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Sphere Peeling: Volume

Sphere Peeling: Volume
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2015-12-29 04:36
Solid GeometryTransformation
In collections Area, Surface Area, Volume, Cross Section, 3D Geometry - Area & Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume GA Chapter 11: Surface Area and Volume Surface Area & Volume Surface Area & Volume
This example demonstrates the process of peeling a sphere, focusing on the concept of volume. Using GeoGebra commands such as Dilate, Translate, Mirror, and Rotate, the process of peeling the sphere is dynamically shown. Suitable for high school mathematics education, particularly in solid geometry and volume calculation.

GeoGebra Commands & Parameters

  1. If(2 < a < 3, Translate[h, (((0.9 * π) * (a - 2)), 0)]) → e_2
  2. Center(e_2) → Q_1
  3. Dilate(D, If[0 < a < 1, 1 - a, 1], B) → D'
  4. Intersect(c, yAxis, 2) → E
  5. Point(yAxis) → F
  6. Translate(c, If[0 < a < 1, ((π * a), 0)]) → c'
  7. Center(c') → G
  8. Mirror(D, f) → D'_1
  9. Mirror(D', f) → D''
  10. Intersect(g, f, 2) → J
  11. Mirror(K, f) → K'
  12. Mirror(L, f) → L'
  13. Mirror(N, f) → N'
  14. Mirror(P, f) → P'
  15. Mirror(Q, f) → Q'
  16. Mirror(R, f) → R'
  17. Mirror(S, f) → S'
  18. Mirror(T, f) → T'
  19. Mirror(U, f) → U'
  20. Mirror(V, f) → V'
  21. Mirror(W, f) → W'
  22. Mirror(Z, f) → Z'
  23. Mirror(A_1, f) → M
  24. Mirror(B_1, f) → O
  25. Mirror(C_1, f) → H_1
  26. Mirror(D_1, f) → I_1
  27. Mirror(E_1, f) → J_1
  28. Mirror(F_1, f) → K_1
  29. Mirror(G_1, f) → L_1
  30. Dilate(J, If[1 < a < 2, 2 - a, 1], K) → J'
  31. If(1 < a < 2, Translate[g, (((0.95 * π) * (a - 1)), 0)]) → t_1
  32. Center(t_1) → N_1
  33. Mirror(J', f) → J''_1
  34. Mirror(J, f) → J'_1
  35. Mirror(J', f) → J''_2
  36. Intersect(g, f, 1) → I
  37. Intersect(h, f) → M_1, O_1
  38. Intersect(k, f) → P_1, R_1
  39. Intersect(p, f) → S_1, T_1
  40. If(1 < a < 2, Translate[I, (((0.95 * π) * (a - 1)), 0)]) → U_1
  41. Rotate(U_1, (((1 - a) * 180))°, N_1) → V_1
  42. CircleArc(N_1, J', V_1) → c_2
  43. Mirror(c_2, f) → d_2
  44. If(2 < a < 3, Translate[M_1, (((0.9 * π) * (a - 2)), 0)]) → W_1
  45. Dilate(O_1, If[2 < a < 3, 3 - a, 1], L) → Z_1
  46. Rotate(W_1, (((2 - a) * 180))°, Q_1) → A_2
  47. CircleArc(Q_1, Z_1, A_2) → f_2
  48. Mirror(f_2, f) → g_2
  49. Mirror(O_1, f) → B_2
  50. Mirror(Z_1, f) → C_2
  51. Intersect(k, f, 1) → D_2
  52. Dilate(R_1, If[3 < a < 4, 4 - a, 1], N) → E_2
  53. If(3 < a < 4, Translate[k, (((0.85 * π) * (a - 3)), 0)]) → h_2
  54. Center(h_2) → F_2
  55. If(3 < a < 4, Translate[D_2, (((0.85 * π) * (a - 3)), 0)]) → G_2
  56. Rotate(G_2, (((3 - a) * 180))°, F_2) → H_2
