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Torus Loxodrome and Related Constructions

Torus Loxodrome and Related Constructions
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2019-12-17 02:25
SequenceSolid GeometryTransformationParametric EquationLocus
① This example demonstrates drawing a loxodrome (a curve making a constant angle with the meridians) on a torus, along with various related geometric objects such as circles, segments, polygons, and surfaces. ② Core mathematical concepts include parametric equations of a torus, space curves (CurveCartesian), surfaces (Surface), geometric transformations (Mirror), sequences (Sequence), and the use of least common multiple (LCM) to create periodic patterns. ③ Construction highlights: GeoGebra commands such as Point, Segment, Circle, CircleArc, Midpoint, Intersect, Line, Mirror, LCM, Sequence, Polygon, Surface, CurveCartesian, OrthogonalLine, Zip, OrthogonalPlane, KeepIf, Unique are used to build the torus, loxodrome, auxiliary circles, tangents, normals, and repeated structures generated by sequences; objects include g_y, J, I, j, F, R, mTr, r, rTr, e, g_x, o, f, K, f_1, A, c, B, C, g_d, eTR, etc. ④ Suitable for high school to university-level mathematics teaching, for exploring 3D geometry, parametric curves and surfaces, transformations and sequences, and as an example for mathematical modeling or computer-aided geometric design.

