boat Game by Woogon Chung

Polar Coordinates Boat Game by Woogon Chung
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2011-10-22 18:18
TrigonometryCoordinate GeometryGeometric ConstructionSlider AnimationLocus
In collections Polar Coordinates
This example demonstrates the use of polar coordinates to locate a hidden boat on a 2D plane. It involves core mathematical concepts such as polar coordinate transformations, circle-line intersection calculations, random number generation, and geometric constructions. GeoGebra commands and objects used include Random, Circle, Ray, OrthogonalLine, Intersect, CircumcircleArc, Midpoint, Sequence, If, Angle, Segment, Polygon, Line, Rotate, Mod, and Point. It is suitable for high school mathematics classes to illustrate the application of polar coordinates, stochastic processes, and analytical geometry, as well as for math clubs and activities.

GeoGebra Commands & Parameters

  1. Random(4.0, 15.0) → rr
  2. Circle(A, (1.5 * v - 1.0) * rr) → c'
  3. Ray(A, (0.0, 0.0)) → a
  4. OrthogonalLine(A, a) → b
  5. Intersect(c', b) → B, C
  6. Intersect(c', a) → D, T
  7. CircumcircleArc(B, D, C) → d
  8. Midpoint(C, D) → E
  9. Midpoint(B, D) → F_2
  10. Ray(A', E) → e
  11. Ray(A', F_2) → f
  12. Intersect(d, e) → G,
  13. Intersect(d, f) → H,
  14. CircumcircleArc(G, D, H) → g
  15. Circle(I, 1.0) → h
  16. Sequence(Text[If[s < 10.0, " ", ""] + (s), (21.6 + 2.0 * Mod[s - 1.0, 3.0], 7.75 - 2.0 * Div[s - 1.0, 3.0])], s, 1.0, 12.0) → liste3
  17. Circle(J, 1.0) → k
  18. Sequence(Text[If[s ≟ 1.0, " ", If[s < 5.0, " ", ""]] + (30.0 * (s - 1.0)) + "º", (21.25 + 2.0 * Mod[s - 1.0, 3.0], -(6.25) - 2.0 * Div[s - 1.0, 3.0])], s, 1.0, 12.0) → liste4
  19. If(yB > -(5.0), 5.0*°, ((xB - 3.0 * yB) / 2.0 - 19.0 - 1.0) * 30.0°) → ang
  20. If(ang ≟ α ∧ dist ≟ rr, true, false) → raz
  21. Angle((1.0, 0.0), (0.0, 0.0), A) → β
  22. Segment(O, N) → m
  23. Circle(A, (2.0 * v - 1.0) * rr - n) → p
  24. Sequence(Circle[A, (1.5 * v - 1.0) * rr - i], i, 0.0, n, n / 10.0) → liste5
  25. Polygon(P, Q, R, S) → poly1, p_1, q, r_1, s
  26. Angle((1.0, 0.0), (0.0, 0.0), A) → γ
  27. Circle((0.0, 0.0), If[2.0 * v * rr > 12.0, 0.0, 2.0 * v * rr]) → c
  28. Midpoint(K, L) → U
  29. Midpoint(U, L) → V
  30. Midpoint(U, V) → W
  31. Midpoint(V, L) → Z
  32. Segment(K, C_1) → l
  33. Line(A, l) → i
  34. Segment(A, U) → t
  35. Intersect(c, t) → A_1, B_1
  36. Segment(U, A_1) → a_1
  37. Circle(U, W) → c_1
  38. Circle(U, V) → d_1
  39. Circle(U, Z) → e_1
  40. Circle(U, L) → f_1
  41. Rotate(C_1, 6.0 * v, U) → L'
  42. Segment(U, L') → b_1
  43. Mod(v * 100.0, 2.0) → ab
  44. Sequence(Circle[(0.0, 0.0), t], t, 1.0, 15.0) → liste1
  45. Sequence(Segment[(0.0, 0.0), (15.0; 2.0 * 3.141592653589793 * t / 36.0)], t, 0.0, 37.0) → liste2
  46. Segment(O, D_1) → j
  47. Circle(O, C_1) → g_1
  48. Point(g_1) → E_1
  49. Segment(O, E_1) → h_1
  50. Circle(U, C_1) → k_1
  51. Circle(F_1, 8.0 - r / 2.5) → q_1
  52. Line(F_1, l) → i_1
  53. Intersect(q_1, i_1) → G_1, H_1
  54. Line(J_1, r_1) → j_1
  55. Point(j_1) → N_1
  56. Line(N_1, l) → l_1
  57. Line(J_1, l) → m_1
  58. Point(m_1) → O_1
  59. Point(l_1) → P_1
  60. Intersect(l_1, r_1) → M_1
  61. Polygon(M_1, N_1, J_1, F) → poly2, m_2, n_1, j_2, f_2
  62. Point(l_1) → Q_1
  63. Point(m_1) → R_1
  64. Polygon(P_1, Q_1, R_1, O_1) → poly3, p_2, q_2, r_2, o_1
  65. Point(j_1) → S_1
  66. Point(o_1) → T_1
  67. Polygon(S_1, T_1, O_1, J_1) → poly4, s_1, t_1, o_2, j_3
  68. Midpoint(s_1) → V_1
  69. Midpoint(o_2) → W_1
  70. Midpoint(S_1, J_1) → Z_1
  71. Midpoint(t_1) → A_2
  72. Segment(V_1, W_1) → a_2
  73. Segment(Z_1, A_2) → b_2
  74. Intersect(b_2, a_2) → B_2
  75. Circle(B_2, V_1) → c_2

AI Drawing Prompt Reference

On a 2D plane, use polar coordinates to locate a hidden boat. Start by drawing a ray and its perpendicular line, then create a circle based on a randomly generated polar coordinate. Find the intersection points of the circle with the ray. Use these points to draw arcs and segments to form the boat's shape. Add a slider to control the circle's size and position, and display angle information as text to illustrate the application of polar coordinates.
The prompt above was compiled from the original drawing commands and is for reference only.