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Solar System: Helio- and Geocentric Views

Solar System: Helio- and Geocentric Views
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2015-05-31 21:35
TrigonometryCoordinate GeometryGeometric ConstructionSlider Animation
This example demonstrates the calculation and visualization of positions for Mercury, Venus, Earth, Mars, Jupiter, and Saturn from 1600 onwards under both heliocentric and geocentric perspectives. Key math concepts include ellipses, trigonometric functions, multiplication, means, unit circles, and incircles. Construction highlights GeoGebra commands such as Rotate, Vector, Translate, Circle, Ellipse, Line, Intersect, Midpoint, Point, Angle, Root, Distance, Ray, Segment, OrthogonalLine, Textfield, Sphere, Plane, IntersectPath, and Tangent. Suitable for high school students and astronomy enthusiasts studying celestial mechanics and mathematical modeling.

GeoGebra Commands & Parameters

  1. Rotate(C, α + (((0.0167 * 180) * ° / 3.141) * sin(α)) + ((0.5 * (((0.0167 * 180) * ° / 3.141))^(2)) * sin((2 * α))), D) → C'
  2. Vector(C', D) → u
  3. Translate(C', u) → C''_1
  4. Circle(A, 0.283464) → r
  5. Ellipse(A, B, C) → c
  6. Line(A, I_4) → n
  7. Intersect(r, n) → J_2, K_2
  8. Midpoint(K_2, A) → L_2
  9. Circle(L_2, 1.524) → s
  10. Intersect(s, n) → M_2, N_2
  11. Ellipse(K_2, A, M_2) → t_2
  12. Intersect(t_2, n) → O_2, P_2
  13. Point(c) → M
  14. Rotate(M, ((2 * π) * t / 365.25), C''_1) → M'_1
  15. Angle(M, C''_1, M'_1) → ϕ
  16. Root(ϕ - x + (0.0167 * sin(x)), 0, 7) → S_3
  17. Distance(C''_1, S_3) → EE
  18. Angle(C, D, C') → β
  19. Point(c) → E
  20. Line(A, L_5) → a
  21. Circle(A, 0.16) → d
  22. Intersect(d, a) → F, G
  23. Midpoint(A, F) → H
  24. Circle(H, 0.387) → e
  25. Intersect(e, a) → I, J
  26. Ellipse(F, A, J) → f
  27. Rotate(J, γ + (((0.206 * 180) * ° / 3.141) * sin(γ)) + ((0.5 * (((0.206 * 180) * ° / 3.141))^(2)) * sin((2 * γ))), H) → J'
  28. Angle(J, H, J') → δ
  29. Ray(H, J') → b
  30. Intersect(f, b) → K,
  31. Segment(C', K) → g
  32. Translate(C', u) → C''
  33. Translate(K, u) → K'
  34. Segment(C'', K') → g'
  35. Segment(D, A) → h
  36. Translate(D, u) → D'
  37. Translate(A, u) → A'
  38. Segment(D', A') → h'
  39. Ray(A, C') → i
  40. Circle(A, 0.723) → k
  41. Rotate(C, ε + (((0.0068 * 180) * ° / 3.141) * sin(ε)) + ((0.5 * (((0.0068 * 180) * ° / 3.141))^(2)) * sin((2 * ε))), A) → C'_1
  42. Angle(C, A, C'_1) → ζ
  43. Ray(A, C'_1) → j
  44. Point(c) → N
  45. Point(c) → O
  46. Point(c) → P
  47. Circle(C'', A') → p
  48. Point(p) → Q
  49. Point(p) → R
  50. Point(p) → S
  51. Point(p) → T
  52. Point(p) → U
  53. Point(c) → V
  54. Point(c) → W
  55. Point(p) → Z
  56. Point(c) → A_1
  57. Point(p) → B_1
  58. Point(c) → C_1
  59. Point(p) → D_1
  60. Point(c) → E_1
  61. Point(p) → F_1
  62. Point(c) → G_1
  63. Point(p) → H_1
  64. Point(c) → I_1
  65. Point(p) → J_1
  66. Point(c) → K_1
  67. Point(p) → L_1
