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Spheres' Hemispheres (3D)

Spheres' Hemispheres (3D)
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2016-01-20 21:23
VectorTrigonometrySolid GeometryParametric Equation
This example demonstrates the interactive construction and observation of two hemispheres of a sphere in three-dimensional space using sliders. Key math concepts include solid geometry, vector operations, trigonometric functions (sine and cosine), parametric equations, and geometric transformations (translation and reflection). Construction highlights the use of GeoGebra commands/objects like Vector, OrthogonalLine, Translate, Mirror, Sequence, CurveCartesian, and Segment. Suitable for high school and university-level students learning 3D geometry, vectors, and parametric equations, or for foundational mathematics education.

GeoGebra Commands & Parameters

  1. Vector(B_1, X) → u
  2. Vector(B_1, Y) → v
  3. OrthogonalLine(View3D, yAxis) → a_1
  4. OrthogonalLine(View3D, xAxis) → b
  5. Vector(B_1, Z) → w
  6. Translate(O, Vector[a * w]) → O'
  7. Mirror(O', O) → O''
  8. Sequence(CurveCartesian[x(O'' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), y(O'' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), s, 0.0, 3.141592653589793 * n / 18.0], t, 0.0, m / 24.0 * 2.0 * 3.141592653589793, 0.2618) → liste1
  9. Sequence(CurveCartesian[x(O'' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), y(O'' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), t, 0.0, 2.0 * m * 3.141592653589793 / 12.0], s, 0.0, 3.141592653589793 * n / 18.0, 0.34907) → liste2
  10. Sequence(CurveCartesian[x(O' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), y(O' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), s, -(3.141592653589793) * n / 18.0, 0.0], t, -(m) / 24.0 * 2.0 * 3.141592653589793, 0.0, 0.2618) → liste3
  11. Sequence(CurveCartesian[x(O' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), y(O' + f_1(s) * cos(-(t)) * u + f_1(s) * sin(t) * v - f_2(s) * w), t, -2.0 * m * 3.141592653589793 / 12.0, 0.0], s, -(3.141592653589793) * n / 18.0, 0.0, 0.34907) → liste4
  12. CurveCartesian(x(O' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), y(O' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), i, 0.0, 360.0) → d
  13. CurveCartesian(x(O'' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), y(O'' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), i, 0.0, 360.0) → e
  14. CurveCartesian(x(O'' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), y(O'' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), i, 0.0, 360.0) → f
  15. CurveCartesian(x(O' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), y(O' + R * sin(i°) * u - R * cos(i°) * v + R * 0.0 * sin(i°) * w), i, 0.0, 360.0) → h
  16. Segment(O', O'') → i