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Step Function and Geometric Construction: Cola de golondrina

Step Function and Geometric Construction: Cola de golondrina
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2023-03-15 19:02
FunctionCoordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the relationship between step functions and geometric constructions. Key mathematical concepts include step functions, logic, lines, arithmetic mean, and averages. Construction involves GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence. Suitable for middle or high school mathematics education to understand the application of function graphs in geometric contexts.

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, yAxis) → zz1
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz1, 1) → ZZ1
  14. Intersect(zz0, zz1, 2) → ZZ2
  15. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  16. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  17. Point(zElipse1) → XY
  18. Angle(ZZ6, C, XY) → zAngRot
  19. OrthogonalLine(C, xAxis) → zz2
  20. Intersect(zz0, zz2, 2) → ZZ3
  21. Intersect(zz0, zz2, 1) → ZZ4
  22. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  23. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  24. Point(zElipse2) → Z
  25. Angle(ZZ8, C, Z) → zAngInc
  26. CurveCartesian(0, ((r * u) * 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 * ((((3 * v^(4)) + (α * v^(2))) * sin(zAngRot)) + (((-4 * v^(3)) - ((2 * α) * v)) * cos(zAngRot)))), (r * ((((-((3 * v^(4)) + (α * v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-4 * v^(3)) - ((2 * α) * v)) * sin(zAngRot)) * sin(zAngInc)) + (α * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r * ((((3 * β^(4)) + (u * β^(2))) * sin(zAngRot)) + (((-4 * β^(3)) - ((2 * u) * β)) * cos(zAngRot)))), (r * ((((-((3 * β^(4)) + (u * β^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-4 * β^(3)) - ((2 * u) * β)) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), u, u1, u2) → a
  34. Sequence(CurveCartesian[(r * ((((3 * v^(4)) + (u * v^(2))) * sin(zAngRot)) + (((-4 * v^(3)) - ((2 * u) * v)) * cos(zAngRot)))), (r * ((((-((3 * v^(4)) + (u * v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-4 * v^(3)) - ((2 * u) * v)) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r * ((((3 * v^(4)) + (u * v^(2))) * sin(zAngRot)) + (((-4 * v^(3)) - ((2 * u) * v)) * cos(zAngRot)))), (r * ((((-((3 * v^(4)) + (u * v^(2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((-4 * v^(3)) - ((2 * u) * v)) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))), v, v1, v2], u, u1, u2, (3.5 * paso)) → list1

AI Drawing Prompt Reference

On a coordinate plane, draw a vertical and a horizontal axis and choose a point C as a circle center. Construct a circle centered at C with a fixed radius (e.g., 0.62), then create two perpendicular reference lines through the axes and obtain two sets of intersection points between the circle and these lines. Use each pair of intersection points to build two ellipses with a given geometric parameter (e.g., 0.65). Intersect the ellipses with the corresponding reference lines to get key angle-defining points, then use the resulting rotation and incremental angle relations to drive parametric equations that generate two Cartesian curves. Finally, sample the parametric curves with a step (a sequence parameter) to produce a stepped, curve-like geometric construction.
The prompt above was compiled from the original drawing commands and is for reference only.