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Steiner Chain on Sphere

Steiner Chain on Sphere
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2015-12-04 04:29
TrigonometrySolid GeometryParametric EquationGeometric Construction
This example demonstrates the geometric construction of a Steiner chain on a sphere. It involves core mathematical concepts such as spherical geometry, equilateral triangles, rotation, and reflection. Key GeoGebra commands and objects used include Ray, Circle, Point, Rotate, If, Mirror, Sphere, Line, and Intersect. Suitable for high school and above students to explore the intersection of spherical geometry and geometric constructions.

GeoGebra Commands & Parameters

  1. Ray(S_i, Vector[(0, 1)]) → b
  2. Circle(S, a) → k0
  3. Point(k0, t) → A
  4. Circle(S, B2) → k1
  5. Circle(S, B4) → k2
  6. Circle(Intersect[k2, Segment[S, A]], B3) → l0
  7. Rotate(l0, (C1 * (2 * π / n)), S) → D1
  8. Rotate(l0, (C2 * (2 * π / n)), S) → D2
  9. Rotate(l0, (C3 * (2 * π / n)), S) → D3
  10. Rotate(l0, (C4 * (2 * π / n)), S) → D4
  11. Rotate(l0, (C5 * (2 * π / n)), S) → D5
  12. Rotate(l0, (C6 * (2 * π / n)), S) → D6
  13. Rotate(l0, (C7 * (2 * π / n)), S) → D7
  14. Rotate(l0, (C8 * (2 * π / n)), S) → D8
  15. Rotate(l0, (C9 * (2 * π / n)), S) → D9
  16. Rotate(l0, (C10 * (2 * π / n)), S) → D10
  17. Rotate(l0, (C11 * (2 * π / n)), S) → D11
  18. Rotate(l0, (C12 * (2 * π / n)), S) → D12
  19. Rotate(l0, (C13 * (2 * π / n)), S) → D13
  20. Rotate(l0, (C14 * (2 * π / n)), S) → D14
  21. Rotate(l0, (C15 * (2 * π / n)), S) → D15
  22. Rotate(l0, (C16 * (2 * π / n)), S) → D16
  23. Rotate(l0, (C17 * (2 * π / n)), S) → D17
  24. Rotate(l0, (C18 * (2 * π / n)), S) → D18
  25. Rotate(l0, (C19 * (2 * π / n)), S) → D19
  26. Rotate(l0, (C20 * (2 * π / n)), S) → D20
  27. If(n - 1 ≥ C1, true, false) → E1
  28. If(n - 1 ≥ C2, true, false) → E2
  29. If(n - 1 ≥ C3, true, false) → E3
  30. If(n - 1 ≥ C4, true, false) → E4
  31. If(n - 1 ≥ C5, true, false) → E5
  32. If(n - 1 ≥ C6, true, false) → E6
  33. If(n - 1 ≥ C7, true, false) → E7
  34. If(n - 1 ≥ C8, true, false) → E8
  35. If(n - 1 ≥ C9, true, false) → E9
  36. If(n - 1 ≥ C10, true, false) → E10
  37. If(n - 1 ≥ C11, true, false) → E11
  38. If(n - 1 ≥ C12, true, false) → E12
  39. If(n - 1 ≥ C13, true, false) → E13
  40. If(n - 1 ≥ C14, true, false) → E14
  41. If(n - 1 ≥ C15, true, false) → E15
  42. If(n - 1 ≥ C16, true, false) → E16
  43. If(n - 1 ≥ C17, true, false) → E17
  44. If(n - 1 ≥ C18, true, false) → E18
  45. If(n - 1 ≥ C19, true, false) → E19
  46. If(n - 1 ≥ C20, true, false) → E20
  47. Point(b) → C
  48. Circle(S_i, C) → c
  49. Mirror(D1, c) → F1
  50. Mirror(D2, c) → F2
  51. Mirror(D3, c) → F3
  52. Mirror(D4, c) → F4
  53. Mirror(D5, c) → F5
  54. Mirror(D6, c) → F6
  55. Mirror(D7, c) → F7
  56. Mirror(D8, c) → F8
  57. Mirror(D9, c) → F9
  58. Mirror(D10, c) → F10
  59. Mirror(D11, c) → F11
  60. Mirror(D12, c) → F12
  61. Mirror(D13, c) → F13
  62. Mirror(D14, c) → F14
  63. Mirror(D15, c) → F15
  64. Mirror(D16, c) → F16
  65. Mirror(D17, c) → F17
  66. Mirror(D18, c) → F18
  67. Mirror(D19, c) → F19
  68. Mirror(D20, c) → F20
  69. Mirror(k0, c) → k0'
  70. Mirror(k1, c) → k1'
  71. Sphere(Center[k0'], Radius[k0']) → d
  72. Line(Center[d], zAxis) → e
  73. Intersect(e, d) → SP, NP
  74. Point(F1, 0) → G1
  75. Point(F1, 0.3333333333333333) → H1
  76. Point(F1, 0.6666666666666666) → I1
  77. Point(F2, 0) → G2
  78. Point(F2, 0.333333333333334) → H2
