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Spherical Model - Cuboctahedron

Spherical Model - Cuboctahedron
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2016-05-15 14:32
CirclePolygonCoordinate GeometrySolid GeometryGeometric Construction
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This example demonstrates how to construct a spherical model of a cuboctahedron, mapping the vertices, edges, and faces of the polyhedron onto a sphere, resulting in a solid-geometry figure composed of circular arcs and spherical polygons. The core mathematical concepts include polyhedral geometry, spherical geometry, Archimedean solids, and intersections of planes with a sphere (great or small circles). Construction highlights: using GeoGebra commands such as Plane to define cutting planes, UnitOrthogonalVector to generate normal vectors, Element to extract objects, Surface to create the sphere, Circle to generate spherical intersection curves, and Segment to connect vertices; it involves numerous point objects (e.g., I, O, A, S1-S11, R1-R11, Q1-Q11) and auxiliary points B, C, P1, building the spherical grid of the cuboctahedron through plane-sphere intersections. Suitable for high school or university teaching of solid geometry, polyhedra, and spherical geometry, as well as for mathematical visualization and exploring properties of Archimedean solids.

GeoGebra Commands & Parameters

  1. Plane(A, I, O) → S1
  2. UnitOrthogonalVector(S1) → S2
  3. Plane(E, I, O) → R1
  4. UnitOrthogonalVector(R1) → R2
  5. Plane(A, E, O) → Q1
  6. UnitOrthogonalVector(Q1) → Q2
  7. Plane(B, C, O) → P1
  8. UnitOrthogonalVector(P1) → P2
  9. Plane(A, Midpoint[E, I], O) → B1
  10. UnitOrthogonalVector(B1) → B2
  11. Element(B12, 1, 2) → B13
  12. Element(B12, 1, 3) → B14
  13. Element(B12, 2, 2) → B15
  14. Element(B12, 2, 3) → B16
  15. Element(B12, 3, 2) → B17
  16. Element(B12, 3, 3) → B18
  17. Surface(((B$13 * (4 + t)) * cos(k)) + ((B$14 * (4 + t)) * sin(k)), ((B$15 * (4 + t)) * cos(k)) + ((B$16 * (4 + t)) * sin(k)), ((B$17 * (4 + t)) * cos(k)) + ((B$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → B19
  18. Circle((0, 0, 0), 4, Line[(0, 0, 0), B2]) → B20
  19. Circle((0, 0, 0), 5, Line[(0, 0, 0), B2]) → B21
  20. Plane(E, Midpoint[A, I], O) → C1
  21. UnitOrthogonalVector(C1) → C2
  22. Element(C12, 1, 2) → C13
  23. Element(C12, 1, 3) → C14
  24. Element(C12, 2, 2) → C15
  25. Element(C12, 2, 3) → C16
  26. Element(C12, 3, 2) → C17
  27. Element(C12, 3, 3) → C18
  28. Surface(((C$13 * (4 + t)) * cos(k)) + ((C$14 * (4 + t)) * sin(k)), ((C$15 * (4 + t)) * cos(k)) + ((C$16 * (4 + t)) * sin(k)), ((C$17 * (4 + t)) * cos(k)) + ((C$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → C19
  29. Circle((0, 0, 0), 4, Line[(0, 0, 0), C2]) → C20
  30. Circle((0, 0, 0), 5, Line[(0, 0, 0), C2]) → C21
  31. Plane(I, Midpoint[A, E], O) → D1
  32. UnitOrthogonalVector(D1) → D2
  33. Element(D12, 1, 2) → D13
  34. Element(D12, 1, 3) → D14
