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Wrapping Function Demo

Wrapping Function Demo
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2017-04-05 08:14
CircleTrigonometryCoordinate GeometryParametric EquationGraphing
This example demonstrates how the wrapping function maps points on a unit circle to a trajectory in Cartesian coordinates. It involves core mathematical concepts such as the unit circle, parametric curves, trigonometric functions (sine and cosine), and coordinate geometry. The construction uses GeoGebra commands like If and Point, along with variables like X, f, g, h to build parametric equations, and slider animations to show dynamic behavior. Suitable for high school students learning parametric equations, trigonometric graphs, and the properties of wrapping functions, or for introductory calculus instruction.

GeoGebra Commands & Parameters

  1. If(X_1 ≥ 0, Arc[c, (1, 0), (cos(X_1), sin(X_1))], Arc[c, (cos(X_1), sin(X_1)), (1, 0)]) → X
  2. If(X_1 > 0, If[((-1)^(floor(X_1 / π)) * x) < ((-1)^(floor(X_1 / π)) * cos(X_1)), (((-cos(X_1))) / sin(X_1) * (x - cos(X_1))) + sin(X_1)]) → f
  3. If(X_1 ≟ 0, x = 1) → g
  4. If(X_1 > 0, Segment[(1, 0), (1, -10)], X_1 < 0, Segment[(1, 0), (1, 10)]) → h
  5. If(X_1 < 0, If[((-1)^(floor(X_1 / π)) * x) > ((-1)^(floor(X_1 / π)) * cos(X_1)), (((-cos(X_1))) / sin(X_1) * (x - cos(X_1))) + sin(X_1)]) → p
  6. If(0 < X_1 < 1, Point[{cos(X_1) - ((1 - X_1) * sin(X_1)), sin(X_1) + ((1 - X_1) * cos(X_1))}]) → B
  7. If(X_1 ≥ 1, Point[{cos(1), sin(1)}]) → C
  8. If(0 < X_1 < 2, Point[{cos(X_1) - ((2 - X_1) * sin(X_1)), sin(X_1) + ((2 - X_1) * cos(X_1))}]) → D
  9. If(X_1 ≤ 0, Point[{1, 2}]) → E
  10. If(X_1 ≤ 0, Point[{1, 1}]) → A
  11. If(X_1 ≥ 2, Point[{cos(2), sin(2)}]) → F
  12. If(X_1 ≤ 0, Point[{1, 3}]) → G
  13. If(0 < X_1 < 3, Point[{cos(X_1) - ((3 - X_1) * sin(X_1)), sin(X_1) + ((3 - X_1) * cos(X_1))}]) → H
  14. If(X_1 ≥ 3, Point[{cos(3), sin(3)}]) → I
  15. If(X_1 ≥ 0, Point[{1, -1}]) → J
  16. If(-1 < X_1 < 0, Point[{cos(X_1) - ((-1 - X_1) * sin(X_1)), sin(X_1) + ((-1 - X_1) * cos(X_1))}]) → K
  17. If(X_1 ≤ -1, Point[{cos(-1), sin(-1)}]) → L
  18. If(X_1 ≥ 0, Point[{1, -2}]) → M
  19. If(-2 < X_1 < 0, Point[{cos(X_1) - ((-2 - X_1) * sin(X_1)), sin(X_1) + ((-2 - X_1) * cos(X_1))}]) → N
  20. If(X_1 ≤ -2, Point[{cos(-2), sin(-2)}]) → O
  21. If(X_1 ≥ 0, Point[{1, -3}]) → P
  22. If(-3 < X_1 < 0, Point[{cos(X_1) - ((-3 - X_1) * sin(X_1)), sin(X_1) + ((-3 - X_1) * cos(X_1))}]) → Q
  23. If(X_1 ≤ -3, Point[{cos(-3), sin(-3)}]) → R
  24. If(X_1 ≥ 0, Point[{1, -4}]) → S
  25. If(-4 < X_1 < 0, Point[{cos(X_1) - ((-4 - X_1) * sin(X_1)), sin(X_1) + ((-4 - X_1) * cos(X_1))}]) → T
  26. If(X_1 ≤ -4, Point[{cos(-4), sin(-4)}]) → U
  27. If(0 < X_1 < 4, Point[{cos(X_1) - ((4 - X_1) * sin(X_1)), sin(X_1) + ((4 - X_1) * cos(X_1))}]) → W
  28. If(X_1 ≥ 4, Point[{cos(4), sin(4)}]) → Z
  29. Point({cos(X_1), sin(X_1)}) → A_1
  30. If(X_1 ≤ 0, Point[{1, 4}]) → V
  31. If(X_1 ≤ 0, Point[{1, 5}]) → B_1
  32. If(0 < X_1 < 5, Point[{cos(X_1) - ((5 - X_1) * sin(X_1)), sin(X_1) + ((5 - X_1) * cos(X_1))}]) → C_1
  33. If(X_1 ≥ 5, Point[{cos(5), sin(5)}]) → D_1
  34. If(X_1 ≤ 0, Point[{1, 6}]) → E_1
  35. If(0 < X_1 < 6, Point[{cos(X_1) - ((6 - X_1) * sin(X_1)), sin(X_1) + ((6 - X_1) * cos(X_1))}]) → F_1
  36. If(X_1 ≥ 6, Point[{cos(6), sin(6)}]) → G_1
  37. Point({1, 0}) → H_1