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Revolution Surfaces (Surfaces of Revolution) Dynamic Construction

Revolution Surfaces (Surfaces of Revolution) Dynamic Construction
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2023-03-15 21:26
Solid GeometryTransformationParametric EquationLocusGraphing
In collections geometriadinamica.es
① This GeoGebra example dynamically demonstrates how a plane curve (such as an ellipse) rotates around an axis to generate a surface of revolution, visualizing the 3D shape of the rotated body. ② Core mathematical concepts include surfaces of revolution, parametric equations, space curves and surfaces, symmetry, and rotational properties of conic sections like ellipses. ③ Construction highlights: from the summary, commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence are used, along with multiple auxiliary points (P1-P4, Q1-Q4), vectors (v1, v2, etc.), a rotation angle variable (zAngRot), and ellipse objects (zElipse1, zElipse2), with sequences generating the rotational traces. ④ Suitable for high school to university-level analytic geometry, solid geometry, or calculus courses, helping students understand the generation principle and parametric representation of surfaces of revolution, and serving as a visual teaching resource.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw a 3D surface of revolution showing how a plane ellipse rotates around an axis to form a solid. The figure includes two ellipses, rotation axes, intersection points, and rotation angle parameters, with parametric curves generating the surface outline. It illustrates the principle of surface generation and spatial curve relationships, suitable for geometry education.
The prompt above was compiled from the original drawing commands and is for reference only.