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Wave-like geometry from parametric curves and ellipses

Wave-like geometry from parametric curves and ellipses
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2023-03-15 20:58
CircleAngleSequenceParametric EquationGeometric Construction
In collections geometriadinamica.es
This example demonstrates a wave-like visual effect by combining parametric curves with ellipse geometry and angle-controlled construction. The key mathematical ideas include parametric equation/curve construction, ellipse properties, angle increments that introduce phase or periodic behavior, and the use of a step size to generate evolving geometric configurations via a sequence. In the GeoGebra build, commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence are used to create foundational points and segments, construct perpendicular lines, define circle intersections, generate the ellipse, and compute/track angular relationships. Finally, CurveCartesian draws the resulting curves while Sequence (with a parameter step like “paso”) produces the successive objects that make the overall “wave” appearance. Suitable for middle school through early university users for geometry modeling, parametric curve visualization, and GeoGebra classroom demonstrations or self-study.

GeoGebra Commands & Parameters

  1. Segment(Q5, Q6) → p5
  2. Point(p5) → P5
  3. Segment(Q3, Q4) → p2
  4. Point(p2) → P3
  5. Segment(P3, Q4) → p4
  6. Point(p4) → P4
  7. Segment(Q1, Q2) → p1
  8. Point(p1) → P1
  9. Segment(P1, Q2) → p3
  10. Point(p3) → P2
  11. OrthogonalLine(C, xAxis) → zz2
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz2, 1) → ZZ4
  14. Intersect(zz0, zz2, 2) → ZZ3
  15. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  16. Point(zElipse2) → Z
  17. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  18. Angle(ZZ8, C, Z) → zAngInc
  19. OrthogonalLine(C, yAxis) → zz1
  20. Intersect(zz0, zz1, 2) → ZZ2
  21. Intersect(zz0, zz1, 1) → ZZ1
  22. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  23. Point(zElipse1) → XY
  24. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  25. Angle(ZZ6, C, XY) → zAngRot
  26. CurveCartesian(((r / 5 * u) * sin(zAngRot)), (r / 5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + (cos(u^(2)) * cos(zAngInc)))), u, u1, u2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r / 5 * ((β * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 5 * ((((-β) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (cos(β^(2) + v^(2)) * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r / 5 * ((u * sin(zAngRot)) + (α * cos(zAngRot)))), (r / 5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((α * sin(zAngRot)) * sin(zAngInc)) + (cos(u^(2) + α^(2)) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 5 * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (cos(u^(2) + v^(2)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 5 * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (cos(u^(2) + v^(2)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Create a wave-like geometric composition featuring two intersecting ellipses as the fundamental framework. These ellipses are constructed from circle intersections with orthogonal lines to the coordinate axes, possessing specific eccentricities. Around both ellipses, multiple sets of parametric curves are generated using angle parameters as controls. These curves distribute along the elliptical trajectories, creating layered wave-like visual effects. The curves employ sine and cosine functions combined with angle increments to produce periodic oscillating transformations. A sequence of continuously varying curves is generated through step-size control, overall presenting a dynamic geometric form resembling flowing waves.
The prompt above was compiled from the original drawing commands and is for reference only.