  35. Element(D12, 2, 2) → D15
  36. Element(D12, 2, 3) → D16
  37. Element(D12, 3, 2) → D17
  38. Element(D12, 3, 3) → D18
  39. Surface(((D$13 * (4 + t)) * cos(k)) + ((D$14 * (4 + t)) * sin(k)), ((D$15 * (4 + t)) * cos(k)) + ((D$16 * (4 + t)) * sin(k)), ((D$17 * (4 + t)) * cos(k)) + ((D$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → D19
  40. Circle((0, 0, 0), 4, Line[(0, 0, 0), D2]) → D20
  41. Circle((0, 0, 0), 5, Line[(0, 0, 0), D2]) → D21
  42. Plane(D, Midpoint[H, I], O) → E1
  43. UnitOrthogonalVector(E1) → E2
  44. Element(E12, 1, 2) → E13
  45. Element(E12, 1, 3) → E14
  46. Element(E12, 2, 2) → E15
  47. Element(E12, 2, 3) → E16
  48. Element(E12, 3, 2) → E17
  49. Element(E12, 3, 3) → E18
  50. Surface(((E$13 * (4 + t)) * cos(k)) + ((E$14 * (4 + t)) * sin(k)), ((E$15 * (4 + t)) * cos(k)) + ((E$16 * (4 + t)) * sin(k)), ((E$17 * (4 + t)) * cos(k)) + ((E$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → E19
  51. Circle((0, 0, 0), 4, Line[(0, 0, 0), E2]) → E20
  52. Circle((0, 0, 0), 5, Line[(0, 0, 0), E2]) → E21
  53. Plane(H, Midpoint[D, I], O) → F1
  54. UnitOrthogonalVector(F1) → F2
  55. Element(F12, 1, 2) → F13
  56. Element(F12, 1, 3) → F14
  57. Element(F12, 2, 2) → F15
  58. Element(F12, 2, 3) → F16
  59. Element(F12, 3, 2) → F17
  60. Element(F12, 3, 3) → F18
  61. Surface(((F$13 * (4 + t)) * cos(k)) + ((F$14 * (4 + t)) * sin(k)), ((F$15 * (4 + t)) * cos(k)) + ((F$16 * (4 + t)) * sin(k)), ((F$17 * (4 + t)) * cos(k)) + ((F$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → F19
  62. Circle((0, 0, 0), 4, Line[(0, 0, 0), F2]) → F20
  63. Circle((0, 0, 0), 5, Line[(0, 0, 0), F2]) → F21
  64. Plane(I, Midpoint[D, H], O) → G1
  65. UnitOrthogonalVector(G1) → G2
  66. Element(G12, 1, 2) → G13
  67. Element(G12, 1, 3) → G14
  68. Element(G12, 2, 2) → G15
  69. Element(G12, 2, 3) → G16
  70. Element(G12, 3, 2) → G17
  71. Element(G12, 3, 3) → G18
  72. Surface(((G$13 * (4 + t)) * cos(k)) + ((G$14 * (4 + t)) * sin(k)), ((G$15 * (4 + t)) * cos(k)) + ((G$16 * (4 + t)) * sin(k)), ((G$17 * (4 + t)) * cos(k)) + ((G$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → G19
  73. Circle((0, 0, 0), 4, Line[(0, 0, 0), G2]) → G20
  74. Circle((0, 0, 0), 5, Line[(0, 0, 0), G2]) → G21
  75. Plane(C, Midpoint[H, L], O) → H1
  76. Plane(H, Midpoint[C, L], O) → I1
  77. Plane(L, Midpoint[C, H], O) → J1
  78. UnitOrthogonalVector(H1) → H2
  79. Element(H12, 1, 2) → H13
  80. Element(H12, 1, 3) → H14
  81. Element(H12, 2, 2) → H15
  82. Element(H12, 2, 3) → H16
  83. Element(H12, 3, 2) → H17
  84. Element(H12, 3, 3) → H18
  85. Surface(((H$13 * (4 + t)) * cos(k)) + ((H$14 * (4 + t)) * sin(k)), ((H$15 * (4 + t)) * cos(k)) + ((H$16 * (4 + t)) * sin(k)), ((H$17 * (4 + t)) * cos(k)) + ((H$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → H19
  86. Circle((0, 0, 0), 4, Line[(0, 0, 0), H2]) → H20
  87. Circle((0, 0, 0), 5, Line[(0, 0, 0), H2]) → H21
  88. UnitOrthogonalVector(I1) → I2
  89. Element(I12, 1, 2) → I13
