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环面斜驶线（Loxodrome）及其相关构造

环面斜驶线（Loxodrome）及其相关构造
几何画板及绘图内容加载中，请稍等……
2019-12-17 02:25
数列立体几何变换参数方程轨迹
① 本范例演示了在三维环面（Torus）上绘制斜驶线（Loxodrome，即与经线成固定角度的曲线）以及与之相关的多种几何对象，如圆、线段、多边形和曲面。② 核心数学概念包括环面的参数方程、空间曲线（CurveCartesian）、曲面（Surface）、几何变换（Mirror）、序列（Sequence）以及最小公倍数（LCM）在构造周期性图案中的应用。③ 作图要点：使用GeoGebra命令如Point、Segment、Circle、CircleArc、Midpoint、Intersect、Line、Mirror、LCM、Sequence、Polygon、Surface、CurveCartesian、OrthogonalLine、Zip、OrthogonalPlane、KeepIf、Unique等，构建了环面、斜驶线、辅助圆、切线、法线以及由序列生成的重复结构；对象包括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等。④ 适合高中至大学阶段的数学教学，用于三维几何、参数曲线曲面、变换与序列的综合探究，也可作为数学建模或计算机辅助几何设计的示例。

GeoGebra 命令及参数

  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 绘图提示语参考

在一个环面上绘制一条等距线，使其与子午线保持恒定角度，并展示相关的几何对象，如圆、线段、多边形和曲面。核心概念包括环面的参数方程、空间曲线、几何变换、序列和最小公倍数创建周期性图案。首先构建环面，然后绘制等距线及其辅助圆、切线和法线，最后通过序列生成重复结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。