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Polar Coordinates Boat Game

Polar Coordinates Boat Game
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2018-10-07 14:32
CirclePolygonAngleTrigonometryCoordinate Geometry
This GeoGebra worksheet demonstrates a boat game based on polar coordinates, where a boat's position and heading are defined by a radius (distance from the origin) and an angle. The core mathematical concepts include polar coordinates, unit circle, angle measurement, rotation, and random generation. Construction uses the commands Random (to set the initial boat position), Circle (to draw circular trajectories), Ray and OrthogonalLine (to create directional lines), Intersect (to find intersections), CircumcircleArc (to draw arcs), Midpoint (to locate midpoints), Sequence (to generate series of points), If (for conditional logic), Angle (to compute angles), Segment (to draw line segments), Polygon (to form polygons), Line (to draw straight lines), Rotate (to rotate objects), Mod (for modulo operations), and Point (to place key points). The example is suitable for high school mathematics teaching of coordinate geometry and trigonometry, as well as for learning GeoGebra construction techniques.

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

Create a polar coordinate-based boat game illustration with a central circle for trajectories, a randomly generated initial boat position, rays and perpendicular lines defining direction, intersection points with the circle, connected by arcs, text labels showing angles and distances, boat paths formed by multiple circles, a background with a unit circle and concentric circles, depicting a dynamic polar coordinate game scene.
The prompt above was compiled from the original drawing commands and is for reference only.