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Double Box-lid (Mouhefanggai) as Intersection of 2 Cylinders

Double Box-lid (Mouhefanggai) as Intersection of 2 Cylinders
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2017-03-08 11:28
Coordinate GeometrySolid Geometry
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes in Ancient Chinese Mathematics 中國古算中的體積 Volumes of Pyramid and Sphere in Ancient Chinese Mathematics Volume of Spheres in Ancient Chinese Mathematics
This example demonstrates the Double Box-lid (Mouhefanggai), a geometric solid formed by the intersection of two cylinders. Key mathematical concepts involved include cylinders, intersection, geometric mean, hyperbolas, and coordinate geometry. The construction utilizes GeoGebra commands and objects such as Sphere, CurveCartesian, Circle, Polygon, and Surface. It is suitable for high school and advanced students studying complex intersection problems in solid geometry and coordinate geometry.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 1), r) → a
  2. CurveCartesian(sqrt(1 - t^(2)), sqrt(1 - t^(2)), t + 1, t, -1, -1 + (2 * h)) → f
  3. CurveCartesian((-sqrt(1 - t^(2))), sqrt(1 - t^(2)), t + 1, t, -1, -1 + (2 * h)) → g
  4. CurveCartesian((-sqrt(1 - t^(2))), (-sqrt(1 - t^(2))), t + 1, t, -1, -1 + (2 * h)) → h_1
  5. CurveCartesian(sqrt(1 - t^(2)), (-sqrt(1 - t^(2))), t + 1, t, -1, -1 + (2 * h)) → i
  6. Circle((0, 0, (2 * h)), sqrt((4 * h) - (4 * h^(2))), zAxis) → k
  7. Polygon((sqrt((4 * h) - (4 * h^(2))), sqrt((4 * h) - (4 * h^(2))), (2 * h)), ((-sqrt((4 * h) - (4 * h^(2)))), sqrt((4 * h) - (4 * h^(2))), (2 * h)), ((-sqrt((4 * h) - (4 * h^(2)))), (-sqrt((4 * h) - (4 * h^(2)))), (2 * h)), (sqrt((4 * h) - (4 * h^(2))), (-sqrt((4 * h) - (4 * h^(2)))), (2 * h))) → poly1, j, l, m, p
  8. Surface(sin(θ), s, 1 - cos(θ), s, -1, 1, θ, (-ϕ), ϕ) → b
  9. Surface(s, sin(θ), 1 - cos(θ), s, -1, 1, θ, (-ϕ), ϕ) → c
  10. CurveCartesian(sin(ϕ), s, 1 - cos(ϕ), s, -1, 1) → d
  11. CurveCartesian(sin((-ϕ)), s, 1 - cos((-ϕ)), s, -1, 1) → e
  12. CurveCartesian(s, sin(ϕ), 1 - cos(ϕ), s, -1, 1) → o
  13. CurveCartesian(s, sin((-ϕ)), 1 - cos((-ϕ)), s, -1, 1) → q
  14. Polygon({}) → poly2
  15. Polygon({(-1, sin(ϕ), 1 - cos(ϕ)), (1, sin(ϕ), 1 - cos(ϕ)), (1, sin((-ϕ)), 1 - cos((-ϕ))), (-1, sin((-ϕ)), 1 - cos((-ϕ)))}) → poly3
  16. Polygon({(sin(ϕ), -1, 1 - cos(ϕ)), (sin(ϕ), 1, 1 - cos(ϕ)), (sin((-ϕ)), 1, 1 - cos((-ϕ))), (sin((-ϕ)), -1, 1 - cos((-ϕ)))}) → poly4

AI Drawing Prompt Reference

Draw the double box-lid (Mouhefanggai) formed by the intersection of two cylinders. Start by drawing the outlines of the two cylinders, then use curves and circles to define the boundaries of the intersection area. Finally, complete the geometric shape by drawing polygons and surfaces.
The prompt above was compiled from the original drawing commands and is for reference only.