  68. Circle(C'', 1.2) → q
  69. Ray(C'', A') → m
  70. Intersect(q, m) → M_1,
  71. Point(q) → N_1
  72. Point(q) → O_1
  73. Intersect(p, xAxis, 1) → P_1
  74. Intersect(p, xAxis, 2) → Q_1
  75. Point(q) → R_1
  76. Point(q) → S_1
  77. Point(q) → T_1
  78. Point(q) → U_1
  79. Point(q) → V_1
  80. Point(q) → W_1
  81. Point(q) → Z_1
  82. Point(q) → A_2
  83. Point(q) → B_2
  84. Point(q) → C_2
  85. Point(q) → D_2
  86. Point(q) → E_2
  87. Point(q) → F_2
  88. Point(q) → G_2
  89. Point(q) → H_2
  90. Rotate(O_2, η + (((0.093 * 180) * ° / 3.141) * sin(η)) + ((0.5 * (((0.093 * 180) * ° / 3.141))^(2)) * sin((2 * η))), L_2) → Q_2
  91. Angle(O_2, L_2, Q_2) → θ_1
  92. Segment(C', Q_2) → a_1
  93. Translate(C', u) → C''_2
  94. Translate(Q_2, u) → R_2
  95. Line(A', C'') → c_1
  96. Intersect(q, c_1) → S_2, T_2
  97. Rotate(T_2, θ, C'') → U_2
  98. Angle(T_2, C'', U_2) → ι
  99. Line(C'', U_2) → d_1
  100. OrthogonalLine(C'', d_1) → e_1
  101. Segment(C'', A') → f_1
  102. Point(c) → V_2
  103. Circle(A, 0.5) → h_1
  104. Line(A, V'_2) → g_1
  105. Intersect(h_1, g_1) → W_2, Z_2
  106. Midpoint(A, Z_2) → A_3
  107. Circle(A_3, 5.2) → k_1
  108. Intersect(k_1, g_1) → B_3, C_3
  109. Ellipse(Z_2, A, C_3) → p_1
  110. Rotate(B_3, ξ + (((0.0484 * 180) * ° / 3.141) * sin(ξ)) + ((0.5 * (((0.0484 * 180) * ° / 3.141))^(2)) * sin((2 * ξ))), A) → D_3
  111. Angle(B_3, A, D_3) → ν
  112. Ray(A_3, D_3) → j_1
  113. Intersect(p_1, j_1) → E_3,
  114. Segment(C', E_3) → l_1
  115. Translate(C', u) → C''_3
  116. Translate(E_3, u) → F_3
  117. Circle(A, 1.033) → q_1
  118. Intersect(q_1, xAxis, 2) → J_3
  119. Rotate(J_3, 170° + (((n_1 - 2015) * 180) * ° / ((3.141 * 25780))), A) → K_3
  120. Line(A, K'_3) → r_1
  121. Angle(J_3, A, K_3) → ρ
  122. Intersect(q_1, r_1) → L_3, M_3
  123. Midpoint(A, L_3) → N_3
  124. Circle(N_3, 9.5367) → s_1
  125. Intersect(s_1, r_1) → O_3, P_3
  126. Ellipse(A, L_3, O_3) → t_1
  127. Distance(A, P_3) → abstandAP3
  128. Distance(A, O_3) → abstandAO3
  129. Rotate(P_3, σ + (((0.0539 * 180) * ° / 3.141) * sin(σ)) + ((0.5 * (((0.0539 * 180) * ° / 3.141))^(2)) * sin((2 * σ))), N_3) → Q_3
  130. Angle(P_3, N_3, Q_3) → τ
  131. Segment(Q_3, C') → a_2
  132. Translate(Q_3, u) → R_3
  133. Translate(C', u) → C''_4
  134. Textfield(n_1) → Textfeld1
  135. Rotate(M, ((2 * π) * t), C''_1) → M'
  136. Textfield(t) → Textfeld2
  137. Rotate(M, (2 * atan((sqrt((1 + 0.0167) / (1 - 0.0167)) * tan((0.5 * EE))))), A) → M'_2
  138. Angle(M, A, M'_2) → χ
  139. Ray(A, M'_2) → f_2
  140. Intersect(c, f_2) → T_3,
  141. Rotate(O_2, ((2 * π) * t / ((365.25 * 1.88085))) + β_1, L_2) → W_3
  142. Angle(O_2, L_2, W_3) → ψ
  143. Root(ψ - x + (0.09341 * sin(x)), 0, 7) → Z_3
  144. Distance(C''_1, Z_3) → EMa
  145. Rotate(O_2, (2 * atan((sqrt((1 + 0.09341) / (1 - 0.09341)) * tan((0.5 * EMa))))) + β_1, A) → B_4
  146. Rotate(O_2, (2 * atan((sqrt((1 + 0.09341) / (1 - 0.09341)) * tan((0.5 * EMa))))), A) → A_4
  147. Angle(O_2, A, A_4) → ω
