MathPlotAI Logo MathPlotAI

Polygonal Wheel Bicycle

Polygonal Wheel Bicycle
Loading the math board and drawing, please wait…
2020-02-03 20:33
PolygonVectorFunctionSequence
This GeoGebra example dynamically demonstrates the rolling motion of a bicycle with polygonal wheels, revealing uniform motion phenomena with non-circular wheel shapes. Core mathematical concepts include polygonal geometry, trajectory planning, vector transformations, and function sequences. Construction heavily utilizes GeoGebra commands such as If, Function, Sequence, Element, Line, Vector, Translate, Segment, Midpoint, and Polygon, along with objects like points, polygons, and vectors to simulate complex motion. Suitable for high school function and geometry units, or for exploring the motion patterns and curve characteristics of non-circular wheels.

GeoGebra Commands & Parameters

  1. If(Mod[n, 2] ≟ 0, (a * n), (a * n) - 1) → aa
  2. Function(cr - (ir * cosh(x / ir)), (-a), a) → f
  3. Function(cr - (ir * cosh((x - (2 * a)) / ir)), a, (3 * a)) → g
  4. Function(cr - (ir * cosh((x - (4 * a)) / ir)), (3 * a), (5 * a)) → h
  5. Function(cr - (ir * cosh((x - (6 * a)) / ir)), (5 * a), (7 * a)) → p
  6. Function(cr - (ir * cosh((x - (8 * a)) / ir)), (7 * a), (9 * a)) → q
  7. Function(cr - (ir * cosh((x - (10 * a)) / ir)), (9 * a), (11 * a)) → r
  8. Sequence(((cr * cos(i)), (cr * sin(i)) + cr), i, -3.14 / 2 + 3.14 / n, (3 * 3.14 / 2) + 3.14 / n, (2 * 3.14 / n)) → S
  9. Sequence((t, 0) + (0, cr) + Rotate[Element[S, i] - (0, cr), ((loop * 2) * 3.14 / n) + atan(sinh((((loop * 2) * a) - t) / ir))], i, 1, n + 1) → P
  10. Element(P, 3) → G
  11. Sequence({Polygon[M, Element[S, i], Element[S, i + 1]]}, i, 1, Length[S] - 1) → L1
  12. Sequence({Polygon[H, Element[P, i], Element[P, i + 1]]}, i, 1, Length[P] - 1) → L2
  13. Sequence(Segment[Element[P, i], Element[P, i + 1]], i, 1, n) → L3
  14. Sequence({Polygon[H, Element[P, i], Element[P, i + 1]]}, i, 1, Length[P] - 1) → Color1
  15. Sequence({Polygon[H, Element[P, i], Element[P, i + 1]]}, i, 1, Length[P] - 1) → Color2
  16. Sequence({Polygon[H, Element[P, i], Element[P, i + 1]]}, i, 1, Length[P] - 1) → Color3
  17. Sequence(Segment[H, Element[P, i]], i, 1, n) → L4
  18. Line(H, xAxis) → b
  19. Sequence(Function[cr - (ir * cosh((x - ((i + 1) * a)) / ir)), (i * a), (i * a)], i, 1, 19, 2) → liste1
  20. Function(cr - (ir * cosh((x - (12 * a)) / ir)), (11 * a), (13 * a)) → s
  21. Function(cr - (ir * cosh((x - (14 * a)) / ir)), (13 * a), (15 * a)) → f_1
  22. Function(cr - (ir * cosh((x - (16 * a)) / ir)), (15 * a), (17 * a)) → g_1
  23. Function(cr - (ir * cosh((x - (18 * a)) / ir)), (17 * a), (19 * a)) → h_1
  24. Function(cr - (ir * cosh((x - (20 * a)) / ir)), (19 * a), (21 * a)) → p_1
  25. Function(cr - (ir * cosh((x - (22 * a)) / ir)), (21 * a), (23 * a)) → q_1
  26. Function(cr - (ir * cosh((x - (24 * a)) / ir)), (23 * a), (25 * a)) → r_1
  27. Function(cr - (ir * cosh((x - (26 * a)) / ir)), (25 * a), (27 * a)) → s_1
  28. Function(cr - (ir * cosh((x - (28 * a)) / ir)), (27 * a), (29 * a)) → t_1
  29. Function(cr - (ir * cosh((x - (30 * a)) / ir)), (29 * a), (31 * a)) → f_2
  30. Function(cr - (ir * cosh((x - (32 * a)) / ir)), (31 * a), (33 * a)) → g_2
  31. Function(cr - (ir * cosh((x - (34 * a)) / ir)), (33 * a), (35 * a)) → h_2
  32. Function(cr - (ir * cosh((x - (36 * a)) / ir)), (35 * a), (37 * a)) → p_2
  33. Vector(H, A) → u
  34. Translate(Color3, u) → Color3'
  35. Segment(C, A) → e
  36. Segment(D, H) → i
  37. Segment(H, E) → j
  38. Segment(D, E) → k
  39. Midpoint(C, A) → F
  40. Midpoint(F, C) → I
  41. Midpoint(I, F) → J
  42. Segment(E, J) → l
  43. Segment(D, I) → m
  44. Segment(K, D) → a_1
  45. Polygon(K, N, O) → poly1, o, k_1, n_1
  46. CircleSector(K, O, L) → c_1
  47. Point(Color3') → Q
  48. Intersect(i, j) → R
  49. Segment(R, E) → b_1
  50. Sequence(Segment[Element[P, i], H], i, 1, Length[P], v) → liste2

AI Drawing Prompt Reference

Draw a bicycle model showing the rolling motion of non-circular wheels. The wheels are composed of multiple polygons, each generated by a cosine hyperbolic function curve. The bicycle frame connects two wheels, and the trajectory of points on the frame forms complex curves as the wheels roll. Display multiple function curves, each corresponding to a wheel's trajectory, connected by segmented functions to form a continuous path. The points on the frame move in sync with the wheel edges, demonstrating the motion characteristics of non-circular wheels.
The prompt above was compiled from the original drawing commands and is for reference only.