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Spheres Intersecting with Sphere

Spheres Intersecting with Sphere
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2017-01-22 20:57
Solid GeometryProbability & StatisticsGeometric Construction
This example demonstrates the geometric figures and phenomena formed by multiple spheres intersecting with a single sphere. Core math concepts include sphere, intersection, median value and standard deviation in probability & statistics, plane figures, and solid geometry. Construction highlights from the summary utilize GeoGebra commands/objects like Intersect, Sphere, IntersectConic, Point, If, and Surface. Suitable for high school and above students learning solid geometry, probability & statistics, and geometric constructions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Sphere(O, r0) → sp0
  3. Sphere(A, ra) → a
  4. IntersectConic(sp0, a) → ca
  5. Point(ca, 0) → A_2
  6. Intersect(Ray[A, O], a) → A_3,
  7. If(y(A) ≥ 0, Angle[(1, 0, 0), O, A_0], (-Angle[(1, 0, 0), O, A_0])) → aau1
  8. If(z(A) ≥ 0, π / 2 + Angle[A_0, O, A], π / 2 - Angle[A_0, O, A]) → aau2
  9. Surface(x(A) + (ra * (((((cos(u) * cos(v)) * cos(aau2)) - (sin(aau2) * sin(v))) * cos(aau1)) - ((sin(aau1) * sin(u)) * cos(v)))), y(A) + (ra * (((((cos(u) * cos(v)) * cos(aau2)) - (sin(aau2) * sin(v))) * sin(aau1)) + ((cos(aau1) * sin(u)) * cos(v)))), z(A) + (ra * (((cos(u) * cos(v)) * sin(aau2)) + (cos(aau2) * sin(v)))), u, 0, (2 * π), v, aav, π / 2) → sa
  10. Sphere(B, rb) → b
  11. IntersectConic(sp0, b) → cb
  12. Point(cb, 0) → B_2
  13. Intersect(Ray[B, O], b) → B_3,
  14. If(y(B) ≥ 0, Angle[(1, 0, 0), O, B_0], (-Angle[(1, 0, 0), O, B_0])) → abu1
  15. If(z(B) ≥ 0, π / 2 + Angle[B_0, O, B], π / 2 - Angle[B_0, O, B]) → abu2
  16. Surface(x(B) + (rb * (((((cos(u) * cos(v)) * cos(abu2)) - (sin(abu2) * sin(v))) * cos(abu1)) - ((sin(abu1) * sin(u)) * cos(v)))), y(B) + (rb * (((((cos(u) * cos(v)) * cos(abu2)) - (sin(abu2) * sin(v))) * sin(abu1)) + ((cos(abu1) * sin(u)) * cos(v)))), z(B) + (rb * (((cos(u) * cos(v)) * sin(abu2)) + (cos(abu2) * sin(v)))), u, 0, (2 * π), v, abv, π / 2) → sb
  17. Sphere(C, rc) → c
  18. IntersectConic(sp0, c) → cc
  19. Point(cc, 0) → C_2
  20. Intersect(Ray[C, O], c) → C_3,
  21. If(y(C) ≥ 0, Angle[(1, 0, 0), O, C_0], (-Angle[(1, 0, 0), O, C_0])) → acu1
  22. If(z(C) ≥ 0, π / 2 + Angle[C_0, O, C], π / 2 - Angle[C_0, O, C]) → acu2
  23. Surface(x(C) + (rc * (((((cos(u) * cos(v)) * cos(acu2)) - (sin(acu2) * sin(v))) * cos(acu1)) - ((sin(acu1) * sin(u)) * cos(v)))), y(C) + (rc * (((((cos(u) * cos(v)) * cos(acu2)) - (sin(acu2) * sin(v))) * sin(acu1)) + ((cos(acu1) * sin(u)) * cos(v)))), z(C) + (rc * (((cos(u) * cos(v)) * sin(acu2)) + (cos(acu2) * sin(v)))), u, 0, (2 * π), v, acv, π / 2) → sc

AI Drawing Prompt Reference

Draw the intersection of three spheres with a central sphere, highlighting their geometric relationships. Start by plotting the origin at the intersection of the coordinate axes, then create a central sphere. Next, draw three spheres of different sizes around the central sphere and calculate their intersections with it. Finally, based on the intersection curves and sphere radii, sketch the spherical surfaces of the intersections to form a complex geometric structure.
The prompt above was compiled from the original drawing commands and is for reference only.