  90. Element(I12, 1, 3) → I14
  91. Element(I12, 2, 2) → I15
  92. Element(I12, 2, 3) → I16
  93. Element(I12, 3, 2) → I17
  94. Element(I12, 3, 3) → I18
  95. Surface(((I$13 * (4 + t)) * cos(k)) + ((I$14 * (4 + t)) * sin(k)), ((I$15 * (4 + t)) * cos(k)) + ((I$16 * (4 + t)) * sin(k)), ((I$17 * (4 + t)) * cos(k)) + ((I$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → I19
  96. Circle((0, 0, 0), 4, Line[(0, 0, 0), I2]) → I20
  97. Circle((0, 0, 0), 5, Line[(0, 0, 0), I2]) → I21
  98. UnitOrthogonalVector(J1) → J2
  99. Element(J12, 1, 2) → J13
  100. Element(J12, 1, 3) → J14
  101. Element(J12, 2, 2) → J15
  102. Element(J12, 2, 3) → J16
  103. Element(J12, 3, 2) → J17
  104. Element(J12, 3, 3) → J18
  105. Surface(((J$13 * (4 + t)) * cos(k)) + ((J$14 * (4 + t)) * sin(k)), ((J$15 * (4 + t)) * cos(k)) + ((J$16 * (4 + t)) * sin(k)), ((J$17 * (4 + t)) * cos(k)) + ((J$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → J19
  106. Circle((0, 0, 0), 4, Line[(0, 0, 0), J2]) → J20
  107. Circle((0, 0, 0), 5, Line[(0, 0, 0), J2]) → J21
  108. Plane(B, Midpoint[L, E], O) → K1
  109. Plane(L, Midpoint[B, E], O) → L1
  110. Plane(E, Midpoint[B, L], O) → M1
  111. UnitOrthogonalVector(K1) → K2
  112. Element(K12, 1, 2) → K13
  113. Element(K12, 1, 3) → K14
  114. Element(K12, 2, 2) → K15
  115. Element(K12, 2, 3) → K16
  116. Element(K12, 3, 2) → K17
  117. Element(K12, 3, 3) → K18
  118. Surface(((K$13 * (4 + t)) * cos(k)) + ((K$14 * (4 + t)) * sin(k)), ((K$15 * (4 + t)) * cos(k)) + ((K$16 * (4 + t)) * sin(k)), ((K$17 * (4 + t)) * cos(k)) + ((K$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → K19
  119. Circle((0, 0, 0), 4, Line[(0, 0, 0), K2]) → K20
  120. Circle((0, 0, 0), 5, Line[(0, 0, 0), K2]) → K21
  121. UnitOrthogonalVector(L1) → L2
  122. Element(L12, 1, 2) → L13
  123. Element(L12, 1, 3) → L14
  124. Element(L12, 2, 2) → L15
  125. Element(L12, 2, 3) → L16
  126. Element(L12, 3, 2) → L17
  127. Element(L12, 3, 3) → L18
  128. Surface(((L$13 * (4 + t)) * cos(k)) + ((L$14 * (4 + t)) * sin(k)), ((L$15 * (4 + t)) * cos(k)) + ((L$16 * (4 + t)) * sin(k)), ((L$17 * (4 + t)) * cos(k)) + ((L$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → L19
  129. Circle((0, 0, 0), 4, Line[(0, 0, 0), L2]) → L20
  130. Circle((0, 0, 0), 5, Line[(0, 0, 0), L2]) → L21
  131. UnitOrthogonalVector(M1) → M2
  132. Element(M12, 1, 2) → M13
  133. Element(M12, 1, 3) → M14
  134. Element(M12, 2, 2) → M15
  135. Element(M12, 2, 3) → M16
  136. Element(M12, 3, 2) → M17
  137. Element(M12, 3, 3) → M18
  138. Surface(((M$13 * (4 + t)) * cos(k)) + ((M$14 * (4 + t)) * sin(k)), ((M$15 * (4 + t)) * cos(k)) + ((M$16 * (4 + t)) * sin(k)), ((M$17 * (4 + t)) * cos(k)) + ((M$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → M19
  139. Circle((0, 0, 0), 4, Line[(0, 0, 0), M2]) → M20
  140. Circle((0, 0, 0), 5, Line[(0, 0, 0), M2]) → M21
  141. Plane(H, E, O) → N1
  142. UnitOrthogonalVector(N1) → N2
  143. Element(N12, 1, 2) → N13
  144. Element(N12, 1, 3) → N14
