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Mathematical Principles of Earth's Sunrise and Sunset Phenomena

Mathematical Principles of Earth's Sunrise and Sunset Phenomena
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2018-09-22 16:10
TrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the mathematical principles behind Earth's sunrise and sunset phenomena, involving concepts such as coordinate geometry, trigonometric functions, and analytic geometry. The construction uses GeoGebra commands like coordinate systems, rotation, circles, and lines to create relevant graphs. Suitable for high school students or college freshmen to explore the intersection of astronomy and mathematics.

GeoGebra Commands & Parameters

  1. If(dia > 287, dia - 287, dia + 78) → dias
  2. If(dia < 13, dia + 19, If[dia < 43, dia - 12, If[dia < 74, dia - 42, If[dia < 104, dia - 73, If[dia < 135, dia - 103, If[dia < 166, dia - 134, If[dia < 196, dia - 165, If[dia < 227, dia - 195, If[dia < 257, dia - 226, If[dia < 288, dia - 256, If[dia < 319, dia - 287, If[dia < 347, dia - 318, dia - 346]]]]]]]]]]]) → diaMes
  3. If(dia < 13 ∨ dia > 346, 3, If[dia < 43, 4, If[dia < 74, 5, If[dia < 104, 6, If[dia < 135, 7, If[dia < 166, 8, If[dia < 196, 9, If[dia < 227, 10, If[dia < 257, 11, If[dia < 288, 12, If[dia < 319, 1, 2]]]]]]]]]]) → mes
  4. If(mes < 3, mes + 12, mes) → mes1
  5. Rotate((0, 5, 0), oblicuidad, (0, 0, 0), xAxis) → Ec
  6. OrthogonalLine((0, 0, 0), Plane[(0, 0, 0), (5, 0, 0), Ec]) → l
  7. Circle(l, (7, 0, 0)) → ecliptica
  8. Segment(PS + (0.075 * PS), PN + (0.075 * PN)) → eje
  9. Circle((0, 0, 0), 5, eje) → ecuador
  10. Ray((0, 0, 0), A) → laserA
  11. Sphere((0, 0, 0), (0.999 * 5)) → tierra
  12. Vector((1; (-rot) + 180°; (-inclinacion))) → dir
  13. Circle((5; 0; (-anguloPolar)), (5; 90°; (-anguloPolar)), (5; 180°; (-anguloPolar))) → antartico
  14. Circle((5; 0; anguloPolar), (5; 90°; anguloPolar), (5; 180°; anguloPolar)) → artico
  15. Circle((5; 0; oblicuidad), (5; 90°; oblicuidad), (5; 180°; oblicuidad)) → cancer
  16. Circle((5; 0; (-oblicuidad)), (5; 90°; (-oblicuidad)), (5; 180°; (-oblicuidad))) → capricornio
  17. Sequence(Circle[(5; 0; (15° * k)), (5; 90°; (15° * k)), (5; 180°; (15° * k))], k, 1, 5) → paralelosN
  18. Sequence(Circle[(5; 0; (-15° * k)), (5; 90°; (-15° * k)), (5; 180°; (-15° * k))], k, 1, 5) → paralelosS
  19. Circle(SolDirecto, 0.25, Vector[SolDirecto, Sol]) → solDirecto
  20. Element(meses, mes) → mesTexto
  21. Vector(SolDirecto) → v
  22. UnitVector(Vector[EQ]) → m
  23. If(tt4 < 10, ("0" * tt4), ("" * tt4)) → t4
  24. Element(tiempo, 2) → segundos
  25. Circle(PN, (5; f; 0), PS) → greenwich
  26. CircumcircleArc(PN, (5; f + lonA; 0), PS) → meridianoA
  27. If(tt3 < 10, ("0" * tt3), ("" * tt3)) → t3
  28. Line(Sol, laserA) → h
  29. OrthogonalPlane((0, 0, 0), laserA) → planoHorizonteA
  30. Intersect(h, planoHorizonteA) → C
