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经纬度—距离的几何可视化与球面模型

经纬度—距离的几何可视化与球面模型
几何画板及绘图内容加载中，请稍等……
2021-01-04 12:44
角向量解析几何几何作图
该范例通过交互式几何构造展示“给定两点经纬度如何对应距离”，并在球面/球体视角下呈现经线、纬线与测地距离相关的空间关系。核心数学概念包括：地理坐标（纬度/经度）、向量表示、线段长度度量、角度与圆弧、以及利用球面几何（Sphere、CircleArc 等）进行距离可视化。作图要点：使用 Length 计算长度量度；Element、Segment、Sequence 构造关键点与连接关系；Sphere 生成球体/地球模型；Point、Vector 表示观测点与方向；Circle 与 CircleArc 画出与经纬相关的圆与圆弧；Angle 用于刻画方位或夹角；Textfield/文本用于显示参数（如经纬度、角度或年份等变量）；KeepIf 用于筛选满足条件的点集；同时通过多条经线/纬线相关对象（如 verMeridianos、verParalelos、verEcuador、verGreenwich 等）与多角度构造（如 Ep0t、Ep90t、Em45t、Em90t 等）组织展示效果，体现从参数到几何图形的映射。适合中学高年级到大学基础课程，用于几何建模、向量与三角关系的演示，以及地理信息相关的数学直观训练。

