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mi-sphere (trigonometric parameterized sphere)

mi-sphere (trigonometric parameterized sphere)
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2017-11-27 06:09
AngleTrigonometrySolid GeometryParametric Equation
This example demonstrates the construction of a sphere using trigonometric parametric equations, visualized in GeoGebra. It highlights how a sphere can be represented as a surface generated by two angular parameters, typically involving sine and cosine functions. The core mathematical concepts include parametric surface representation, spherical coordinates, and the role of trigonometric functions in 3D geometry. The construction uses GeoGebra commands such as PointIn to place a point P on the sphere, Angle to define the angular parameters a1 and a2, and Surface to render the sphere s from these parameters. Additional objects like n, m, sP2, and sP21 support the visualization. This is suitable for high school or undergraduate students studying solid geometry, parametric equations, or 3D graphing, and serves as an interactive teaching tool for understanding spherical surfaces.

GeoGebra Commands & Parameters

  1. PointIn(Sphere[(0, 0, 0), 1]) → P
  2. Angle((1, 0), (0, 0), (x(P), y(P))) → a1
  3. Surface(((cos(u) * cos(v)) * cos(a2)) - (sin(a2) * sin(v)), (sin(u) * cos(v)), ((cos(u) * cos(v)) * sin(a2)) + (cos(a2) * sin(v)), u, 0, (2 * π), v, n, π / 2) → sP2
  4. Surface((cos(u) * cos(v)), (sin(u) * cos(v)), sin(v), u, 0, (2 * π), v, m, π / 2) → s
  5. Surface(((((cos(u) * cos(v)) * cos(a2)) - (sin(a2) * sin(v))) * cos(a1)) - ((sin(a1) * sin(u)) * cos(v)), ((((cos(u) * cos(v)) * cos(a2)) - (sin(a2) * sin(v))) * sin(a1)) + ((cos(a1) * sin(u)) * cos(v)), ((cos(u) * cos(v)) * sin(a2)) + (cos(a2) * sin(v)), u, 0, (2 * π), v, n, π / 2) → sP21

AI Drawing Prompt Reference

Draw a unit sphere with its surface generated by two angular parameters using trigonometric parametric equations. Place a point on the sphere and show two angles associated with it: one horizontal and one vertical. The sphere is rendered with a grid of latitude and longitude lines. Additionally, display a rotated auxiliary surface to illustrate how the sphere changes under angular rotation. The overall figure emphasizes the parametric representation of the sphere and the relationship between angles and points on the surface.
The prompt above was compiled from the original drawing commands and is for reference only.