  57. CircleArc(F_2, E_2, H_2) → k_2
  58. Mirror(R_1, f) → I_2
  59. Mirror(E_2, f) → J_2
  60. Intersect(f, yAxis) → K_2
  61. Intersect(f, yAxis) → L_2
  62. Intersect(f, yAxis) → M_2
  63. Intersect(c, f) → N_2, O_2
  64. If(0 < a < 1, Translate[N_2, ((a * π), 0)]) → P_2
  65. Rotate(P_2, If[0 < a < 1, (((-a) * 180))°], C) → H
  66. CircleArc(C, D', H) → e
  67. Mirror(e, f) → e'
  68. Intersect(q, f) → Q_2, R_2
  69. Intersect(f, yAxis) → S_2
  70. Intersect(f, yAxis) → T_2
  71. Intersect(f, yAxis) → U_2
  72. Intersect(f, yAxis) → V_2
  73. Intersect(f, yAxis) → W_2
  74. Intersect(f, yAxis) → Z_2
  75. Intersect(f, yAxis) → A_3
  76. Intersect(r, s) → B_3, , ,
  77. Intersect(f, yAxis) → C_3
  78. Intersect(r, f, 1) → D_3
  79. Intersect(s, f, 1) → E_3
  80. Intersect(t, f, 1) → F_3
  81. Intersect(c_1, f, 1) → G_3
  82. Intersect(d_1, f, 1) → H_3
  83. Intersect(e_1, f, 1) → I_3
  84. Intersect(f_1, f, 1) → J_3
  85. Intersect(g_1, f, 1) → K_3
  86. Intersect(h_1, f, 1) → L_3
  87. Intersect(k_1, f, 1) → M_3
  88. Intersect(p_1, f, 1) → N_3
  89. Intersect(q_1, f, 1) → O_3
  90. Intersect(r_1, f, 1) → P_3
  91. Intersect(s_1, f, 1) → Q_3
  92. Intersect(s_1, f, 2) → R_3
  93. Intersect(r_1, f, 2) → S_3
  94. Intersect(q_1, f, 2) → T_3
  95. Intersect(f, yAxis) → U_3
  96. Intersect(p_1, f, 2) → V_3
  97. Intersect(k_1, f, 2) → W_3
  98. Intersect(h_1, f, 2) → Z_3
  99. Intersect(g_1, f, 2) → A_4
  100. Intersect(f_1, f, 2) → B_4
  101. Intersect(e_1, f, 2) → C_4
  102. Intersect(d_1, f, 2) → D_4
  103. Intersect(c_1, f, 2) → E_4
  104. Intersect(t, f, 2) → F_4
  105. Intersect(s, f, 2) → G_4
  106. Intersect(r, f, 2) → H_4
  107. Dilate(T_1, If[4 < a < 5, 5 - a, 1], P) → I_4
  108. Dilate(R_2, If[5 < a < 6, 6 - a, 1], Q) → J_4
  109. Mirror(T_1, f) → K_4
  110. Mirror(I_4, f) → L_4
  111. Dilate(H_4, If[6 < a < 7, 7 - a, 1], R) → M_4
  112. Dilate(G_4, If[7 < a < 8, 8 - a, 1], S) → N_4
  113. Dilate(F_4, If[8 < a < 9, 9 - a, 1], T) → O_4
  114. Dilate(E_4, If[9 < a < 10, 10 - a, 1], U) → P_4
  115. Dilate(D_4, If[10 < a < 11, 11 - a, 1], V) → Q_4
  116. Dilate(C_4, If[11 < a < 12, 12 - a, 1], W) → R_4
  117. Dilate(B_4, If[12 < a < 13, 13 - a, 1], Z) → S_4
  118. Dilate(A_4, If[13 < a < 14, 14 - a, 1], A_1) → T_4
  119. Dilate(Z_3, If[14 < a < 15, 15 - a, 1], B_1) → U_4
  120. Dilate(W_3, If[15 < a < 16, 16 - a, 1], C_1) → V_4
  121. Dilate(V_3, If[16 < a < 17, 17 - a, 1], D_1) → W_4
  122. Dilate(T_3, If[17 < a < 18, 18 - a, 1], E_1) → Z_4
  123. Dilate(S_3, If[18 < a < 19, 19 - a, 1], F_1) → A_5
  124. Dilate(R_3, If[19 < a < 20, 20 - a, 1], G_1) → B_5
  125. If(4 < a < 5, Translate[p, (((0.8 * π) * (a - 4)), 0)]) → d_4
  126. If(4 < a < 5, Translate[S_1, (((0.8 * π) * (a - 4)), 0)]) → C_5