GeoGebra 命令及参数

  1. Length(capitalesLista) → capitalesNum
  2. Element(tiempo, 7) → año
  3. Segment(PS + (0.075 * PS), PN + (0.075 * PN)) → eje
  4. Length(ciudadesLista) → ciudadesNum
  5. Sequence(If[Element[ciudadesLista, k] > 0, k, 0], k, 1, ciudadesNum) → ciudadesElegidas1, {1, 0, 3, 0, 0, 0, 0, 0, 9, 0, 0, 0, 13, 0, 0, 0}
  6. KeepIf(x > 0, ciudadesElegidas1) → ciudadesElegidas2
  7. Length(ciudadesElegidas2) → ciudadesNumElegidas
  8. Sequence((5; Element[ciudadesLongitud, Element[ciudadesElegidas2, k]]°; Element[ciudadesLatitud, Element[ciudadesElegidas2, k]]°), k, 1, ciudadesNumElegidas) → ciudades, {(5; 316.77*°; -22.9*°), (5; 353.88691*°; 43.48818*°), (4.999999999999999; 241.75*°; 34.05*°), (5; 286*°; 40.72*°)}
  9. Sequence((1.5 * Element[ciudades, k]), k, 1, ciudadesNumElegidas) → baseCiudades, {(5.033887424956106, -4.732097411365642, -2.9184296260515468), (5.4104311124157975, -0.5794585197841322, 5.1615368836861695), (-2.94126498214301, -5.473960329575279, 4.199371216629386), (1.5668054176673767, -5.464099844261399, 4.892722521777698)}
  10. Sequence(Text[Element[ciudadesNombres, Element[ciudadesElegidas2, k]], Element[baseCiudades, k] + (0.03, 0.03, 0.03)], k, 1, ciudadesNumElegidas) → ciudadesNombresVisibles, {"Rio de Janeiro", "Madrid", "Los Angeles", "New York"}
  11. Sequence(Segment[(0, 0, 0), Element[baseCiudades, k]], k, 1, ciudadesNumElegidas) → laserOtros, {7.5, 7.5, 7.499999999999999, 7.5}
  12. Sphere((0, 0, 0), (0.999 * 5)) → tierra
  13. Segment(PN, PS) → eje2
  14. Point(eje2) → L
  15. Vector((1; (-rot) + 180°; (-inclinacion))) → dir
  16. Circle((0, 0, 0), 5, eje) → ecuador2
  17. Circle((5; 0; (-anguloPolar)), (5; 90°; (-anguloPolar)), (5; 180°; (-anguloPolar))) → antartico
  18. Circle((5; 0; anguloPolar), (5; 90°; anguloPolar), (5; 180°; anguloPolar)) → artico
  19. Circle((5; 0; anguloEcliptica), (5; 90°; anguloEcliptica), (5; 180°; anguloEcliptica)) → cancer
  20. Circle((5; 0; (-anguloEcliptica)), (5; 90°; (-anguloEcliptica)), (5; 180°; (-anguloEcliptica))) → capricornio
  21. Circle((0, 0, 0), 5, eje) → ecuador
  22. Circle(PN, (5; 0; 0), PS) → greenwich
  23. Sequence(Circle[PN, (5; (15° * k); 0), PS], k, 1, 11) → meridianos, {X = (-2.0825185391870887E-15, -4.4408920985006257E-16, 0) + (3.4150635094610964 cos(t) - 3.4150635094610973 sin(t), 0.9150635094610965 cos(t) - 0.9150635094610966 sin(t), -3.5355339059327373 cos(t) - 3.5355339059327378 sin(t)), X = (1.379275100632243E-16, 3.7493994566546446E-32, 0) + (3.0618621784789735 cos(t) - 3.061862178478973 sin(t), 1.7677669529663687 cos(t) - 1.7677669529663687 sin(t), -3.5355339059327378 cos(t) - 3.5355339059327378 sin(t)), X = (-3.061616997868383E-16, 4.440892098500626E-16, 0) + (2.500000000000001 cos(t) - 2.500000000000001 sin(t), 2.5 cos(t) - 2.4999999999999996 sin(t), -3.5355339059327378 cos(t) - 3.5355339059327378 sin(t)), X = (-8.4117094861807E-17, 4.440892098500626E-16, 0) + (1.7677669529663698 cos(t) - 1.7677669529663693 sin(t), 3.061862178478973 cos(t) - 3.061862178478972 sin(t), -3.5355339059327378 cos(t) - 3.5355339059327378 sin(t)), X = (-5.282063047118696E-16, -8.881784197001252E-16, 0) + (0.9150635094610976 cos(t) - 0.9150635094610977 sin(t), 3.4150635094610964 cos(t) - 3.4150635094610973 sin(t), -3.5355339059327386 cos(t) - 3.5355339059327378 sin(t)), X = (-3.0616169978683836E-16, 3.7493994566546446E-32, 0) + (0, 3.5355339059327378 cos(t) - 3.5355339059327378 sin(t), -3.5355339059327378 cos(t) - 3.5355339059327378 sin(t)), X = (-3.061616997868383E-16, 3.7493994566546446E-32, 