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Wheel Paradox

Wheel Paradox
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2019-02-06 19:33
FunctionTrigonometryCoordinate GeometryGeometric Construction
In collections MalinC's GeoGebra-book
This example demonstrates the relationship between the path traced by a point on the circumference of a rolling wheel and the contact point with the ground, revealing geometric illusions in motion. It involves core mathematical concepts like cosine functions, intersection calculations, area measurement, confidence intervals, and quadratic functions. The construction utilizes GeoGebra commands such as conditional statements (If), orthogonal line creation, intersection finding, midpoint calculation, circle arc drawing, segment connection, and circle generation. Suitable for high school and above students to explore dynamic relationships between analytical geometry, function graphs, and geometric construction techniques.

GeoGebra Commands & Parameters

  1. If(n < 50, 0, If[n < 200, (((n - 50) * 2) * π / 150), 6.28319]) → a
  2. OrthogonalLine(M, xAxis) → b
  3. Intersect(b, xAxis) → C1
  4. Midpoint(M, C1) → C3
  5. Midpoint(M, C3) → C4
  6. CircleArc(M, C1, B1) → D1
  7. CircleArc(M, C3, B3) → D3
  8. CircleArc(M, C4, B4) → D4
  9. Midpoint(C1, C3) → C2
  10. CircleArc(M, C2, B2) → D2
  11. Segment((0, 1 - A1), C1) → E1
  12. Segment((0, 1 - A2), C2) → E2
  13. Segment((0, 1 - A3), C3) → E3
  14. Segment((0, 1 - A4), C4) → E4
  15. Circle((0, 1), A1) → F1
  16. Circle((0, 1), A2) → F2
  17. Circle((0, 1), A3) → F3
  18. Circle((0, 1), A4) → F4
  19. Circle(M, C1) → G1
  20. Circle(M, C2) → G2
  21. Circle(M, C3) → G3
  22. Circle(M, C4) → G4
  23. Segment((0, 1 - A1), ((2 * π), 1 - A1)) → H1
  24. Segment((0, 1 - A2), ((2 * π), 1 - A2)) → H2
  25. Segment((0, 1 - A3), ((2 * π), 1 - A3)) → H3
  26. Segment((0, 1 - A4), ((2 * π), 1 - A4)) → H4
  27. Segment((0, 0), (0, 1)) → f
  28. Segment(((2 * π), 0), ((2 * π), 1)) → g
  29. Segment(M, B1) → h

AI Drawing Prompt Reference

Draw a rolling wheel showing the path traced by a point on its circumference and the contact point with the ground. The wheel rolls along the ground, with arcs representing the path of a point on the edge. Clearly mark the contact point where the wheel touches the ground. The path should reveal a complex curve, demonstrating geometric illusions in motion. Include midpoint lines from the contact point to the wheel's center, along with relevant辅助 lines and circles to illustrate the geometric construction and calculations.
The prompt above was compiled from the original drawing commands and is for reference only.