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Lissajous Curve and Sine Function

Lissajous Curve and Sine Function
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2020-09-23 20:16
FunctionTrigonometryCoordinate GeometryParametric EquationGraphing
This example demonstrates how Lissajous curves are generated by the addition of sine functions. Key math concepts include sine functions, trigonometry, parametric equations, and coordinate geometry. Construction highlights involve commands/objects like Polygon, CurveCartesian, Line, Point, Circle, Intersect, Rotate, Segment, Angle, and Translate from GeoGebra. Suitable for high school mathematics teaching to understand the叠加 and transformation patterns of function graphs.

GeoGebra Commands & Parameters

  1. Polygon(E, D, B, C) → q1, e, d, b, c
  2. CurveCartesian(x(O) + (A2 * sin((w2 * k) + p2)), y(O) + (A1 * sin((w1 * k) + p1)), k, 0, t) → a
  3. Line(O, yAxis) → f
  4. Line(O, xAxis) → g
  5. Point(f) → F
  6. Point(g) → G
  7. Circle(F, A2, xOyPlane) → h
  8. Circle(G, A1, xOyPlane) → p
  9. Intersect(f, h) → L, M
  10. Rotate(M, (w2 * t) + p2, F) → M'
  11. Segment(M', A) → i
  12. Segment(F, M') → l
  13. Intersect(g, p) → H, I
  14. Rotate(H, (w1 * t) + p1, G) → H'
  15. Segment(G, H') → j
  16. Segment(H', A) → m
  17. Segment(G, H) → n
  18. Segment(F, M) → q
  19. Angle(M, F, M', xOyPlane) → α
  20. Angle(H, G, H', xOyPlane) → β
  21. Segment(G, O) → f_1
  22. Line(G, O) → h_1
  23. Line(F, O) → g_1
  24. CurveCartesian(((-A2) * sin((w2 * i) - p2 - (w2 * t))), i, i, (-6 * π), 0) → k
  25. Translate(s, Vector[O, O + (A2, 0)]) → s_{1}
  26. Line(A, f_1) → i_1
  27. Intersect(i_1, d) → J
  28. Segment(A, J) → j_1
  29. Line(A, q) → k_1
  30. Intersect(e, k_1) → K
  31. Translate(k, Vector[O, O - (0, A1)]) → k'
  32. Segment(A, K) → l_1

AI Drawing Prompt Reference

Draw a Lissajous curve and sine function graph. Start by drawing a unit circle and the x-axis and y-axis. Then, generate two circles by rotating and moving, and calculate their intersection points. Use these points and the origin to construct the parametric equation of the Lissajous curve. Finally, add angle markings and auxiliary lines to demonstrate the curve generation process.
The prompt above was compiled from the original drawing commands and is for reference only.