  79. Point(F2, 0.666666666666668) → I2
  80. Point(F3, 0) → G3
  81. Point(F3, 0.333333333333334) → H3
  82. Point(F3, 0.666666666666668) → I3
  83. Point(F4, 0) → G4
  84. Point(F4, 0.333333333333334) → H4
  85. Point(F4, 0.666666666666668) → I4
  86. Point(F5, 0) → G5
  87. Point(F5, 0.333333333333334) → H5
  88. Point(F5, 0.666666666666668) → I5
  89. Point(F6, 0) → G6
  90. Point(F6, 0.333333333333334) → H6
  91. Point(F6, 0.666666666666668) → I6
  92. Point(F7, 0) → G7
  93. Point(F7, 0.333333333333334) → H7
  94. Point(F7, 0.666666666666668) → I7
  95. Point(F8, 0) → G8
  96. Point(F8, 0.333333333333334) → H8
  97. Point(F8, 0.666666666666668) → I8
  98. Point(F9, 0) → G9
  99. Point(F9, 0.333333333333334) → H9
  100. Point(F9, 0.666666666666668) → I9
  101. Point(F10, 0) → G10
  102. Point(F10, 0.333333333333334) → H10
  103. Point(F10, 0.666666666666668) → I10
  104. Point(F11, 0) → G11
  105. Point(F11, 0.333333333333334) → H11
  106. Point(F11, 0.666666666666668) → I11
  107. Point(F12, 0) → G12
  108. Point(F12, 0.333333333333334) → H12
  109. Point(F12, 0.666666666666668) → I12
  110. Point(F13, 0) → G13
  111. Point(F13, 0.333333333333334) → H13
  112. Point(F13, 0.666666666666668) → I13
  113. Point(F14, 0) → G14
  114. Point(F14, 0.333333333333334) → H14
  115. Point(F14, 0.666666666666668) → I14
  116. Point(F15, 0) → G15
  117. Point(F15, 0.333333333333334) → H15
  118. Point(F15, 0.666666666666668) → I15
  119. Point(F16, 0) → G16
  120. Point(F16, 0.333333333333334) → H16
  121. Point(F16, 0.666666666666668) → I16
  122. Point(F17, 0) → G17
  123. Point(F17, 0.333333333333334) → H17
  124. Point(F17, 0.666666666666668) → I17
  125. Point(F18, 0) → G18
  126. Point(F18, 0.333333333333334) → H18
  127. Point(F18, 0.666666666666668) → I18
  128. Point(F19, 0) → G19
  129. Point(F19, 0.333333333333334) → H19
  130. Point(F19, 0.666666666666668) → I19
  131. Point(F20, 0) → G20
  132. Point(F20, 0.333333333333334) → H20
  133. Point(F20, 0.666666666666668) → I20
  134. Ray(SP, G1) → J1
  135. Ray(SP, H1) → K1
  136. Ray(SP, I1) → L1
  137. Ray(SP, G2) → J2
  138. Ray(SP, H2) → K2
  139. Ray(SP, I2) → L2
  140. Ray(SP, G3) → J3
  141. Ray(SP, H3) → K3
  142. Ray(SP, I3) → L3
  143. Ray(SP, G4) → J4
  144. Ray(SP, H4) → K4
  145. Ray(SP, I4) → L4
  146. Ray(SP, G5) → J5
  147. Ray(SP, H5) → K5
  148. Ray(SP, I5) → L5
  149. Ray(SP, G6) → J6
  150. Ray(SP, H6) → K6
  151. Ray(SP, I6) → L6
  152. Ray(SP, G7) → J7
  153. Ray(SP, H7) → K7
  154. Ray(SP, I7) → L7
  155. Ray(SP, G8) → J8
  156. Ray(SP, H8) → K8
  157. Ray(SP, I8) → L8
  158. Ray(SP, G9) → J9
  159. Ray(SP, H9) → K9
  160. Ray(SP, I9) → L9
  161. Ray(SP, G10) → J10
  162. Ray(SP, H10) → K10
  163. Ray(SP, I10) → L10
  164. Ray(SP, G11) → J11
  165. Ray(SP, H11) → K11
  166. Ray(SP, I11) → L11
  167. Ray(SP, G12) → J12
  168. Ray(SP, H12) → K12
  169. Ray(SP, I12) → L12
  170. Ray(SP, G13) → J13
  171. Ray(SP, H13) → K13
  172. Ray(SP, I13) → L13
  173. Ray(SP, G14) → J14
  174. Ray(SP, H14) → K14
  175. Ray(SP, I14) → L14
  176. Ray(SP, G15) → J15
  177. Ray(SP, H15) → K15
  178. Ray(SP, I15) → L15
  179. Ray(SP, G16) → J16
  180. Ray(SP, H16) → K16
  181. Ray(SP, I16) → L16
  182. Ray(SP, G17) → J17
  183. Ray(SP, H17) → K17
  184. Ray(SP, I17) → L17
  185. Ray(SP, G18) → J18
  186. Ray(SP, H18) → K18
  187. Ray(SP, I18) → L18
  188. Ray(SP, G19) → J19
  189. Ray(SP, H19) → K19
  190. Ray(SP, I19) → L19
  191. Ray(SP, G20) → J20
  192. Ray(SP, H20) → K20
