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Cylinder-Paraboloid Intersection Demo

Cylinder-Paraboloid Intersection Demo
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2018-06-28 18:06
FunctionCoordinate GeometrySolid GeometryParametric EquationGraphing
In collections intersection: surface-surface, curve-curve
This example demonstrates the intersection of a cylinder and a paraboloid, showcasing spatial geometry concepts. Core math involves intersection calculations and 3D graphing. Construction uses commands like Surface, CurveCartesian, and parameters a, b, h, along with objects like Text1 and Zyl. Suitable for high school and university students learning spatial geometry and 3D shapes.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw the intersection of a cylinder and a paraboloid. First, create a paraboloid with the equation z=v, x=sqrt(v)*cos(u), y=sqrt(v)*sin(u), where u ranges from 0 to 2π and v from 0 to h. Then, create a cylinder with the equation x=a*cos(u), y=b*sin(u), z=v, where u ranges from 0 to 2π and v from -h to h. Finally, plot the intersection curve defined by the parametric equations x=sqrt(b*sqrt(1-t^2)-a^2*t^2), y=a*t, z=b*sqrt(1-t^2), with t ranging from -π to π.
The prompt above was compiled from the original drawing commands and is for reference only.