  127. Center(d_4) → D_5
  128. Rotate(C_5, (((4 - a) * 180))°, D_5) → E_5
  129. CircleArc(D_5, I_4, E_5) → e_4
  130. If(5 < a < 6, Translate[q, (((0.75 * π) * (a - 5)), 0)]) → f_4
  131. If(5 < a < 6, Translate[Q_2, (((0.75 * π) * (a - 5)), 0)]) → F_5
  132. Center(f_4) → G_5
  133. Rotate(F_5, (((5 - a) * 180))°, G_5) → H_5
  134. CircleArc(G_5, J_4, H_5) → g_4
  135. If(6 < a < 7, Translate[r, (((0.7 * π) * (a - 6)), 0)]) → h_4
  136. If(6 < a < 7, Translate[D_3, (((0.7 * π) * (a - 6)), 0)]) → I_5
  137. Center(h_4) → J_5
  138. Rotate(I_5, (((6 - a) * 180))°, J_5) → K_5
  139. CircleArc(J_5, M_4, K_5) → k_4
  140. If(7 < a < 8, Translate[s, (((0.65 * π) * (a - 7)), 0)]) → p_4
  141. If(7 < a < 8, Translate[E_3, (((0.65 * π) * (a - 7)), 0)]) → L_5
  142. Center(p_4) → M_5
  143. Rotate(L_5, (((7 - a) * 180))°, M_5) → N_5
  144. CircleArc(M_5, N_4, N_5) → q_4
  145. If(8 < a < 9, Translate[t, (((0.6 * π) * (a - 8)), 0)]) → r_4
  146. If(8 < a < 9, Translate[F_3, (((0.6 * π) * (a - 8)), 0)]) → O_5
  147. Center(r_4) → P_5
  148. Rotate(O_5, (((8 - a) * 180))°, P_5) → Q_5
  149. CircleArc(P_5, O_4, Q_5) → s_4
  150. If(9 < a < 10, Translate[c_1, (((0.55 * π) * (a - 9)), 0)]) → t_4
  151. If(9 < a < 10, Translate[G_3, (((0.55 * π) * (a - 9)), 0)]) → R_5
  152. Center(t_4) → S_5
  153. Rotate(R_5, (((9 - a) * 180))°, S_5) → T_5
  154. CircleArc(S_5, P_4, T_5) → c_5
  155. If(10 < a < 11, Translate[d_1, (((0.5 * π) * (a - 10)), 0)]) → d_5
  156. If(10 < a < 11, Translate[H_3, (((0.5 * π) * (a - 10)), 0)]) → U_5
  157. Center(d_5) → V_5
  158. Rotate(U_5, (((10 - a) * 180))°, V_5) → W_5
  159. CircleArc(V_5, Q_4, W_5) → e_5
  160. If(11 < a < 12, Translate[e_1, (((0.45 * π) * (a - 11)), 0)]) → f_5
  161. If(11 < a < 12, Translate[I_3, (((0.45 * π) * (a - 11)), 0)]) → Z_5
  162. Center(f_5) → A_6
  163. Rotate(Z_5, (((11 - a) * 180))°, A_6) → B_6
  164. CircleArc(A_6, R_4, B_6) → g_5
  165. If(12 < a < 13, Translate[f_1, (((0.4 * π) * (a - 12)), 0)]) → h_5
  166. If(12 < a < 13, Translate[J_3, (((0.4 * π) * (a - 12)), 0)]) → C_6
  167. Center(h_5) → D_6
  168. Rotate(C_6, (((12 - a) * 180))°, D_6) → E_6
  169. CircleArc(D_6, S_4, E_6) → k_5
  170. If(13 < a < 14, Translate[g_1, (((0.35 * π) * (a - 13)), 0)]) → p_5
  171. If(13 < a < 14, Translate[K_3, (((0.35 * π) * (a - 13)), 0)]) → F_6
  172. Center(p_5) → G_6
  173. Rotate(F_6, (((13 - a) * 180))°, G_6) → H_6
  174. CircleArc(G_6, T_4, H_6) → q_5
  175. If(14 < a < 15, Translate[h_1, (((0.3 * π) * (a - 14)), 0)]) → r_5
  176. If(14 < a < 15, Translate[L_3, (((0.3 * π) * (a - 14)), 0)]) → I_6
  177. Center(r_5) → J_6
  178. Rotate(I_6, (((14 - a) * 180))°, J_6) → K_6