  145. Element(N12, 2, 2) → N15
  146. Element(N12, 2, 3) → N16
  147. Element(N12, 3, 2) → N17
  148. Element(N12, 3, 3) → N18
  149. Surface(((N$13 * (4 + t)) * cos(k)) + ((N$14 * (4 + t)) * sin(k)), ((N$15 * (4 + t)) * cos(k)) + ((N$16 * (4 + t)) * sin(k)), ((N$17 * (4 + t)) * cos(k)) + ((N$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → N19
  150. Circle((0, 0, 0), 4, Line[(0, 0, 0), N2]) → N20
  151. Circle((0, 0, 0), 5, Line[(0, 0, 0), N2]) → N21
  152. Plane(I, L, O) → O1
  153. UnitOrthogonalVector(O1) → O2
  154. Element(O12, 1, 2) → O13
  155. Element(O12, 1, 3) → O14
  156. Element(O12, 2, 2) → O15
  157. Element(O12, 2, 3) → O16
  158. Element(O12, 3, 2) → O17
  159. Element(O12, 3, 3) → O18
  160. Surface(((O$13 * (4 + t)) * cos(k)) + ((O$14 * (4 + t)) * sin(k)), ((O$15 * (4 + t)) * cos(k)) + ((O$16 * (4 + t)) * sin(k)), ((O$17 * (4 + t)) * cos(k)) + ((O$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → O19
  161. Circle((0, 0, 0), 4, Line[(0, 0, 0), O2]) → O20
  162. Circle((0, 0, 0), 5, Line[(0, 0, 0), O2]) → O21
  163. Element(P8, 1, 2) → P13
  164. Element(P8, 1, 3) → P14
  165. Element(P8, 2, 2) → P15
  166. Element(P8, 2, 3) → P16
  167. Element(P8, 3, 2) → P17
  168. Element(P8, 3, 3) → P18
  169. Surface(((P$13 * (4 + t)) * cos(k)) + ((P$14 * (4 + t)) * sin(k)), ((P$15 * (4 + t)) * cos(k)) + ((P$16 * (4 + t)) * sin(k)), ((P$17 * (4 + t)) * cos(k)) + ((P$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → P19
  170. Circle((0, 0, 0), 4, Line[(0, 0, 0), P2]) → P20
  171. Circle((0, 0, 0), 5, Line[(0, 0, 0), P2]) → P21
  172. Element(Q12, 1, 2) → Q13
  173. Element(Q12, 1, 3) → Q14
  174. Element(Q12, 2, 2) → Q15
  175. Element(Q12, 2, 3) → Q16
  176. Element(Q12, 3, 2) → Q17
  177. Element(Q12, 3, 3) → Q18
  178. Surface(((Q$13 * (4 + t)) * cos(k)) + ((Q$14 * (4 + t)) * sin(k)), ((Q$15 * (4 + t)) * cos(k)) + ((Q$16 * (4 + t)) * sin(k)), ((Q$17 * (4 + t)) * cos(k)) + ((Q$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → Q19
  179. Circle((0, 0, 0), 4, Line[(0, 0, 0), Q2]) → Q20
  180. Circle((0, 0, 0), 5, Line[(0, 0, 0), Q2]) → Q21
  181. Element(R12, 1, 2) → R13
  182. Element(R12, 1, 3) → R14
  183. Element(R12, 2, 2) → R15
  184. Element(R12, 2, 3) → R16
  185. Element(R12, 3, 2) → R17
  186. Element(R12, 3, 3) → R18
  187. Surface(((R$13 * (4 + t)) * cos(k)) + ((R$14 * (4 + t)) * sin(k)), ((R$15 * (4 + t)) * cos(k)) + ((R$16 * (4 + t)) * sin(k)), ((R$17 * (4 + t)) * cos(k)) + ((R$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → R19
  188. Circle((0, 0, 0), 4, Line[(0, 0, 0), R2]) → R20
  189. Circle((0, 0, 0), 5, Line[(0, 0, 0), R2]) → R21
  190. Element(S12, 1, 2) → S13
  191. Element(S12, 1, 3) → S14
  192. Element(S12, 2, 2) → S15
  193. Element(S12, 2, 3) → S16
  194. Element(S12, 3, 2) → S17
  195. Element(S12, 3, 3) → S18
  196. Surface(((S$13 * (4 + t)) * cos(k)) + ((S$14 * (4 + t)) * sin(k)), ((S$15 * (4 + t)) * cos(k)) + ((S$16 * (4 + t)) * sin(k)), ((S$17 * (4 + t)) * cos(k)) + ((S$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → S19
