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Hyperboloid of One Sheet as a Locus

Hyperboloid of One Sheet as a Locus
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2017-08-27 03:10
VectorSolid GeometryTransformationGeometric ConstructionLocus
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources Multivariable Calculus Quadratic Surfaces
This example demonstrates the dynamic locus of a special hyperboloid of one sheet through rotation and intersection constructions. It involves core concepts like rotation, intersection, vector, translation, dilation, and geometric transformations. GeoGebra commands such as Rotate, IntersectPath, Point, Focus, Segment, Sphere, Intersect, Vector, Translate, Dilate, Plane, Angle, Distance, and Line are used to construct the model. Suitable for high school and advanced mathematics education to explore spatial geometry and locus problems.

GeoGebra Commands & Parameters

  1. Rotate(e, (d * π / 180), Line[(0, 0, 0), (0, 0, 1)]) → e'
  2. IntersectPath(e', c) → f
  3. Point(f) → I
  4. Focus(f) → G, F1
  5. Segment(I, G) → j
  6. Segment(F1, I) → i
  7. Sphere(F1, m) → o
  8. Intersect(i, o) → J,
  9. Intersect(g, f) → A, F
  10. Vector(F1, F) → u_1
  11. Translate(J, u_1) → J'_1
  12. Segment(E, D) → h
  13. IntersectPath(e, c) → k
  14. Segment(F, A) → l
  15. Dilate(F1, If[0 < n ≤ 1, 1 - n, 0], J) → H'
  16. Segment(F1, H') → p
  17. Dilate(J, If[1 < n ≤ 2, 2 - n, 0], I) → J'
  18. Segment(J, J') → q
  19. Dilate(G, If[2 < n ≤ 3, 3 - n, 0], I) → G'
  20. Segment(G, G') → r
  21. Plane(F1, I, G) → s
  22. Angle(F1, I, G) → α
  23. Rotate(G, If[3 < n ≤ 4, ((n - 3) * α), α], I, s) → G'_1
  24. Rotate(G', If[3 < n ≤ 4, ((n - 3) * α), α], I, s) → G''
  25. Segment(G'_1, G'') → r'
  26. IntersectPath(z = 0, c) → t
  27. Dilate(F1, If[4 < n ≤ 5, 5 - n, 0], F) → H'_1
  28. Vector(F1, H'_1) → u
  29. Translate(F1, u) → H'_2
  30. Translate(H', u) → H''
  31. Angle(A, F, J'_1) → β
  32. Segment(H'_2, H'') → p'
  33. Plane(A, F, J) → v
  34. Rotate(H'_2, If[n < 5 ≤ 6, ((n - 5) * β), β], F, v) → K
  35. Rotate(H'', If[5 < n ≤ 6, ((n - 5) * β), β], F, v) → H'''
  36. Segment(K, H''') → p''
  37. Distance(A, F) → w
  38. Line((0, 0, 0), (1, 0, 0)) → f_1
  39. Line((0, 0, 0), (0, 1, 0)) → g_1

AI Drawing Prompt Reference

Draw a dynamic locus of a hyperboloid of one sheet, showing geometric relationships through rotation and intersection constructions. Rotate a line, intersect it with a curve to get a point, then use foci, vectors, translations, and dilations to construct multiple辅助 lines and spheres, forming a complex spatial geometric solid that demonstrates core concepts like rotation, intersection, vector, translation, dilation, and geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.