  179. CircleArc(J_6, U_4, K_6) → s_5
  180. If(15 < a < 16, Translate[k_1, (((0.25 * π) * (a - 15)), 0)]) → t_5
  181. If(15 < a < 16, Translate[M_3, (((0.25 * π) * (a - 15)), 0)]) → L_6
  182. Center(t_5) → M_6
  183. Rotate(L_6, (((15 - a) * 180))°, M_6) → N_6
  184. CircleArc(M_6, V_4, N_6) → c_6
  185. If(16 < a < 17, Translate[p_1, (((0.2 * π) * (a - 16)), 0)]) → d_6
  186. If(16 < a < 17, Translate[N_3, (((0.2 * π) * (a - 16)), 0)]) → O_6
  187. Center(d_6) → P_6
  188. Rotate(O_6, (((16 - a) * 180))°, P_6) → Q_6
  189. CircleArc(P_6, W_4, Q_6) → e_6
  190. If(17 < a < 18, Translate[q_1, (((0.15 * π) * (a - 17)), 0)]) → f_6
  191. If(17 < a < 18, Translate[O_3, (((0.15 * π) * (a - 17)), 0)]) → R_6
  192. Center(f_6) → S_6
  193. Rotate(R_6, (((17 - a) * 180))°, S_6) → T_6
  194. CircleArc(S_6, Z_4, T_6) → g_6
  195. If(18 < a < 19, Translate[r_1, (((0.1 * π) * (a - 18)), 0)]) → h_6
  196. If(18 < a < 19, Translate[P_3, (((0.1 * π) * (a - 18)), 0)]) → U_6
  197. Center(h_6) → V_6
  198. Rotate(U_6, (((18 - a) * 180))°, V_6) → W_6
  199. CircleArc(V_6, A_5, W_6) → k_6
  200. If(19 < a < 20, Translate[s_1, (((0.05 * π) * (a - 19)), 0)]) → p_6
  201. If(19 < a < 20, Translate[Q_3, (((0.05 * π) * (a - 19)), 0)]) → Z_6
  202. Center(p_6) → A_7
  203. Rotate(Z_6, (((19 - a) * 180))°, A_7) → B_7
  204. CircleArc(A_7, B_5, B_7) → q_6
  205. Mirror(R_2, f) → C_7
  206. Mirror(J_4, f) → D_7
  207. Mirror(H_4, f) → E_7
  208. Mirror(M_4, f) → F_7
  209. Mirror(G_4, f) → G_7
  210. Mirror(N_4, f) → H_7
  211. Mirror(F_4, f) → I_7
  212. Mirror(O_4, f) → J_7
  213. Mirror(E_4, f) → K_7
  214. Mirror(P_4, f) → L_7
  215. Mirror(D_4, f) → M_7
  216. Mirror(Q_4, f) → N_7
  217. Mirror(C_4, f) → O_7
  218. Mirror(R_4, f) → P_7
  219. Mirror(B_4, f) → Q_7
  220. Mirror(S_4, f) → R_7
  221. Mirror(A_4, f) → S_7
  222. Mirror(T_4, f) → T_7
  223. Mirror(Z_3, f) → U_7
  224. Mirror(U_4, f) → V_7
  225. Mirror(W_3, f) → W_7
  226. Mirror(V_4, f) → Z_7
  227. Mirror(V_3, f) → A_8
  228. Mirror(W_4, f) → B_8
  229. Mirror(T_3, f) → C_8
  230. Mirror(Z_4, f) → D_8
  231. Mirror(S_3, f) → E_8
  232. Mirror(A_5, f) → F_8
  233. Mirror(R_3, f) → G_8
  234. Mirror(B_5, f) → H_8
  235. Sequence(Cylinder[(0, k, 0), (0, k + 0.05, 0), ((1 - k) * π)], k, 0, 0.95, 0.05) → baselist
  236. Element(baselist, 1) → cyl1
  237. Line((0, 0, 0), (0, 1, 0)) → m_6
  238. Sequence(Rotate[e, α, m_6], α, 1*°, 359*°, 1*°) → arc_1
  239. Cylinder((0, 0, 0), (0, 0.05, 0), (a * π)) → cyl1_a, d_8, e_8, o
  240. Sequence(If[a ≤ 0, Rotate[c, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_1
  241. Sequence(If[a ≤ 1, Rotate[g, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_2
  242. Sequence(Rotate[c_2, α, m_6], α, 1*°, 359*°, 1*°) → arc_2