  148. Ray(A, A_4) → g_2
  149. Intersect(t_2, g_2) → C_4,
  150. Rotate(W_2, ((2 * π) * t / ((365.25 * 11.862615))) + ε_1, A_3) → E_4
  151. Angle(W_2, A_3, E_4) → ζ_1
  152. Root(ζ_1 - x + (0.04892 * sin(x)), 0, 7) → F_4
  153. Distance(C''_1, F_4) → EJ
  154. Rotate(W_2, (2 * atan((sqrt((1 + 0.04892) / (1 - 0.04892)) * tan((0.5 * EJ))))), A) → G_4
  155. Angle(W_2, A, G_4) → η_1
  156. Ray(A, G_4) → i_2
  157. Intersect(p_1, i_2) → H_4,
  158. Rotate(P_3, ((2 * π) * t / ((365.25 * 29.47498))) + σ_1, N_3) → J_4
  159. Angle(K'_3, N_3, J_4) → σ_2
  160. Root(σ_2 - x + (0.05772 * sin(x)), 0, 7) → K_4
  161. Distance(C''_1, K_4) → ES
  162. Rotate(P_3, (2 * atan((sqrt((1 + 0.05772) / (1 - 0.05772)) * tan((0.5 * ES))))), A) → L_4
  163. Ray(A, L_4) → j_2
  164. Intersect(t_1, j_2) → M_4,
  165. Angle(K'_3, A, L_4) → σ_3
  166. Rotate(C, 30° + (360° * t / ((365.2425 * 25784))), A) → C'_2
  167. Angle(C, A, C'_2) → ι_1
  168. Line(A, C'_2) → k_2
  169. Circle(A, 0.009822897447999999) → p_2
  170. Intersect(p_2, k_2) → N_4, O_4
  171. Midpoint(A, N_4) → P_4
  172. Circle(P_4, 0.7233356) → q_2
  173. Intersect(q_2, k_2) → Q_4, R_4
  174. Ellipse(N_4, A, R_4) → r_2
  175. Rotate(R_4, ((2 * π) * t / ((365.25 * 0.61519726))) + κ_1, P_4) → S_4
  176. Angle(R_4, P_4, S_4) → λ_1
  177. Root(λ_1 - x + (0.00679 * sin(x)), 0, 7) → T_4
  178. Distance(C''_1, T_4) → EV
  179. Rotate(R_4, (2 * atan((sqrt((1 + 0.00679) / (1 - 0.00679)) * tan((0.5 * EV))))), A) → U_4
  180. Angle(R_4, A, U_4) → μ_1
  181. Ray(A, U_4) → l_2
  182. Intersect(q_2, l_2) → V_4,
  183. Vector(T_3, A) → w_1
  184. Translate(T_3, w_1) → W_4
  185. Translate(A, w_1) → A'_1
  186. Translate(T_3, w_1) → A_5
  187. Ray(W_4, A'_1) → s_2
  188. Segment(T_3, V_4) → a_3
  189. Translate(V_4, w_1) → C_5
  190. Segment(W_4, C_5) → b_3
  191. Angle(b_3, s_2) → μ
  192. Segment(T_3, C_4) → c_3
  193. Translate(C_4, w_1) → D_5
  194. Segment(W_4, D_5) → d_3
  195. Ray(D_5, W_4) → e_3
  196. Angle(e_3, s_2) → λ
  197. Point(p) → I_2
  198. Intersect(f, a) → L, D_4
  199. Rotate(L, ((2 * π) * t / ((365.25 * 0.2408467))) + α_1, H) → L'
  200. Angle(L, H, L') → κ
  201. Root(κ - x + (0.20564 * sin(x)), 0, 7) → Z_4
  202. Distance(C''_1, Z_4) → EMe
  203. Rotate(L_5, (2 * atan((sqrt((1 + 0.20564) / (1 - 0.20564)) * tan((0.5 * EMe))))), A) → L'_1
  204. Ray(A, L'_1) → l
  205. Intersect(f, l) → B_5,
  206. Angle(L_5, A, L'_1) → γ_1
  207. Segment(T_3, B_5) → b_1
  208. Translate(B_5, w_1) → E_5
  209. Segment(W_4, E_5) → e_2
  210. Angle(e_2, s_2) → δ_1
  211. Segment(F_5, G_5) → h_2
  212. Segment(T_3, H_4) → m_2
  213. Translate(H_4, w_1) → H_5
  214. Ray(H_5, W_4) → n_2
  215. Point(n_2) → J_5
  216. Angle(n_2, s_2) → ν_1
  217. Point(n_2) → I_5
  218. Circle(W_4, I_5) → f_3
  219. Intersect(f_3, n_2) → K_5, M_5
  220. Circle(W_4, 1.3) → g_3
  221. Intersect(g_3, n_2) → N_5, P_5
  222. Segment(A_1, M_4) → m_1
  223. Translate(M_4, w_1) → G_3
  224. Ray(G_3, W_4) → d_2