GeoGebra Commands & Parameters

  1. Point(g_y) → J
  2. Point(g_y) → I
  3. Segment(I, J) → j
  4. Point(j) → F
  5. Circle(mTr, rTr) → e
  6. Point(g_x) → o
  7. Circle(o, F) → f
  8. Point(f) → K
  9. CircleArc(o, F, K) → f_1
  10. Midpoint(o, mTr) → A
  11. Circle(A, mTr) → c
  12. Intersect(e, c) → B, C, ,
  13. Line(o, C) → g_d
  14. Mirror(eTR, xAxis) → eTR'_1
  15. Mirror(eTR, g_x) → eTR'
  16. Mirror(eTR', xAxis) → eTR''
  17. LCM(n', m') → kgvAnz
  18. Sequence(k, k, (-kgvAnz), kgvAnz) → Listkgv
  19. Mirror(g_d, g_x) → g'_d
  20. Intersect(g'_d, gl1) → E
  21. Polygon(eTR, eTR', eTR'', eTR'_1) → v1, h, i, l, p_1
  22. Segment(eTR'', eTR) → d_{+}
  23. Mirror(mTr, g_x) → mTr'
  24. Mirror(rTr, g_x) → rTr'
  25. Mirror(e, g_x) → e'
  26. Mirror(C, g_x) → C'
  27. Mirror(B, g_x) → B'
  28. Circle(o, C) → g
  29. Surface(((R + (r * cos(σ))) * cos(τ)), ((R + (r * cos(σ))) * sin(τ)), (r * sin(σ)), σ, (-π), π, τ, (-π), π) → Torus
  30. CurveCartesian(((R + (r * cos((m / R * τ)))) * cos((n / r' * τ))), ((R + (r * cos((m / R * τ)))) * sin((n / r' * τ))), (r * sin((m / R * τ))), τ, (((-kgvAnz) * R) * π), ((kgvAnz * R) * π)) → KurveVers
  31. CurveCartesian((((R + (r * cos((τ / y(eTR) * π)))) * cos(τ)), ((R + (r * cos((τ / y(eTR) * π)))) * sin(τ)), (r * sin((τ / y(eTR) * π)))), τ, ((-R) * π), (R * π)) → Diagonale
  32. Point(d_{+}) → d
  33. CurveCartesian((sqrt(R^(2) - r^(2)) / (R - (r * cos(γ))) * ((sqrt(R^(2) - r^(2)) * cos(γ)), (sqrt(R^(2) - r^(2)) * sin(γ)), (r * sin(γ)))), γ, (-π), π) → wLox
  34. OrthogonalLine(o, g_d) → g_n
  35. Circle(m_v, rTr) → k
  36. Circle(o, m_v) → s
  37. Point(wLox) → H
  38. Mirror(rTr, mTr) → rTr'_1
  39. Mirror(rTr'_1, o) → z_1
  40. Circle(o, rTr'_1) → c_1
  41. Line(m_v, o) → t
  42. Segment(eTR'_1, eTR') → d_{-}
  43. CurveCartesian((Rr / (R - (r * cos((m' / n' * γ)))) * ((Rr * cos(γ)), (Rr * sin(γ)), (r * sin((m' / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wLoxVar0
  44. CurveCartesian((Rr / (R - (r * cos((m' / n' * γ)))) * ((Rr * cos(γ + σ_V)), (Rr * sin(γ + σ_V)), (r * sin((m' / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wLoxVar_σ
  45. CurveCartesian((((r + (R * cos(φ))) * cos(σ_V)) + ((Rr * sin(φ)) * sin(σ_V)), ((r + (R * cos(φ))) * sin(σ_V)) - ((Rr * sin(φ)) * cos(σ_V)), (r * sin(φ))), φ, (-π), π) → Vill01_σ
  46. CurveCartesian(((((-r) + (R * cos(φ))) * cos(σ_V)) + ((Rr * sin(φ)) * sin(σ_V)), (((-r) + (R * cos(φ))) * sin(σ_V)) - ((Rr * sin(φ)) * cos(σ_V)), (r * sin(φ))), φ, (-π), π) → Vill02_σ
  47. Sequence(k, k, 0, (2 * m')) → L_j
  48. Sequence(k, k, 0, (2 * n')) → Lj_1
  49. CurveCartesian((Rr / (R - (r * cos((((-m')) / n' * γ)))) * ((Rr * cos(γ + σ_V)), (Rr * sin(γ + σ_V)), (r * sin((((-m')) / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wLoxVar'_σ
  50. Point(zAxis) → G
  51. Line(o, G) → g_z
  52. CurveCartesian(((R + (r * cos(σ))) * cos(π / m')), ((R + (r * cos(σ))) * sin(π / m')), (r * sin(σ)), σ, (-π), π) → QVersuch
  53. Zip(CurveCartesian[((R + (r_1 * cos(σ))) * cos((k / m' * π) + σ_V)), ((R + (r_1 * cos(σ))) * sin((k / m' * π) + σ_V)), (r_1 * sin(σ)), σ, (-π), π], k, L_j) → ListQkreise
  54. Sequence(k, k, 0, 5) → L_6
  55. Zip(CurveCartesian[(((r + (R * cos(φ))) * cos(σ_V + (k * π / 3))) + ((Rr * sin(φ)) * sin(σ_V + (k * π / 3))), ((r + (R * cos(φ))) * sin(σ_V + (k * π / 3))) - ((Rr * sin(φ)) * cos(σ_V + (k * π / 3))), (r * sin(φ))), φ, (-π), π], k, L_6) → ListVill1