  243. Cylinder((0, 0.05, 0), (0, 0.1, 0), (((a - 1) * 0.95) * π)) → cyl2_a, f_8, g_8, u
  244. Element(baselist, 2) → cyl2
  245. Sequence(If[a ≤ 2, Rotate[h, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_3
  246. Sequence(Rotate[f_2, α, m_6], α, 1*°, 359*°, 1*°) → arc_3
  247. Cylinder((0, 0.1, 0), (0, 0.15, 0), (((a - 2) * 0.9) * π)) → cyl3_a, h_8, k_8, v
  248. Element(baselist, 3) → cyl_3
  249. Sequence(If[a ≤ 3, Rotate[k, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_4
  250. Sequence(Rotate[k_2, α, m_6], α, 1*°, 359*°, 1*°) → arc_4
  251. Cylinder((0, 0.15, 0), (0, 0.2, 0), (((a - 3) * 0.85) * π)) → cyl4_a, p_2, p_8, w
  252. Element(baselist, 4) → cyl_4
  253. Sequence(If[a ≤ 4, Rotate[p, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_5
  254. Sequence(Rotate[e_4, α, m_6], α, 1*°, 359*°, 1*°) → arc_5
  255. Cylinder((0, 0.2, 0), (0, 0.25, 0), (((a - 4) * 0.8) * π)) → cyl5_a, r_6, q_8, o_1
  256. Element(baselist, 5) → cyl_5
  257. Sequence(If[a ≤ 5, Rotate[q, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_6
  258. Sequence(Rotate[g_4, α, m_6], α, 1*°, 359*°, 1*°) → arc_6
  259. Cylinder((0, 0.25, 0), (0, 0.3, 0), (((a - 5) * 0.75) * π)) → cyl6_a, s_6, r_8, u_1
  260. Element(baselist, 6) → cyl_6
  261. Sequence(If[a ≤ 6, Rotate[r, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_7
  262. Sequence(Rotate[k_4, α, m_6], α, 1*°, 359*°, 1*°) → arc_7
  263. Cylinder((0, 0.3, 0), (0, 0.35, 0), (((a - 6) * 0.7) * π)) → cyl7_a, t_6, s_8, v_1
  264. Element(baselist, 7) → cyl_7
  265. Sequence(If[a ≤ 7, Rotate[s, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_8
  266. Sequence(Rotate[q_4, α, m_6], α, 1*°, 359*°, 1*°) → arc_8
  267. Cylinder((0, 0.35, 0), (0, 0.4, 0), (((a - 7) * 0.65) * π)) → cyl8_a, c_7, t_8, w_1
  268. Element(baselist, 8) → cyl_8
  269. Sequence(If[a ≤ 8, Rotate[t, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_9
  270. Sequence(Rotate[s_4, α, m_6], α, 1*°, 359*°, 1*°) → arc_9
  271. Cylinder((0, 0.4, 0), (0, 0.45, 0), (((a - 8) * 0.6) * π)) → cyl9_a, d_7, c_9, o_2
  272. Element(baselist, 9) → cyl_9
  273. Sequence(If[a ≤ 9, Rotate[c_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_10
  274. Sequence(Rotate[c_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_10
  275. Cylinder((0, 0.45, 0), (0, 0.5, 0), (((a - 9) * 0.55) * π)) → cyl10_a, e_7, d_9, u_2
  276. Element(baselist, 10) → cyl_10
  277. Sequence(If[a ≤ 10, Rotate[d_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_11
  278. Sequence(Rotate[e_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_11
  279. Cylinder((0, 0.5, 0), (0, 0.55, 0), (((a - 10) * 0.5) * π)) → cyl11_a, f_7, e_9, v_2