  225. Angle(s_2, d_2) → ο
  226. Intersect(g_3, d_2) → H_3, I_3
  227. Point(d_2) → R_5
  228. Segment(H_5, W_4) → b_2
  229. Intersect(g_3, b_2) → U_3,
  230. Point(b_2) → V_3
  231. Vector(U_3, V_3) → v_1
  232. Segment(W_4, G_3) → c_2
  233. Intersect(g_3, c_2) → Q_5,
  234. Point(c_2) → S_5
  235. Vector(Q_5, S_5) → u_1
  236. Rotate(C, 233*°, A) → C'_3
  237. Rotate(C, 333*°, A) → C'_4
  238. Angle(C, A, C'_4) → υ
  239. Line(A, C'_4) → h_3
  240. Rotate(q_2, 3.4*°, h_3) → k_3
  241. Rotate(V_4, 3.4*°, h_3) → T_5
  242. Angle(C, A, C'_5) → ξ_1
  243. Line(A, C'_5) → i_3
  244. Rotate(p_1, 1.3*°, i_3) → p_3
  245. Rotate(H_4, 1.3*°, i_3) → U_5
  246. Rotate(C, 306*°, A) → C'_6
  247. Angle(C, A, C'_6) → ο_1
  248. Line(A, C'_6) → j_3
  249. Rotate(f, 7*°, j_3) → f'
  250. Rotate(B_5, 7*°, j_3) → V_5
  251. Rotate(C, 309*°, A) → C'_7
  252. Angle(C, A, C'_7) → ρ_1
  253. Line(A, C'_7) → l_3
  254. Rotate(t_2, 1.85*°, l_3) → q_3
  255. Rotate(C_4, 1.85*°, l_3) → W_5
  256. Rotate(C, 12*°, A) → C'_8
  257. Angle(C, A, C'_8) → τ_1
  258. Line(A, C'_8) → m_3
  259. Rotate(t_1, 2.49*°, m_3) → r_3
  260. Rotate(M_4, 2.49*°, m_3) → Z_5
  261. Vector(T_3, A) → u_2
  262. Translate(A, u_2) → A'_2
  263. Translate(V_5, u_2) → A_6
  264. Translate(T_5, u_2) → B_6
  265. Translate(W_5, u_2) → C_6
  266. Translate(xOyPlane, Vector[(0, 0, 0)]) → xyEbene'
  267. Translate(U_5, u_2) → D_6
  268. Translate(Z_5, u_2) → E_6
  269. Sphere(W_4, 1) → w_2
  270. Ray(W_4, B_6) → n_3
  271. Intersect(n_3, w_2) → F_6, G_6
  272. Ray(W_4, A_6) → s_3
  273. Intersect(s_3, w_2) → H_6, I_6
  274. Ray(W_4, C_6) → t_3
  275. Intersect(t_3, w_2) → J_6, K_6
  276. Ray(W_4, D_6) → a_4
  277. Intersect(a_4, w_2) → L_6, M_6
  278. Ray(W_4, E_6) → b_4
  279. Intersect(b_4, w_2) → N_6, O_6
  280. Segment(W_4, B_6) → c_4
  281. Segment(W_4, A_6) → d_4
  282. Segment(W_4, A_6) → e_4
  283. Segment(W_4, C_6) → f_4
  284. Segment(W_4, B_6) → g_4
  285. Segment(W_4, D_6) → h_4
  286. Segment(W_4, E_6) → i_4
  287. Rotate(C_7, 78*°, A) → P_6
  288. Angle(C_7, A, P_6) → υ_1
  289. Line(A, P_6) → j_4
  290. Intersect(c, j_4) → Q_6, R_6
  291. Plane(Q_6, j_4) → k_4
  292. Plane(Q_6, R_6, C_7) → l_4
  293. Rotate(l_4, -23.4*°, j_4) → m_4
  294. IntersectPath(m_4, w_2) → p_4
  295. Tangent(T_3, q_2) → n_4, q_4
  296. Translate(n_4, u_2) → r_4
  297. Translate(q_4, u_2) → s_4
  298. Tangent(T_3, f) → t_4, a_5
  299. Translate(a_5, u_2) → b_5
  300. Translate(t_4, u_2) → c_5

AI Drawing Prompt Reference

Create a solar system model showing the positions of six planets from 1600 onwards under both heliocentric and geocentric views. Use ellipses, trigonometric functions, multiplication, means, unit circles, and incircles to visualize orbits and positions, employing rotations, vectors, translations, and geometric constructions suitable for high school students and astronomy enthusiasts.
The prompt above was compiled from the original drawing commands and is for reference only.