  197. Circle((0, 0, 0), 4, Line[(0, 0, 0), S2]) → S20
  198. Circle((0, 0, 0), 5, Line[(0, 0, 0), S2]) → S21
  199. Plane(D, H, O) → T1
  200. Plane(D, I, O) → U1
  201. Plane(H, I, O) → V1
  202. UnitOrthogonalVector(T1) → T2
  203. Element(T12, 1, 2) → T13
  204. Element(T12, 1, 3) → T14
  205. Element(T12, 2, 2) → T15
  206. Element(T12, 2, 3) → T16
  207. Element(T12, 3, 2) → T17
  208. Element(T12, 3, 3) → T18
  209. Surface(((T$13 * (4 + t)) * cos(k)) + ((T$14 * (4 + t)) * sin(k)), ((T$15 * (4 + t)) * cos(k)) + ((T$16 * (4 + t)) * sin(k)), ((T$17 * (4 + t)) * cos(k)) + ((T$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → T19
  210. Circle((0, 0, 0), 4, Line[(0, 0, 0), T2]) → T20
  211. Circle((0, 0, 0), 5, Line[(0, 0, 0), T2]) → T21
  212. UnitOrthogonalVector(U1) → U2
  213. Element(U12, 1, 2) → U13
  214. Element(U12, 1, 3) → U14
  215. Element(U12, 2, 2) → U15
  216. Element(U12, 2, 3) → U16
  217. Element(U12, 3, 2) → U17
  218. Element(U12, 3, 3) → U18
  219. Surface(((U$13 * (4 + t)) * cos(k)) + ((U$14 * (4 + t)) * sin(k)), ((U$15 * (4 + t)) * cos(k)) + ((U$16 * (4 + t)) * sin(k)), ((U$17 * (4 + t)) * cos(k)) + ((U$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → U19
  220. Circle((0, 0, 0), 4, Line[(0, 0, 0), U2]) → U20
  221. Circle((0, 0, 0), 5, Line[(0, 0, 0), U2]) → U21
  222. UnitOrthogonalVector(V1) → V2
  223. Element(V12, 1, 2) → V13
  224. Element(V12, 1, 3) → V14
  225. Element(V12, 2, 2) → V15
  226. Element(V12, 2, 3) → V16
  227. Element(V12, 3, 2) → V17
  228. Element(V12, 3, 3) → V18
  229. Surface(((V$13 * (4 + t)) * cos(k)) + ((V$14 * (4 + t)) * sin(k)), ((V$15 * (4 + t)) * cos(k)) + ((V$16 * (4 + t)) * sin(k)), ((V$17 * (4 + t)) * cos(k)) + ((V$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → V19
  230. Circle((0, 0, 0), 4, Line[(0, 0, 0), V2]) → V20
  231. Circle((0, 0, 0), 5, Line[(0, 0, 0), V2]) → V21
  232. Plane(C, H, O) → W1
  233. Plane(C, L, O) → X1
  234. Plane(H, L, O) → Y1
  235. UnitOrthogonalVector(W1) → W2
  236. Element(W12, 1, 2) → W13
  237. Element(W12, 1, 3) → W14
  238. Element(W12, 2, 2) → W15
  239. Element(W12, 2, 3) → W16
  240. Element(W12, 3, 2) → W17
  241. Element(W12, 3, 3) → W18
  242. Surface(((W$13 * (4 + t)) * cos(k)) + ((W$14 * (4 + t)) * sin(k)), ((W$15 * (4 + t)) * cos(k)) + ((W$16 * (4 + t)) * sin(k)), ((W$17 * (4 + t)) * cos(k)) + ((W$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → W19
  243. Circle((0, 0, 0), 4, Line[(0, 0, 0), W2]) → W20
  244. Circle((0, 0, 0), 5, Line[(0, 0, 0), W2]) → W21
  245. UnitOrthogonalVector(X1) → X2
  246. Element(X12, 1, 2) → X13
  247. Element(X12, 1, 3) → X14
  248. Element(X12, 2, 2) → X15
  249. Element(X12, 2, 3) → X16
  250. Element(X12, 3, 2) → X17
  251. Element(X12, 3, 3) → X18
  252. Surface(((X$13 * (4 + t)) * cos(k)) + ((X$14 * (4 + t)) * sin(k)), ((X$15 * (4 + t)) * cos(k)) + ((X$16 * (4 + t)) * sin(k)), ((X$17 * (4 + t)) * cos(k)) + ((X$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → X19