  280. Element(baselist, 11) → cyl_11
  281. Sequence(If[a ≤ 11, Rotate[e_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_12
  282. Sequence(Rotate[g_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_12
  283. Cylinder((0, 0.55, 0), (0, 0.6, 0), (((a - 11) * 0.45) * π)) → cyl12_a, g_7, f_9, w_2
  284. Element(baselist, 12) → cyl_12
  285. Sequence(If[a ≤ 12, Rotate[f_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_13
  286. Sequence(Rotate[k_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_13
  287. Cylinder((0, 0.6, 0), (0, 0.65, 0), (((a - 12) * 0.4) * π)) → cyl13_a, h_7, g_9, o_3
  288. Element(baselist, 13) → cyl_13
  289. Sequence(If[a ≤ 13, Rotate[g_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_14
  290. Sequence(Rotate[q_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_14
  291. Cylinder((0, 0.65, 0), (0, 0.7, 0), (((a - 13) * 0.35) * π)) → cyl14_a, k_7, h_9, u_3
  292. Element(baselist, 14) → cyl_14
  293. Sequence(If[a ≤ 14, Rotate[h_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_15
  294. Sequence(Rotate[s_5, α, m_6], α, 1*°, 359*°, 1*°) → arc_15
  295. Cylinder((0, 0.7, 0), (0, 0.75, 0), (((a - 14) * 0.3) * π)) → cyl15_a, p_7, k_9, v_3
  296. Element(baselist, 15) → cyl_15
  297. Sequence(If[a ≤ 15, Rotate[k_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_16
  298. Sequence(Rotate[c_6, α, m_6], α, 1*°, 359*°, 1*°) → arc_16
  299. Cylinder((0, 0.75, 0), (0, 0.8, 0), (((a - 15) * 0.25) * π)) → cyl16_a, q_7, p_9, w_3
  300. Element(baselist, 16) → cyl_16
  301. Sequence(If[a ≤ 16, Rotate[p_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_17
  302. Sequence(Rotate[e_6, α, m_6], α, 1*°, 359*°, 1*°) → arc_17
  303. Cylinder((0, 0.8, 0), (0, 0.85, 0), (((a - 16) * 0.2) * π)) → cyl17_a, r_7, q_9, o_4
  304. Element(baselist, 17) → cyl_17
  305. Sequence(If[a ≤ 17, Rotate[q_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_18
  306. Sequence(Rotate[g_6, α, m_6], α, 1*°, 359*°, 1*°) → arc_18
  307. Cylinder((0, 0.85, 0), (0, 0.9, 0), (((a - 17) * 0.15) * π)) → cyl18_a, s_7, r_9, u_4
  308. Element(baselist, 18) → cyl_18
  309. Sequence(If[a ≤ 18, Rotate[r_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_19
  310. Sequence(Rotate[k_6, α, m_6], α, 1*°, 359*°, 1*°) → arc_19
  311. Cylinder((0, 0.9, 0), (0, 0.95, 0), (((a - 18) * 0.1) * π)) → cyl19_a, t_7, s_9, v_4
  312. Element(baselist, 19) → cyl_19
  313. Sequence(If[a ≤ 19, Rotate[s_1, α, m_6]], α, 1*°, 180*°, 1*°) → sphere_20
  314. Sequence(Rotate[q_6, α, m_6], α, 1*°, 359*°, 1*°) → arc_20
  315. Cylinder((0, 0.95, 0), (0, 1, 0), (((a - 19) * 0.05) * π)) → cyl20_a, c_8, t_9, w_4
  316. Element(baselist, 20) → cyl_20
  317. Point(yAxis) → I_8
  318. Sphere(I_8, 1) → d