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Dandelin's Spheres: Conic Section Focus Demonstration

Dandelin's Spheres: Conic Section Focus Demonstration
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2018-08-02 00:16
CircleCoordinate GeometrySolid GeometryGeometric ConstructionLocus
①This GeoGebra example demonstrates the classic Dandelin's spheres construction, showing that when a plane intersects a cone, the resulting conic section (an ellipse) has its foci at the tangency points of two inscribed spheres with the plane.②Core mathematical concepts include conic sections, foci and directrix properties, spatial geometry of tangent spheres, and the intersection path between a plane and a cone.③Construction details: using GeoGebra commands such as Line, Cone, Sphere, IntersectPath, Focus, OrthogonalLine, Segment, Intersect, Point, Ray, and Circle, the model builds a cone, two inscribed spheres, a cutting plane, the intersection curve, the foci, and auxiliary segments/points. Sliders or dynamic points allow adjusting the plane's angle to observe how the ellipse changes.④This is suitable for high school or undergraduate courses in analytic geometry and solid geometry, as well as for mathematical exploration activities, helping students intuitively grasp the geometric essence of conic sections.

GeoGebra Commands & Parameters

  1. Line((0, 0, -3), (0, 0, r_2)) → zg
  2. Cone((0, 0, -3), (0, 0, r_2), r_1) → kg, c, a
  3. IntersectPath(E_0, kg) → d
  4. Focus(d) → F_1, F_2
  5. OrthogonalLine(F_1, d) → a_1
  6. Corner(d) → A, B, C, D
  7. Segment((0, 0, -3), (0, 0, r_2)) → za
  8. Intersect(a_1, zg) → C_1
  9. Sphere(C_1, F_1) → Dk_1
  10. OrthogonalLine(F_2, d) → a_2
  11. Intersect(a_2, zg) → C_2
  12. Sphere(C_2, F_2) → Dk_2
  13. Point(d) → P
  14. Segment(C_1, F_1) → cf_1
  15. Segment(C_2, F_2) → cf_2
  16. Ray(Z, Q) → zq
  17. OrthogonalLine(C_2, zq, space) → sc_2
  18. Intersect(zq, sc_2) → Bk_2
  19. Circle((0, 0, z(Bk_2)), x(Bk_2), space) → kr_2
  20. Circle(Bk_{2c}, Bk_{2b}, Bk_2) → e_2
  21. OrthogonalLine(C_1, zq, space) → sc_1
  22. Intersect(zq, sc_1) → Bk_1
  23. Circle(Bk_{1c}, Bk_{1b}, Bk_1) → e_1
  24. Intersect(zq, c) → E,
  25. Ray(Z, P) → zp
  26. Intersect(zp, c) → F,
  27. IntersectPath(E_0, E_1) → b
  28. OrthogonalLine(P_1, b, space) → e
  29. Intersect(b, e) → L
  30. Line(L, P) → f
  31. Segment(P, P_1) → s_{01}
  32. Segment(P_1, L) → s_{02}
  33. Segment((0, 0, z(Bk_1)), P_1) → s_{m2}
  34. Intersect(s_{m2}, e_1) → P_3,

AI Drawing Prompt Reference

Draw a cone and place two inscribed spheres inside it, with the spheres tangent to the cutting plane. The intersection of the cutting plane with the cone forms an ellipse, whose foci are the points where the spheres are tangent to the plane. First, draw the cone, then draw the two spheres ensuring they are internally tangent to the cone, followed by drawing the cutting plane and marking the foci of the ellipse. Finally, add auxiliary lines and points to illustrate the properties of the ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.