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Ultimate Symmetry in Cone-Sphere System

Ultimate Symmetry in Cone-Sphere System
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2019-05-11 10:36
Solid GeometryTransformationGeometric Construction
This example demonstrates the ultimate symmetry between cones and spheres. It involves key concepts such as cones, spheres, and geometric transformations. Using GeoGebra commands like Point, Sphere, Circle, and Intersect, complex geometric shapes can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → STAR
  2. Sphere(STAR, a) → b
  3. Circle(STAR, a) → c
  4. Point(c) → planet
  5. PointIn(b) → Pointb_1
  6. PointIn(b) → Pointb_2
  7. PointIn(b) → Pointb_3
  8. PointIn(b) → Pointb
  9. Circle(zAxis, STAR) → k
  10. Point(zAxis) → E
  11. Point(zAxis) → F
  12. Intersect(zAxis, b, 2) → A
  13. Point(zAxis) → B
  14. Cone(B, STAR, d) → e, f, g
  15. Cone(B, A, d) → h, p, i
  16. Intersect(c, xAxis, 2) → C
  17. Intersect(c, xAxis, 2) → D
  18. Line(C, zAxis) → l
  19. Intersect(c, xAxis, 1) → G
  20. Line(G, zAxis) → m
  21. Intersect(l, f, 1) → H
  22. Intersect(m, f, 1) → I
  23. Segment(H, I) → n
  24. Segment(H, planet) → j
  25. Segment(I, planet) → q

AI Drawing Prompt Reference

In 3D space, place a point on an axis as the center, and draw a sphere and a coplanar circle with the same center. Pick a point on the circle and several random points inside the sphere. Along the vertical axis, choose two points as apexes to construct two cones whose bases intersect the sphere. From two intersection points of the circle with the horizontal axis, draw lines up to the vertical axis; these lines meet the side of one cone at two points. Connect these two points to each other and to the point on the circle, forming three segments that reveal the symmetry between the cone and sphere.
The prompt above was compiled from the original drawing commands and is for reference only.