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Helicoide Transformation Demo

Helicoide Transformation Demo
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2023-03-15 20:22
FunctionTrigonometrySolid GeometryParametric EquationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the formation and transformation of a helicoide. It involves core mathematical concepts like lines, cosine, rational numbers, sine, and linear programming. Construction highlights include using GeoGebra commands and objects such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence. Suitable for high school and above students to understand spatial geometry and parametric equations applications.

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. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 3 * (((((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (α * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2) → curva2
  32. CurveCartesian((r / 3 * (((((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * sin(zAngRot)) * sin(zAngInc)) + (((β * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → a1
  33. CurveCartesian((r / 3 * ((((-sin(a)) * cosh(v)) * sin(zAngRot)) - ((cos(a) * sinh(v)) * cos(zAngRot)))), (r / 3 * ((((sin(a) * cosh(v)) * cos(zAngRot)) * sin(zAngInc)) - (((cos(a) * sinh(v)) * sin(zAngRot)) * sin(zAngInc)) + (((3.14159 * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → curva1A
  34. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a helicoide surface using a series of ellipses and curves. Start by creating a series of points and line segments, then construct ellipses and curves using these points and segments. Next, use rotation and translation operations to generate multiple cross-sections of the helicoide, and finally depict the overall shape of the helicoide using these cross-sections.
The prompt above was compiled from the original drawing commands and is for reference only.