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Visualizing Points on a Sphere: (R;φ;θ)→(x,y,z) and (x,y,z)→(R;φ;θ)

Visualizing Points on a Sphere: (R;φ;θ)→(x,y,z) and (x,y,z)→(R;φ;θ)
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2019-01-22 00:10
VectorTrigonometryCoordinate GeometrySolid GeometryParametric Equation
This example demonstrates the visualization of points on a sphere through the conversion between spherical coordinates (R;φ;θ) and Cartesian coordinates (x,y,z). Key mathematical concepts include spherical coordinates, Cartesian coordinates, vectors, trigonometric functions, and transformations. The construction utilizes GeoGebra commands/objects such as Sphere, Vector, Angle, Length, Textfield, CurveCartesian, and CircleSector. It is suitable for high school students and above, as well as university-level foundational mathematics education.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), R) → a
  2. Vector(B) → u_B
  3. Angle(Vector[(0, 0, R)], u_B) → θ'
  4. Length(B) → läB
  5. Textfield(R) → InputBox1
  6. Textfield(φ) → InputBox2
  7. Textfield(θ) → InputBox3
  8. CurveCartesian(((R * cos(θ)) * cos(φ)), ((R * cos(θ)) * sin(φ)), (R * sin(θ)), φ, (-π), π) → b
  9. CircleSector((0, 0, (R * sin(θ))), A, B, xOyPlane) → d
  10. Vector((0, 0, (R * sin(θ))), B) → u_A
  11. Vector((0, 0, (R * sin(θ))), A) → u_A0
  12. Angle(u_Bxy, Vector[B]) → γ