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Intersection of Cylinder and Paraboloid: 3D Surface Construction

Intersection of Cylinder and Paraboloid: 3D Surface Construction
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2018-06-28 02:10
Coordinate GeometrySolid GeometryParametric EquationGeometric ConstructionGraphing
In collections intersection: surface-surface, curve-curve
①This example demonstrates the penetration (Durchdringung) of a cylinder and a paraboloid in 3D space, showing how the two surfaces intersect and form a spatial curve. ②Core mathematical concepts include solid geometry, parametric representation of surfaces, and the construction of intersection curves. ③Construction details: using GeoGebra commands Surface and CurveCartesian, the example creates the cylinder (Zyl) and paraboloid (Par), with parameters a, b, h and auxiliary objects a_1, a_2, and Text1 for control and labeling. ④It is suitable for high school or early college students learning 3D geometry, analytic geometry, or computer-assisted mathematics, enhancing understanding of parametric surfaces and their intersections.

GeoGebra Commands & Parameters

  1. Surface((sqrt(v) * cos(u)), (sqrt(v) * sin(u)), v, u, 0, (2 * π), v, 0, h) → Par
  2. Surface((a * cos(u)), (b * sin(u)), v, u, 0, (2 * π), v, 0, h) → Zyl
  3. CurveCartesian((a * t), (b * sqrt(1 - t^(2))), b^(2) + ((a^(2) - b^(2)) * t^(2)), t, (-π), π) → a_1
  4. CurveCartesian((a * t), ((-b) * sqrt(1 - t^(2))), b^(2) + ((a^(2) - b^(2)) * t^(2)), t, (-π), π) → a_2