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Dini Surface Construction

Dini Surface Construction
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2023-03-15 19:41
TrigonometrySolid GeometryParametric EquationGeometric ConstructionGraphing
In collections geometriadinamica.es
This example demonstrates the construction and dynamic manipulation of a Dini surface in 3D. It involves core mathematical concepts such as parametric equations, trigonometric functions, polar coordinate transformations, and 3D spatial geometry. Key GeoGebra commands and objects used in the construction include Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence. Suitable for high school students, university beginners, and users needing to visualize complex surfaces.

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 / 1.5 * ((-sin(v)) * sin(zAngRot))), (r / 1.5 * (((sin(v) * cos(zAngRot)) * sin(zAngInc)) + ((-1 + ln(tan(v / 2)) + (a * 3.14159)) * cos(zAngInc)))) + 2, v, v1, v2) → 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 / 1.5 * (((cos(u) * sin(β)) * sin(zAngRot)) + ((sin(u) * sin(β)) * cos(zAngRot)))), (r / 1.5 * (((((-cos(u)) * sin(β)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * sin(β)) * sin(zAngRot)) * sin(zAngInc)) + ((cos(u) + ln(tan(β / 2)) + (a * u)) * cos(zAngInc)))) + 2, u, u1, u2) → curva2
  33. CurveCartesian((r / 1.5 * (((cos((2 * α)) * sin(v)) * sin(zAngRot)) + ((sin((2 * α)) * sin(v)) * cos(zAngRot)))), (r / 1.5 * (((((-cos((2 * α))) * sin(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin((2 * α)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((cos((2 * α)) + ln(tan(v / 2)) + ((a * 2) * α)) * cos(zAngInc)))) + 2, v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r / 1.5 * (((cos(u) * sin(v)) * sin(zAngRot)) + ((sin(u) * sin(v)) * cos(zAngRot)))), (r / 1.5 * (((((-cos(u)) * sin(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((cos(u) + ln(tan(v / 2)) + (a * u)) * cos(zAngInc)))) + 2, v, v1, v2], u, u1, u2, (2 * paso)) → list2
  35. Sequence(CurveCartesian[(r / 1.5 * (((cos(u) * sin(v)) * sin(zAngRot)) + ((sin(u) * sin(v)) * cos(zAngRot)))), (r / 1.5 * (((((-cos(u)) * sin(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((cos(u) + ln(tan(v / 2)) + (a * u)) * cos(zAngInc)))) + 2, u, u1, u2], v, v1, v2, (2 * paso)) → list1

AI Drawing Prompt Reference

Create a complex 3D surface consisting of two ellipses and two curves. The two ellipses intersect at two points, and the curves rotate around the axes of the ellipses. The curve equations involve trigonometric functions, logarithmic functions, and parameters u, v, α, β. Please generate a 3D surface graph with two ellipses and two curves, and adjust the curve colors and line widths as needed.
The prompt above was compiled from the original drawing commands and is for reference only.