  253. Circle((0, 0, 0), 4, Line[(0, 0, 0), X2]) → X20
  254. Circle((0, 0, 0), 5, Line[(0, 0, 0), X2]) → X21
  255. UnitOrthogonalVector(Y1) → Y2
  256. Element(Y12, 1, 2) → Y13
  257. Element(Y12, 1, 3) → Y14
  258. Element(Y12, 2, 2) → Y15
  259. Element(Y12, 2, 3) → Y16
  260. Element(Y12, 3, 2) → Y17
  261. Element(Y12, 3, 3) → Y18
  262. Surface(((Y$13 * (4 + t)) * cos(k)) + ((Y$14 * (4 + t)) * sin(k)), ((Y$15 * (4 + t)) * cos(k)) + ((Y$16 * (4 + t)) * sin(k)), ((Y$17 * (4 + t)) * cos(k)) + ((Y$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → Y19
  263. Circle((0, 0, 0), 4, Line[(0, 0, 0), Y2]) → Y20
  264. Circle((0, 0, 0), 5, Line[(0, 0, 0), Y2]) → Y21
  265. Plane(B, L, O) → Z1
  266. Plane(B, E, O) → AA1
  267. Plane(L, E, O) → AB1
  268. UnitOrthogonalVector(Z1) → Z2
  269. Element(Z12, 1, 2) → Z13
  270. Element(Z12, 1, 3) → Z14
  271. Element(Z12, 2, 2) → Z15
  272. Element(Z12, 2, 3) → Z16
  273. Element(Z12, 3, 2) → Z17
  274. Element(Z12, 3, 3) → Z18
  275. Surface(((Z$13 * (4 + t)) * cos(k)) + ((Z$14 * (4 + t)) * sin(k)), ((Z$15 * (4 + t)) * cos(k)) + ((Z$16 * (4 + t)) * sin(k)), ((Z$17 * (4 + t)) * cos(k)) + ((Z$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → Z19
  276. Circle((0, 0, 0), 4, Line[(0, 0, 0), Z2]) → Z20
  277. Circle((0, 0, 0), 5, Line[(0, 0, 0), Z2]) → Z21
  278. UnitOrthogonalVector(AA1) → AA2
  279. Element(AA12, 1, 2) → AA13
  280. Element(AA12, 1, 3) → AA14
  281. Element(AA12, 2, 2) → AA15
  282. Element(AA12, 2, 3) → AA16
  283. Element(AA12, 3, 2) → AA17
  284. Element(AA12, 3, 3) → AA18
  285. Surface(((AA$13 * (4 + t)) * cos(k)) + ((AA$14 * (4 + t)) * sin(k)), ((AA$15 * (4 + t)) * cos(k)) + ((AA$16 * (4 + t)) * sin(k)), ((AA$17 * (4 + t)) * cos(k)) + ((AA$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → AA19
  286. Circle((0, 0, 0), 4, Line[(0, 0, 0), AA2]) → AA20
  287. Circle((0, 0, 0), 5, Line[(0, 0, 0), AA2]) → AA21
  288. UnitOrthogonalVector(AB1) → AB2
  289. Element(AB12, 1, 2) → AB13
  290. Element(AB12, 1, 3) → AB14
  291. Element(AB12, 2, 2) → AB15
  292. Element(AB12, 2, 3) → AB16
  293. Element(AB12, 3, 2) → AB17
  294. Element(AB12, 3, 3) → AB18
  295. Surface(((AB$13 * (4 + t)) * cos(k)) + ((AB$14 * (4 + t)) * sin(k)), ((AB$15 * (4 + t)) * cos(k)) + ((AB$16 * (4 + t)) * sin(k)), ((AB$17 * (4 + t)) * cos(k)) + ((AB$18 * (4 + t)) * sin(k)), t, 0, 1, k, 0, (2 * π)) → AB19
  296. Circle((0, 0, 0), 4, Line[(0, 0, 0), AB2]) → AB20
  297. Circle((0, 0, 0), 5, Line[(0, 0, 0), AB2]) → AB21
  298. Segment(O + (4 * AD1), O + (5 * AD1)) → AE1
  299. Segment(O + (4 * AD2), O + (5 * AD2)) → AE2
  300. Segment(O + (4 * AD3), O + (5 * AD3)) → AE3
  301. Segment(O + (4 * AD4), O + (5 * AD4)) → AE4
  302. Segment(O + (4 * AD5), O + (5 * AD5)) → AE5
  303. Segment(O + (4 * AD6), O + (5 * AD6)) → AE6
  304. Segment(O + (4 * AD7), O + (5 * AD7)) → AE7
  305. Segment(O + (4 * AD8), O + (5 * AD8)) → AE8
