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Nephroid

Nephroid
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2018-06-14 21:23
CircleParametric EquationGeometric Construction
This example demonstrates the generation of a nephroid, a special type of roulette formed by the trajectory of a point on a circle rolling inside another circle. Key construction commands include Circle, Point, Segment, Line, OrthogonalLine, Intersect, Midpoint, and Mirror. Suitable for high school mathematics education and coordinate geometry learning.

GeoGebra Commands & Parameters

  1. Circle(A, 5) → c
  2. Point(c) → B
  3. Point(c) → C
  4. Segment(B, C) → f
  5. Line(E, f) → g
  6. Point(g) → F
  7. OrthogonalLine(F, f) → h
  8. Intersect(c, h, 1) → G
  9. Intersect(h, f) → H
  10. Intersect(c, h, 2) → I
  11. Midpoint(G, H) → J
  12. Midpoint(J, G) → K
  13. Midpoint(J, K) → L
  14. Midpoint(K, G) → M
  15. Midpoint(J, H) → N
  16. Midpoint(H, N) → O
  17. Midpoint(N, J) → P
  18. Midpoint(H, I) → Q
  19. Midpoint(Q, I) → R
  20. Midpoint(R, I) → S
  21. Midpoint(Q, R) → T
  22. Midpoint(H, Q) → U
  23. Midpoint(U, Q) → V
  24. Midpoint(U, H) → W
  25. Midpoint(G, M) → Z
  26. Midpoint(M, K) → A_1
  27. Midpoint(K, L) → B_1
  28. Midpoint(L, J) → C_1
  29. Midpoint(J, P) → D_1
  30. Midpoint(P, N) → E_1
  31. Midpoint(N, O) → F_1
  32. Midpoint(O, H) → G_1
  33. Midpoint(H, W) → H_1
  34. Midpoint(W, U) → I_1
  35. Midpoint(U, V) → J_1
  36. Midpoint(V, Q) → K_1
  37. Midpoint(Q, T) → L_1
  38. Midpoint(T, R) → M_1
  39. Midpoint(R, S) → N_1
  40. Midpoint(S, I) → O_1
  41. OrthogonalLine(M, h) → i
  42. OrthogonalLine(Z, h) → j
  43. OrthogonalLine(A_1, h) → k
  44. OrthogonalLine(K, h) → l
  45. OrthogonalLine(B_1, h) → m
  46. OrthogonalLine(L, h) → n
  47. OrthogonalLine(C_1, h) → p
  48. OrthogonalLine(J, h) → q
  49. OrthogonalLine(D_1, h) → r
  50. OrthogonalLine(P, h) → s
  51. OrthogonalLine(E_1, h) → t
  52. OrthogonalLine(N, h) → a
  53. OrthogonalLine(F_1, h) → b
  54. OrthogonalLine(O, h) → d
  55. OrthogonalLine(G_1, h) → e
  56. OrthogonalLine(H_1, h) → f_1
  57. OrthogonalLine(W, h) → g_1
  58. OrthogonalLine(I_1, h) → h_1
  59. OrthogonalLine(U, h) → i_1
  60. OrthogonalLine(J_1, h) → j_1
  61. OrthogonalLine(V, h) → k_1
  62. OrthogonalLine(K_1, h) → l_1
  63. OrthogonalLine(Q, h) → m_1
  64. OrthogonalLine(L_1, h) → n_1
  65. OrthogonalLine(T, h) → p_1
  66. OrthogonalLine(M_1, h) → q_1
  67. OrthogonalLine(R, h) → r_1
  68. OrthogonalLine(N_1, h) → s_1
  69. OrthogonalLine(S, h) → t_1
  70. OrthogonalLine(O_1, h) → a_1
  71. OrthogonalLine(G, h) → b_1
  72. OrthogonalLine(I, h) → c_1
  73. Intersect(c, j, 2) → P_1
  74. Circle(P_1, Z) → d_1
  75. Intersect(c, i, 2) → Q_1
  76. Circle(Q_1, M) → e_1
  77. Intersect(c, k, 2) → R_1
  78. Circle(R_1, A_1) → c_2
  79. Intersect(c, l, 2) → S_1
  80. Intersect(c, m, 2) → T_1
  81. Intersect(c, n, 2) → U_1
  82. Intersect(c, p, 2) → V_1
  83. Intersect(c, f_1, 2) → C_2
  84. Intersect(c, q, 2) → D_2
  85. Circle(S_1, K) → d_2
  86. Circle(T_1, B_1) → e_2
  87. Circle(U_1, L) → f_2
  88. Circle(V_1, C_1) → g_2
  89. Circle(D_2, J) → h_2
  90. Intersect(c, r, 2) → W_1
  91. Circle(W_1, D_1) → k_2
  92. Intersect(c, s, 2) → Z_1
  93. Circle(Z_1, P) → p_2
  94. Intersect(c, t, 2) → A_2
