Sphere-Cylinder-Plane Transformation: 3D Geometric Mapping

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2018-12-03 19:56

This example demonstrates the geometric transformation among a sphere, a cylinder, and a plane, showcasing how parametric curves and surfaces map between these 3D shapes. Core mathematical concepts include solid geometry, parametric equations, surface and curve construction, and spatial transformations such as scaling, projection, or unwrapping. Construction highlights: using GeoGebra commands CurveCartesian for parametric curves and Surface for surfaces; objects include a button (button1), sliders (R, t), spheres (Sph, SPH), cylinder-related lines (SphLine, Circl2, etc.), and multiple text labels (text1 to text5) for dynamic parameter control and result display. Suitable for high school to undergraduate mathematics education, particularly in solid geometry, analytic geometry, or computer-aided geometry courses, helping students intuitively understand mappings between 3D shapes and their parametric representations.

GeoGebra Commands & Parameters

CurveCartesian(((R * sin(((-t) * π))), 0, ((-R) * cos(((-t) * π)))), t, 0, 1) → Sph
CurveCartesian(((t * (-R)) + ((1 - t) * (R * sin((((-k)) / 4 * π)))), 0, (t * (((k / 4 * 2) * R) - R)) + ((1 - t) * ((-R) * cos(((-k) * π / 4))))), k, 0, 4) → SphLine
Surface(SphLine, 360°, zAxis) → SPH
CurveCartesian(((R * cos(((2 * (-t)) * π))), (R * sin(((2 * (-t)) * π)))), t, 0, 1) → Circl2
CurveCartesian((((t - 1) * (-R)) + ((2 - t) * (R * cos(((2 * (-k)) * π)))), ((t - 1) * ((((k * 2) * π) * R) - (π * R))) + ((2 - t) * (R * sin(((2 * (-k)) * π))))), k, 0, 1) → SphLine2
Surface(((t - 1) * (-R)) + ((2 - t) * (R * cos(((2 * ((-u)) / 8) * π)))), ((t - 1) * ((((u / 8 * 2) * π) * R) - (π * R))) + ((2 - t) * (R * sin(((2 * ((-u)) / 8) * π)))), ((v / 4 * 2) * R) - R, u, 0, 8, v, 0, 4) → SPH2
CurveCartesian(((-R), (((t * 2) * π) * R) - (π * R), R), t, 0, 1) → Lin2
CurveCartesian(((-R), (-π), ((t * 2) * R) - R), t, 0, 1) → Line