  56. Zip(CurveCartesian[((((-r) + (R * cos(φ))) * cos(σ_V + (k * π / 3))) + ((Rr * sin(φ)) * sin(σ_V + (k * π / 3))), (((-r) + (R * cos(φ))) * sin(σ_V + (k * π / 3))) - ((Rr * sin(φ)) * cos(σ_V + (k * π / 3))), (r * sin(φ))), φ, (-π), π], k, L_6) → ListVill2
  57. Line(o, N) → Normale
  58. OrthogonalPlane(o, Normale) → EVill
  59. CurveCartesian((rR / (R + (r * cosh((m' / n' * γ)))) * ((rR * cos(γ + σ_V)), (rR * sin(γ + σ_V)), (r * sinh((m' / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wSpinLoxVar_σ
  60. CurveCartesian((rR / ((-R) + (r * cosh((m' / n' * γ)))) * ((rR * cos(γ + σ_V)), (rR * sin(γ + σ_V)), (r * sinh((m' / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wSpinLox2Var_σ
  61. CurveCartesian(((2 * R / (1 + ((m' / n' * γ))^(2))) * (cos(γ + σ_V), sin(γ + σ_V), (m' / n' * γ))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wLoxHorn_σ
  62. CurveCartesian(((2 * R / (1 + ((m' / n' * γ))^(2))) * (cos(γ + σ_V), sin(γ + σ_V), (((-m')) / n' * γ))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wLoxHorn'_σ
  63. Point(p_1) → D
  64. Mirror(D, g_x) → D'
  65. Segment(D, D') → q
  66. CurveCartesian(((2 * x(eTR)) * ((t * n) - floor((t * n)) - 1 / 2)) + (((2 * y(eTR)) * ((t * m) - floor((t * m)) - 1 / 2)) * ί), t, (-1 / n * kgvAnz), (1 / n * kgvAnz)) → LinRechtEckLoxos
  67. CurveCartesian(((2 * x(eTR)) * (((-t) * n) - floor(((-t) * n)) - 1 / 2)) + (((2 * y(eTR)) * ((t * m) - floor((t * m)) - 1 / 2)) * ί), t, (-1 / n * kgvAnz), (1 / n * kgvAnz)) → LinRechtEckLoxos'
  68. Point(h) → L
  69. Mirror(L, g_y) → L'
  70. Segment(L, L') → a
  71. Intersect(a, q) → p
  72. CurveCartesian(((R + (r * cos(σ))) * cos((x(L) / x(eTR) * π))), ((R + (r * cos(σ))) * sin((x(L) / x(eTR) * π))), (r * sin(σ)), σ, (-π), π) → Querkreis
  73. CurveCartesian((((R + (r * cos((y(D) / y(eTR) * π)))) * cos(τ)), ((R + (r * cos((y(D) / y(eTR) * π)))) * sin(τ)), (r * sin((y(D) / y(eTR) * π)))), τ, (-π), π) → Laengskreis
  74. Zip(((Rr^(2) * cos((k * π)) / (R - (r_1 * cos(((m' / n' * k) * π))))), 0, ((Rr * r_1) * sin(((m' / n' * k) * π)) / (R - (r_1 * cos(((m' / n' * k) * π)))))), k, Listkgv) → QPkteNeu
  75. KeepIf(x(p) ≥ 0, p, QPkteNeu) → QPkteNeu0
  76. Zip(((Rr^(2) * cos((k * π)) / (R - (r_1 * cos((((k * m') + 1) / n' * π))))), 0, ((Rr * r_1) * sin((((k * m') + 1) / n' * π)) / (R - (r_1 * cos((((k * m') + 1) / n' * π)))))), k, Listkgv) → QPkteNeu2
  77. KeepIf(x(p) ≥ 0, p, QPkteNeu2) → QPkteNeu20
  78. Unique(Join[{QPkteNeu0, QPkteNeu20}]) → QPkteTot
  79. CurveCartesian((rR / ((-R) + (r * cosh((m' / n' * γ)))) * ((rR * cos((-γ) + σ_V)), (rR * sin((-γ) + σ_V)), (r * sinh((m' / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wSpinLoxVar'_σ
  80. CurveCartesian((rR / (R + (r * cosh((((-m')) / n' * γ)))) * ((rR * cos(γ + σ_V)), (rR * sin(γ + σ_V)), (r * sinh((((-m')) / n' * γ))))), γ, ((-kgvAnz) * π), (kgvAnz * π)) → wSpinLox2Var'_σ
  81. Zip(Circle[g_z, P], P, QPkteTot) → LaengsKreiseNeu

AI Drawing Prompt Reference

Draw a loxodrome on a torus, ensuring it maintains a constant angle with the meridians, while displaying related geometric objects like circles, segments, polygons, and surfaces. Key concepts involve the parametric equations of a torus, space curves, geometric transformations, sequences, and using the least common multiple to create periodic patterns. Begin by constructing the torus, then draw the loxodrome along with its auxiliary circles, tangents, and normals, finally generating repeated structures through sequences.
The prompt above was compiled from the original drawing commands and is for reference only.