  193. Ray(SP, I20) → L20
  194. Intersect(J1, d, 2) → M1
  195. Intersect(K1, d, 2) → N1
  196. Intersect(L1, d, 2) → O1
  197. Intersect(J2, d, 2) → M2
  198. Intersect(K2, d, 2) → N2
  199. Intersect(L2, d, 2) → O2
  200. Intersect(J3, d, 2) → M3
  201. Intersect(K3, d, 2) → N3
  202. Intersect(L3, d, 2) → O3
  203. Intersect(J4, d, 2) → M4
  204. Intersect(K4, d, 2) → N4
  205. Intersect(L4, d, 2) → O4
  206. Intersect(J5, d, 2) → M5
  207. Intersect(K5, d, 2) → N5
  208. Intersect(L5, d, 2) → O5
  209. Intersect(J6, d, 2) → M6
  210. Intersect(K6, d, 2) → N6
  211. Intersect(L6, d, 2) → O6
  212. Intersect(J7, d, 2) → M7
  213. Intersect(K7, d, 2) → N7
  214. Intersect(L7, d, 2) → O7
  215. Intersect(J8, d, 2) → M8
  216. Intersect(K8, d, 2) → N8
  217. Intersect(L8, d, 2) → O8
  218. Intersect(J9, d, 2) → M9
  219. Intersect(K9, d, 2) → N9
  220. Intersect(L9, d, 2) → O9
  221. Intersect(J10, d, 2) → M10
  222. Intersect(K10, d, 2) → N10
  223. Intersect(L10, d, 2) → O10
  224. Intersect(J11, d, 2) → M11
  225. Intersect(K11, d, 2) → N11
  226. Intersect(L11, d, 2) → O11
  227. Intersect(J12, d, 2) → M12
  228. Intersect(K12, d, 2) → N12
  229. Intersect(L12, d, 2) → O12
  230. Intersect(J13, d, 2) → M13
  231. Intersect(K13, d, 2) → N13
  232. Intersect(L13, d, 2) → O13
  233. Intersect(J14, d, 2) → M14
  234. Intersect(K14, d, 2) → N14
  235. Intersect(L14, d, 2) → O14
  236. Intersect(J15, d, 2) → M15
  237. Intersect(K15, d, 2) → N15
  238. Intersect(L15, d, 2) → O15
  239. Intersect(J16, d, 2) → M16
  240. Intersect(K16, d, 2) → N16
  241. Intersect(L16, d, 2) → O16
  242. Intersect(J17, d, 2) → M17
  243. Intersect(K17, d, 2) → N17
  244. Intersect(L17, d, 2) → O17
  245. Intersect(J18, d, 2) → M18
  246. Intersect(K18, d, 2) → N18
  247. Intersect(L18, d, 2) → O18
  248. Intersect(J19, d, 2) → M19
  249. Intersect(K19, d, 2) → N19
  250. Intersect(L19, d, 2) → O19
  251. Intersect(J20, d, 2) → M20
  252. Intersect(K20, d, 2) → N20
  253. Intersect(L20, d, 2) → O20
  254. Circle(M1, N1, O1) → P1
  255. Circle(M2, N2, O2) → P2
  256. Circle(M3, N3, O3) → P3
  257. Circle(M4, N4, O4) → P4
  258. Circle(M5, N5, O5) → P5
  259. Circle(M6, N6, O6) → P6
  260. Circle(M7, N7, O7) → P7
  261. Circle(M8, N8, O8) → P8
  262. Circle(M9, N9, O9) → P9
  263. Circle(M10, N10, O10) → P10
  264. Circle(M11, N11, O11) → P11
  265. Circle(M12, N12, O12) → P12
  266. Circle(M13, N13, O13) → P13
  267. Circle(M14, N14, O14) → P14
  268. Circle(M15, N15, O15) → P15
  269. Circle(M16, N16, O16) → P16
  270. Circle(M17, N17, O17) → P17
  271. Circle(M18, N18, O18) → P18
  272. Circle(M19, N19, O19) → P19
  273. Circle(M20, N20, O20) → P20
  274. Point(F21, 0) → G21
  275. Point(F21, 0.333333333333334) → H21
  276. Point(F21, 0.66666666666667) → I21
  277. Ray(SP, G21) → J21
  278. Ray(SP, H21) → K21
  279. Ray(SP, I21) → L21
  280. Intersect(J21, d, 2) → M21
  281. Intersect(K21, d, 2) → N21
  282. Intersect(L21, d, 2) → O21
  283. Circle(M21, N21, O21) → P21

AI Drawing Prompt Reference

On a sphere, draw a ray pointing upwards, then a circle centered at the sphere's center, and mark a point on this circle. Draw another circle centered at this point. Next, draw multiple circles centered at the sphere's center with varying radii, connecting one of these circles to the sphere's center and a point on the first circle. Finally, rotate and arrange these circles at specific angles to form a Steiner chain on the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.