0) + (-0.915063509461096 cos(t) + 0.915063509461096 sin(t), 3.4150635094610964 cos(t) - 3.4150635094610973 sin(t), -3.5355339059327373 cos(t) - 3.5355339059327378 sin(t)), X = (-3.061616997868383E-16, -4.4408920985006257E-16, 0) + (-1.7677669529663682 cos(t) + 1.7677669529663682 sin(t), 3.0618621784789735 cos(t) - 3.061862178478974 sin(t), -3.5355339059327378 cos(t) - 3.5355339059327378 sin(t)), X = (-8.4117094861807E-17, -2.2204460492503126E-16, 0) + (-2.4999999999999996 cos(t) + 2.5 sin(t), 2.5 cos(t) - 2.5000000000000004 sin(t), -3.5355339059327373 cos(t) - 3.5355339059327378 sin(t)), X = (-3.4147861687372766E-15, 1.7763568394002505E-15, 0) + (-3.0618621784789726 cos(t) + 3.0618621784789717 sin(t), 1.76776695296637 cos(t) - 1.7677669529663695 sin(t), -3.5355339059327373 cos(t) - 3.5355339059327378 sin(t)), X = (5.82016719913287E-16, -2.2204460492503126E-16, 0) + (-3.4150635094610964 cos(t) + 3.4150635094610973 sin(t), 0.9150635094610976 cos(t) - 0.9150635094610977 sin(t), -3.5355339059327386 cos(t) - 3.5355339059327378 sin(t))}
  24. Sequence(Circle[(5; 0; (15° * k)), (5; 90°; (15° * k)), (5; 180°; (15° * k))], k, 1, 5) → paralelosN, {X = (1.7763568394002505E-15, 0, 1.2940952255126037) + (-3.4150635094610955 cos(t) - 3.4150635094610955 sin(t), 3.4150635094610955 cos(t) - 3.4150635094610955 sin(t), 0), X = (8.881784197001252E-16, 4.440892098500626E-16, 2.4999999999999996) + (-3.0618621784789726 cos(t) - 3.0618621784789726 sin(t), 3.0618621784789726 cos(t) - 3.0618621784789726 sin(t), 0), X = (0, 0, 3.5355339059327373) + (-2.5000000000000004 cos(t) - 2.5000000000000004 sin(t), 2.5000000000000004 cos(t) - 2.5000000000000004 sin(t), 0), X = (4.440892098500626E-16, 2.220446049250313E-16, 4.330127018922193) + (-1.7677669529663689 cos(t) - 1.7677669529663689 sin(t), 1.7677669529663689 cos(t) - 1.7677669529663689 sin(t), 0), X = (2.220446049250313E-16, 0, 4.829629131445341) + (-0.9150635094610973 cos(t) - 0.9150635094610973 sin(t), 0.9150635094610973 cos(t) - 0.9150635094610973 sin(t), 0)}
  25. Sequence(Circle[(5; 0; (-15° * k)), (5; 90°; (-15° * k)), (5; 180°; (-15° * k))], k, 1, 5) → paralelosS, {X = (1.7763568394002505E-15, 0, -1.2940952255126037) + (-3.4150635094610955 cos(t) - 3.4150635094610955 sin(t), 3.4150635094610955 cos(t) - 3.4150635094610955 sin(t), 0), X = (8.881784197001252E-16, 4.440892098500626E-16, -2.4999999999999996) + (-3.0618621784789726 cos(t) - 3.0618621784789726 sin(t), 3.0618621784789726 cos(t) - 3.0618621784789726 sin(t), 0), X = (0, 0, -3.5355339059327373) + (-2.5000000000000004 cos(t) - 2.5000000000000004 sin(t), 2.5000000000000004 cos(t) - 2.5000000000000004 sin(t), 0), X = (4.440892098500626E-16, 2.220446049250313E-16, -4.330127018922193) + (-1.7677669529663689 cos(t) - 1.7677669529663689 sin(t), 1.7677669529663689 cos(t) - 1.7677669529663689 sin(t), 0), X = (2.220446049250313E-16, 0, -4.829629131445341) + (-0.9150635094610973 cos(t) - 0.9150635094610973 sin(t), 0.9150635094610973 cos(t) - 0.9150635094610973 sin(t), 0)}
  26. Segment((0, 0, 0), (5; 0; 0)) → radio
  27. Sphere((0, 0, 0), 5) → esfera
  28. Textfield(ALon) → InputBox1
  29. Textfield(ALat) → InputBox2
  30. Textfield(BLon) → InputBox3
  31. Textfield(BLat) → InputBox4
  32. Angle(B, C, A) → Central
  33. Segment(A, C) → f
  34. Segment(B, C) → g
  35. CircleArc(C, B, A) → c

AI 绘图提示语参考

绘制一个地球仪，上面标记四个城市的位置，分别是里约热内卢（316.77°E，-22.9°S）、马德里（353.89°E，43.49°N）、洛杉矶（286°E，40.72°N）和纽约（316.77°E，40.72°N）。使用经纬度定义这些点，并用圆弧和大圆弧连接它们，展示赤道、极圈和回归线。在地球仪上绘制格林尼治子午线，并标注这些城市的名称。
以上提示语为根据原始绘图命令整理而来，仅供参考。