  306. Segment(O + (4 * AD9), O + (5 * AD9)) → AE9
  307. Segment(O + (4 * AD10), O + (5 * AD10)) → AE10
  308. Segment(O + (4 * AD11), O + (5 * AD11)) → AE11
  309. Segment(O + (4 * AD12), O + (5 * AD12)) → AE12
  310. Segment(O + (4 * AD13), O + (5 * AD13)) → AE13
  311. Segment(O + (4 * AD14), O + (5 * AD14)) → AE14
  312. Segment(O + (4 * AD15), O + (5 * AD15)) → AE15
  313. Segment(O + (4 * AD16), O + (5 * AD16)) → AE16
  314. Segment(O + (4 * AD17), O + (5 * AD17)) → AE17
  315. Segment(O + (4 * AD18), O + (5 * AD18)) → AE18
  316. Segment(O + (4 * AD19), O + (5 * AD19)) → AE19
  317. Segment(O + (4 * AD20), O + (5 * AD20)) → AE20
  318. Segment(O + (4 * AD21), O + (5 * AD21)) → AE21
  319. Segment(O + (4 * AD22), O + (5 * AD22)) → AE22
  320. Segment(O + (4 * AD23), O + (5 * AD23)) → AE23
  321. Segment(O + (4 * AD24), O + (5 * AD24)) → AE24
  322. Segment(O + (4 * AD25), O + (5 * AD25)) → AE25
  323. Segment(O + (4 * AD26), O + (5 * AD26)) → AE26
  324. Segment(O + (4 * AD27), O + (5 * AD27)) → AE27
  325. Segment(O + (4 * AD28), O + (5 * AD28)) → AE28
  326. Segment(O + (4 * AD30), O + (5 * AD30)) → AE30
  327. Segment(O + (4 * AD31), O + (5 * AD31)) → AE31
  328. Segment(O + (4 * AD32), O + (5 * AD32)) → AE32
  329. Segment(O + (4 * AD33), O + (5 * AD33)) → AE33
  330. Segment(O + (4 * AD34), O + (5 * AD34)) → AE34
  331. Segment(O + (4 * AD35), O + (5 * AD35)) → AE35
  332. Segment(O + (4 * AD36), O + (5 * AD36)) → AE36
  333. Segment(O + (4 * AD37), O + (5 * AD37)) → AE37
  334. Segment(O + (4 * AD38), O + (5 * AD38)) → AE38
  335. Segment(O + (4 * AD39), O + (5 * AD39)) → AE39
  336. Segment(O + (4 * AD40), O + (5 * AD40)) → AE40
  337. Segment(O + (4 * AD41), O + (5 * AD41)) → AE41
  338. Segment(O + (4 * AD42), O + (5 * AD42)) → AE42
  339. Segment(O + (4 * AD43), O + (5 * AD43)) → AE43
  340. Segment(O + (4 * AD44), O + (5 * AD44)) → AE44
  341. Segment(O + (4 * AD45), O + (5 * AD45)) → AE45
  342. Segment(O + (4 * AD46), O + (5 * AD46)) → AE46
  343. Segment(O + (4 * AD47), O + (5 * AD47)) → AE47
  344. Segment(O + (4 * AD48), O + (5 * AD48)) → AE48
  345. Segment(O + (4 * AD49), O + (5 * AD49)) → AE49
  346. Segment(O + (4 * AD50), O + (5 * AD50)) → AE50

AI Drawing Prompt Reference

Create a spherical model of a cuboctahedron by mapping its vertices, edges, and faces onto a sphere, resulting in a solid geometry figure composed of circular arcs and spherical polygons. Define multiple cutting planes and generate orthogonal unit vectors, then create spherical intersection curves (great or small circles) through plane-sphere intersections to form the polygons. Finally, connect the vertices to form surfaces and circles. This visualization highlights polyhedral geometry, spherical geometry, Archimedean solids, and the intersection of planes with a sphere, suitable for teaching and exploring these mathematical concepts.
The prompt above was compiled from the original drawing commands and is for reference only.