  31. OrthogonalLine(PN, planoHorizonteA) → j
  32. Intersect(j, planoHorizonteA) → PN'
  33. Vector(PN') → vectorPN
  34. Vector(C) → vectorSol
  35. Segment(Sol, C) → p
  36. Segment(PN, PN') → q
  37. Angle(C, (0, 0, 0), Sol) → elevacion0
  38. Segment((0, 0, 0), Sol) → i
  39. Surface((((5.03 * cos(u)) * sin(v)) * x1) + (((5.03 * sin(u)) * sin(v)) * x3) - (((5.03 * cos(v)) * y(m)) * sin(β)), (((5.03 * cos(u)) * sin(v)) * x3) + (((5.03 * sin(u)) * sin(v)) * x2) + (((5.03 * cos(v)) * x(m)) * sin(β)), ((((5.03 * cos(u)) * sin(v)) * y(m)) * sin(β)) - ((((5.03 * sin(u)) * sin(v)) * x(m)) * sin(β)) + ((5.03 * cos(v)) * cos(β)), u, 0, π, v, π / 2, (3 * π / 2)) → a
  40. Circle((0, 0, z(Sol)), Sol, Vector[(0, 0, 1)]) → circuloAparenteSol
  41. Intersect(planoHorizonteA, circuloAparenteSol) → Sunset, Sunrise
  42. CircumcircleArc(Sunset, Sol, Sunrise) → trayectoriaAparenteSol
  43. CircumcircleArc(PN1, (((2 * (hora - UT ≤ 12)) - 1) * M), Sol1) → arcoAcimut
  44. Circle((0, 0, 0), 7, laserA) → horizonteA
  45. Sequence(DJ0 - 2451545 + k - If[k > 287, 365, 0], k, 1, 365) → secN
  46. Sequence(Mod[280.46 + (0.98565 * Element[secN, k]), 360]°, k, 1, 365) → secL
  47. Sequence(Mod[357.528° + (0.9856° * Element[secN, k]), 360]°, k, 1, 365) → secG
  48. Sequence(Element[secL, k] + (1.915° * sin(Element[secG, k])) + (0.02° * sin((2 * Element[secG, k]))), k, 1, 365) → secLanda
  49. Sequence(asin((sin(oblicuidad) * sin(Element[secLanda, k]))), k, 1, 365) → secDeclinacion
  50. Sequence(atan2((cos(oblicuidad) * sin(Element[secLanda, k])), cos(Element[secLanda, k])), k, 1, 365) → secAR
  51. Sequence(((2 * π / 365) * (If[k > 287, k - 287, k + 78] - 1 + (hora - UT - 12) / 24)), k, 1, 365) → secTiempo1
  52. Sequence(((229.18 * (0.00075 + (0.00187 * cos(Element[secTiempo1, k])) - (0.03208 * sin(Element[secTiempo1, k])) - (0.01462 * cos((2 * Element[secTiempo1, k]))) - (0.04085 * sin((2 * Element[secTiempo1, k])))) / 4) * π / 180), k, 1, 365) → secTiempo
  53. Sequence((7; AR + Element[secTiempo, k]; Element[secDeclinacion, k]), k, 1, 365) → secAnalema
  54. If(lonA1 < -180°, lonA1 + 180°, If[lonA1 > 180°, lonA1 - 360°, lonA1]) → lonA2
  55. Sequence(Circle[PN, (5; f + (15° * k); 0), PS], k, 1, 11) → meridianos
  56. CircumcircleSector(Aries, (7; (AR + ecTiempo) / 2; 0), Sol2) → arcoAR
  57. CircumcircleSector(Sol2, (7; AR + ecTiempo; declinacion / 2), Sol) → arcoDeclinacion

AI Drawing Prompt Reference

Draw a 3D sphere model of the Earth with the equator, poles, and lines of latitude and longitude. Include points for sunrise and sunset, the sun's position, the path of the sun, and arcs of the Earth's surface and visible face. The diagram illustrates the relationship between Earth's rotation and solar illumination, combining astronomy and mathematics.
The prompt above was compiled from the original drawing commands and is for reference only.