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Spherical Coordinates Demonstration by 古徳純一

Spherical Coordinates Demonstration by 古徳純一
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2016-08-12 16:46
TrigonometryCoordinate GeometrySolid GeometryParametric EquationGeometric Construction
This example demonstrates how to draw a brick shape in spherical coordinates. The core mathematical concepts involved include the definition and application of spherical coordinates. Key construction points involve using GeoGebra commands and objects such as PointIn, If, CurveCartesian, Point, Surface, Intersect, Segment, CircumcircleArc, and CircumcircleSector to construct a 3D shape. Suitable for high school students and above to learn about solid geometry and spherical coordinates.

GeoGebra Commands & Parameters

  1. PointIn(o) → A
  2. If(x(A) > 0, atan(y(A) / x(A)), atan(y(A) / x(A)) + π) → α_1
  3. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 3.141592653589793) → c
  4. Point(c) → B
  5. If(x(B) > 0, atan(y(B) / x(B)), atan(y(B) / x(B)) + π) → α_2
  6. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, π - β_1) → d
  7. Point(d) → C
  8. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, Δα) → e
  9. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_1, β_2, t, α_1, α_2) → a
  10. Intersect(yAxis, zAxis) → E
  11. Segment(A, E) → f
  12. Point(f) → F
  13. Segment(E, F) → h
  14. Segment(F, A) → i
  15. Surface(((h * sin(u)) * cos(t)), ((h * sin(u)) * sin(t)), (h * cos(u)), u, β_1, β_2, t, α_1, α_2) → b
  16. Surface(((t * sin(β_1)) * cos(u)), ((t * sin(β_1)) * sin(u)), (t * cos(β_1)), u, α_1, α_2, t, h, 1) → j
  17. Surface(((t * sin(β_2)) * cos(u)), ((t * sin(β_2)) * sin(u)), (t * cos(β_2)), u, α_1, α_2, t, h, 1) → k
  18. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, h, 1) → l
  19. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, h, 1) → m
  20. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, Δα) → g
  21. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → n
  22. CurveCartesian((cos(α_2) * sin(β_1 + t)), (sin(α_2) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → p
  23. Segment(B, B') → q
  24. Segment(C, C') → r
  25. Segment(D, D') → s
  26. CurveCartesian(((h * cos(α_1 + t)) * sin(β_1)), ((h * sin(α_1 + t)) * sin(β_1)), (h * cos(β_1)), t, 0, Δα) → t
  27. CurveCartesian(((h * cos(α_1 + t)) * sin(β_2)), ((h * sin(α_1 + t)) * sin(β_2)), (h * cos(β_2)), t, 0, Δα) → u
  28. CurveCartesian(((h * cos(α_1)) * sin(β_1 + t)), ((h * sin(α_1)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → v
  29. CurveCartesian(((h * cos(α_2)) * sin(β_1 + t)), ((h * sin(α_2)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → w
  30. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 6.283185307179586) → a_1
  31. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, 6.28319) → b_1
  32. CurveCartesian((cos(α_1) * sin(t)), (sin(α_1) * sin(t)), cos(t), t, 0, 3.141592653589793) → c_1
  33. CurveCartesian((cos(α_2) * sin(t)), (sin(α_2) * sin(t)), cos(t), t, 0, 3.141592653589793) → d_1
  34. Segment(B, E) → f_1
  35. Segment(C, E) → g_1
  36. Segment(D, E) → h_1
  37. CircumcircleArc(H, C, G) → k_1
  38. CircumcircleArc(H, B, G) → p_1
  39. Segment(I, E) → i_1
  40. Segment(J, E) → j_1
  41. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, 0, β_1, t, α_1, α_2) → m_1
  42. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, 0, h) → n_1
  43. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, 0, β_1, t, 0, 1) → o_1
  44. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_2, 1.5707963267948966, t, 0, 1) → q_1
  45. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, 0, β_1, t, 0, 1) → r_1
  46. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_2, 1.5708, t, 0, 1) → s_1
  47. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, 0, h) → t_1
  48. CurveCartesian(cos(t), sin(t), 0, t, 0, 1.5707963267948966) → e_1
  49. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_2, 3.141592653589793, t, α_1, α_2) → l_1
  50. Surface((t * cos(α_1)), (t * sin(α_1)), u, t, 0, 1, u, -1, 1) → u_1
  51. Surface((t * cos(α_2)), (t * sin(α_2)), u, t, 0, 1, u, -1, 1) → v_1
  52. CircumcircleSector(I, K, J) → c_2

AI Drawing Prompt Reference

Draw a brick shape using spherical coordinates to demonstrate the definition and application of spherical coordinates. Start by plotting curves generated by spherical coordinate parametric equations, then use these curves to construct a 3D brick shape. Key steps involve determining point positions, forming the faces of the brick with curves and surfaces, and refining the shape through intersection and connection operations, suitable for learning solid geometry and spherical coordinates.
The prompt above was compiled from the original drawing commands and is for reference only.