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Serpentine Sinusoids

Serpentine Sinusoids
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2020-11-29 04:28
FunctionTrigonometryCoordinate GeometrySolid Geometry
This example demonstrates the generation and transformation of serpentine sinusoids, allowing dynamic parameter adjustments to observe their shapes. Core mathematical concepts include sine functions, modes, arithmetic operations, solid geometry, ratios, and secant lines. The construction utilizes GeoGebra commands/objects such as Vector, Function, Sequence, Polygon, Element, Segment, and CurveCartesian. It is suitable for high school and above students to explore the application of function graph transformations combined with solid geometry.

GeoGebra Commands & Parameters

  1. Vector(B_1, X) → u
  2. Vector(B_1, Z) → w
  3. Vector(B_1, Y) → v
  4. Function((A1 * sin((w1 * t) - (k1 * x))), 0, 25.13274) → f1
  5. Function((A2 * sin((w2 * t) - (k2 * x))), 0, 25.13274) → f2
  6. Function(f1(x) + f2(x) + (h(x) * x), 0, 25.13274) → f_c
  7. Function(f_c, (Vit * t), (Vit * t) + 150) → f
  8. Sequence((i * u) + (f(i) * v), i, (Vit * t), (Vit * t) + 30, 0.25) → liste1
  9. Sequence(Circle[Element[liste1, i], i / ((10 * R))], i, 1, Length[liste1] + 45) → liste2
  10. Polygon(B, D, C, E) → poly1, b, d, e, i
  11. Element(liste1, Length[liste1]) → M
  12. Segment(I, M) → j
  13. Segment(I, J) → l
  14. Segment(I, K) → m
  15. Function(f2c, 150, 175) → f3
  16. Function(f3, 150 - (Vit2 * t), 175 - (Vit2 * t)) → p
  17. Sequence((i * v) + (125 * u) + (f3(i) * u), i, 150 - (Vit2 * t), 175 - (Vit2 * t), 0.25) → liste3
  18. Sequence(Circle[Element[liste3, Length[liste3] - i], i / ((10 * R))], i, 1, Length[liste3] + 45) → liste4
  19. CurveCartesian(x(N - (i * u) + (60 * v) + (f3(i) * v)), y(N - (i * u) + (60 * v) + (f3(i) * v)), i, 0, 62.8319) → q
  20. CurveCartesian(x(L + i), y(L + (2 * f(i))), i, 0, 125.6638) → o

AI Drawing Prompt Reference

In a 3D coordinate space, start from a common point and define three direction vectors. Build a composite function by combining two sine functions with parameters, then apply a vertical shift and restrict it to a parameter interval so its phase can be adjusted. Using the chosen step, generate a sequence of points from the direction vectors and the modulated function values. From each point, construct circles and collect two point sets sampled along those circles. Use selected points to form connecting line segments and intermediate reference points. Finally, draw two parametric planar curves: one whose coordinates come from the direction-weighted parameter expression involving the composite function, and another obtained by translating along a base offset while adding the function value to the corresponding coordinate, producing serpentine sinusoid–like trajectories.
The prompt above was compiled from the original drawing commands and is for reference only.