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Sine Surface (Seno superficie) Dynamic Construction

Sine Surface (Seno superficie) Dynamic Construction
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2023-03-15 21:30
FunctionTrigonometrySolid GeometryParametric EquationGraphing
In collections geometriadinamica.es
① This GeoGebra example constructs a three-dimensional sine surface (Seno superficie), visually demonstrating the variation of the sine function across a surface. ② Core mathematical concepts include the sine function, parametric surfaces, spatial geometry, and the interaction between curves and surfaces, involving trigonometry and parametric equations. ③ Construction highlights: using commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence, it creates auxiliary points (e.g., P1, Q1), vectors (u1, v1), and ellipse objects, generating dynamic trajectories via sequences to form the sine surface. ④ Suitable for high school to undergraduate students, this example helps understand the spatial extension of sine functions and parametric surface construction, and serves as a visual teaching tool in mathematics.

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. 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 * ((sin(v) * sin(zAngRot)) + (sin(α) * cos(zAngRot)))), (r * ((((-sin(v)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(α) * sin(zAngRot)) * sin(zAngInc)) + (sin(v + α) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian(((r * sin(u)) * cos(zAngRot)), (r * (((sin(u) * sin(zAngRot)) * sin(zAngInc)) + (sin(3.14159 + u) * cos(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r * ((sin(β) * sin(zAngRot)) + (sin(u) * cos(zAngRot)))), (r * ((((-sin(β)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(u) * sin(zAngRot)) * sin(zAngInc)) + (sin(β + u) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r * ((sin(v) * sin(zAngRot)) + (sin(u) * cos(zAngRot)))), (r * ((((-sin(v)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(u) * sin(zAngRot)) * sin(zAngInc)) + (sin(v + u) * cos(zAngInc)))), u, u1, u2], v, v1, v2, (2 * paso)) → list3
  35. Sequence(CurveCartesian[(r * ((sin(v) * sin(zAngRot)) + (sin(u) * cos(zAngRot)))), (r * ((((-sin(v)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(u) * sin(zAngRot)) * sin(zAngInc)) + (sin(v + u) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Create a dynamic 3D sine surface. Set up two perpendicular reference axes, then build auxiliary geometry from a circle and two ellipses associated with those axes to obtain intersection points that determine rotation and inclination directions. Define the needed angles from these directions. Use parametric curve equations with trigonometric sine expressions to generate spatial curve trajectories, then sample them over two parameter ranges using sequences, so the resulting family of curves forms a sine-like surface. Keep the main intersection points, angle relationships, and the auxiliary ellipses as visible geometric references to emphasize how the sine variation extends across the surface.
The prompt above was compiled from the original drawing commands and is for reference only.