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Spherical Coordinates (robphy) - 3D Visualization

Spherical Coordinates (robphy) - 3D Visualization
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2019-02-01 06:30
VectorCoordinate GeometrySolid GeometryCalculusParametric Equation
① This GeoGebra example dynamically illustrates the complete structure of spherical coordinates in 3D space, including the radial, polar, and azimuthal unit vector fields, along with grid lines and coordinate curves on a sphere. ② Core mathematical concepts include conversion between spherical and Cartesian coordinates, decomposition of 3D vectors, geometric meaning of unit vectors (r-hat, θ-hat, φ-hat), and parametric surface construction of a sphere. ③ Construction highlights: uses Surface to draw the sphere, Vector to generate vector fields in three directions, CurveCartesian for coordinate curves, Segment for reference lines; employs Zip, Flatten, Simplify for efficient list management of many vectors, and sliders (r, theta, phi) for interactive exploration. ④ Suitable for university-level calculus, multivariable functions, or vector analysis courses to teach spherical coordinates and curvilinear coordinate systems, as well as for advanced high school 3D geometry demonstrations.

GeoGebra Commands & Parameters

  1. Simplify(sin(theta(x, y, z))) → sinTheta
  2. Flatten(Sequence[Sequence[Sequence[(i, j, k), i, (-N), N], j, (-N), N], k, (-N), N]) → grid
  3. Zip(Vector[(1, 0, 0)], P, grid) → xHats
  4. Simplify(cos(theta(x, y, z))) → rZ
  5. Simplify((sinTheta(x, y, z) * sinPhi(x, y, z))) → rY
  6. Simplify((sinTheta(x, y, z) * cosPhi(x, y, z))) → rX
  7. Zip(Vector[(rX(P), rY(P), rZ(P))], P, grid) → rHats
  8. Zip(Vector[P, P + If[abs(v) > maxMag, Vector[(maxMag * UnitVector[v])], Vector[v]]], P, grid, Vector[v], rHats) → fieldOFrHAT
  9. Simplify(cos(theta(x, y, z))) → cosTheta
  10. Zip(Vector[(thetaX(P), thetaY(P), thetaZ(P))], P, grid) → thetaHats
  11. Zip(Vector[P, P + If[abs(v) > maxMag, Vector[(maxMag * UnitVector[v])], Vector[v]]], P, grid, Vector[v], thetaHats) → fieldOFthetaHAT
  12. Zip(Vector[(phiX(P), phiY(P), phiZ(P))], P, grid) → phiHats
  13. Zip(Vector[P, P + If[abs(v) > maxMag, Vector[(maxMag * UnitVector[v])], Vector[v]]], P, grid, Vector[v], phiHats) → fieldOFphiHAT
  14. Surface((u * cos(PhiValue)), (u * sin(PhiValue)), v, u, 0, 5, v, -5, 5) → constPhiCyl
  15. Vector(Q, Q + ((phiX(Q), phiY(Q), phiZ(Q)) * 2)) → QphiHat
  16. Vector(Q, Q + ((rX(Q), rY(Q), rZ(Q)) * 2)) → QrHat
  17. Vector(Q, Q + ((thetaX(Q), thetaY(Q), thetaZ(Q)) * 2)) → QthetaHat
  18. CurveCartesian((((rValue * sin(u)) * cos(PhiValue)), ((rValue * sin(u)) * sin(PhiValue)), (rValue * cos(u))), u, 0, ThetaValue) → ThetaVar
  19. CurveCartesian((((rValue * sin(ThetaValue)) * cos(u)), ((rValue * sin(ThetaValue)) * sin(u)), (rValue * cos(ThetaValue))), u, 0, PhiValue) → PhiVar
  20. Surface(((u * cos(PhiValue)) * sin(v)), ((u * sin(PhiValue)) * sin(v)), (u * cos(v)), u, 0, rValue + 2, v, 0, π) → constPhiRad
  21. Surface(((u * cos(v)) * sin(ThetaValue)), ((u * sin(v)) * sin(ThetaValue)), (u * cos(ThetaValue)), u, 0, rValue + 2, v, 0, (2 * π)) → constTheta
  22. CurveCartesian((((u * sin(ThetaValue)) * cos(PhiValue)), ((u * sin(ThetaValue)) * sin(PhiValue)), (u * cos(ThetaValue))), u, 0, rValue) → rVar
  23. Surface((u * sin(v)), ((u * 0) * sin(v)), (u * cos(v)), u, rValue, rValue + 0.1, v, 0, π) → zeroPhiRad
  24. Flatten(Sequence[Sequence[Sequence[(i, j, k), i, (-N), N], j, (-N), N], k, (-N), N]) → gridRECT
  25. Flatten(Sequence[Sequence[Sequence[(((r * sin((t * π / ((2 * N))))) * cos(((p * 2) * π / ((2 * N))))), ((r * sin((t * π / ((2 * N))))) * sin(((p * 2) * π / ((2 * N))))), (r * cos((t * π / ((2 * N)))))), r, 1, (2 * N)], t, 0, (2 * N)], p, 0, (2 * N)]) → gridSPH
  26. CurveCartesian((((rValue * sin(ThetaValue)) * cos(u)), ((rValue * sin(ThetaValue)) * sin(u)), 0), u, 0, PhiValue) → PhiVarXYPplane
  27. Segment((0, 0, 0), (((rValue * sin(ThetaValue)) * cos(PhiValue)), ((rValue * sin(ThetaValue)) * sin(PhiValue)), 0)) → RadiusProjectionOnXYplane
  28. Segment(Q, (((rValue * sin(ThetaValue)) * cos(PhiValue)), ((rValue * sin(ThetaValue)) * sin(PhiValue)), 0)) → VerticalDroppedFromQ

AI Drawing Prompt Reference

Visualize the structure of spherical coordinates in 3D space, including radial, polar, and azimuthal unit vector fields, along with grid lines and coordinate curves on a sphere. Show the sphere surface, represent unit vectors in three directions, draw coordinate curves, and add reference lines. Enable interactive exploration of radial distance, polar angle, and azimuthal angle using sliders, and illustrate the conversion between spherical and Cartesian coordinates.
The prompt above was compiled from the original drawing commands and is for reference only.