  95. Circle(A_2, E_1) → q_2
  96. Intersect(c, a, 2) → B_2
  97. Circle(B_2, N) → r_2
  98. Intersect(c, b, 2) → E_2
  99. Circle(E_2, F_1) → s_2
  100. Intersect(c, d, 2) → F_2
  101. Circle(F_2, O) → t_2
  102. Intersect(c, e, 2) → G_2
  103. Circle(G_2, G_1) → c_3
  104. Circle(B, H) → d_3
  105. Circle(C_2, H_1) → e_3
  106. Intersect(c, g_1, 2) → H_2
  107. Circle(H_2, W) → f_3
  108. Intersect(c, h_1, 2) → I_2
  109. Circle(I_2, I_1) → g_3
  110. Intersect(c, i_1) → J_2, K_2
  111. Circle(K_2, U) → h_3
  112. Intersect(c, j_1, 2) → L_2
  113. Circle(L_2, J_1) → k_3
  114. Intersect(c, k_1, 2) → M_2
  115. Circle(M_2, V) → p_3
  116. Intersect(c, l_1, 2) → N_2
  117. Circle(N_2, K_1) → q_3
  118. Intersect(c, m_1, 2) → O_2
  119. Circle(O_2, Q) → r_3
  120. Intersect(c, n_1, 2) → P_2
  121. Circle(P_2, L_1) → s_3
  122. Intersect(c, p_1, 2) → Q_2
  123. Circle(Q_2, T) → t_3
  124. Intersect(c, q_1, 2) → R_2
  125. Circle(R_2, M_1) → c_4
  126. Intersect(c, r_1, 2) → S_2
  127. Circle(S_2, R) → d_4
  128. Intersect(c, s_1, 2) → T_2
  129. Circle(T_2, N_1) → e_4
  130. Intersect(c, t_1, 2) → U_2
  131. Circle(U_2, S) → f_4
  132. Intersect(c, a_1, 2) → V_2
  133. Circle(V_2, O_1) → g_4
  134. Intersect(c, j, 1) → Z_2
  135. Intersect(c, i, 1) → A_3
  136. Intersect(c, k, 1) → B_3
  137. Intersect(c, l, 1) → C_3
  138. Intersect(c, m, 1) → D_3
  139. Intersect(c, n, 1) → E_3
  140. Intersect(c, n, 1) → F_3
  141. Intersect(c, p) → G_3, H_3
  142. Intersect(c, p, 1) → I_3
  143. Intersect(c, q, 1) → J_3
  144. Intersect(c, r, 1) → K_3
  145. Intersect(c, s, 1) → L_3
  146. Intersect(c, t) → M_3, N_3
  147. Intersect(c, a, 1) → O_3
  148. Intersect(c, b) → P_3, Q_3
  149. Intersect(c, b, 1) → R_3
  150. Intersect(c, d) → S_3, T_3
  151. Intersect(c, e, 1) → U_3
  152. Intersect(c, f_1, 1) → V_3
  153. Intersect(c, g_1, 1) → W_3
  154. Intersect(c, h_1, 1) → Z_3
  155. Intersect(c, j_1, 1) → A_4
  156. Intersect(c, k_1, 1) → B_4
  157. Intersect(c, l_1, 1) → C_4
  158. Intersect(l_1, m_1) → D_4
  159. Intersect(c, m_1, 1) → E_4
  160. Intersect(c, n_1, 1) → F_4
  161. Intersect(c, p_1, 1) → G_4
  162. Intersect(c, q_1, 1) → H_4
  163. Intersect(c, r_1, 1) → I_4
  164. Intersect(r_1, s_1) → J_4
  165. Intersect(c, s_1) → K_4, L_4
  166. Intersect(c, s_1, 1) → M_4
  167. Intersect(c, t_1, 1) → N_4
  168. Intersect(c, a_1, 1) → O_4
  169. Circle(Z_2, Z) → h_4
  170. Circle(A_3, M) → k_4
  171. Circle(B_3, A_1) → p_4
  172. Circle(C_3, K) → q_4
  173. Circle(D_3, B_1) → r_4
  174. Circle(E_3, L) → s_4
  175. Circle(G_3, C_1) → t_4
  176. Circle(J_3, J) → c_5
  177. Circle(K_3, D_1) → d_5
  178. Circle(L_3, P) → e_5
  179. Circle(M_3, E_1) → f_5
  180. Circle(P_3, F_1) → g_5
  181. Circle(O_3, N) → h_5
  182. Circle(U_3, G_1) → k_5
  183. Circle(S_3, O) → p_5
  184. Circle(V_3, H_1) → q_5
  185. Circle(W_3, W) → r_5
  186. Circle(Z_3, I_1) → s_5
  187. Circle(J_2, U) → t_5
  188. Circle(A_4, J_1) → c_6
  189. Circle(B_4, V) → d_6
  190. Circle(C_4, K_1) → e_6
  191. Circle(E_4, Q) → f_6
  192. Circle(F_4, L_1) → g_6
  193. Circle(G_4, T) → h_6
  194. Circle(H_4, M_1) → k_6
  195. Circle(I_4, R) → p_6
  196. Circle(K_4, N_1) → q_6
  197. Circle(N_4, S) → r_6
  198. Circle(O_4, O_1) → s_6
  199. Circle(C, H) → t_6
  200. Mirror(Z, G) → Z'
  201. Mirror(G, Z') → G'
  202. Mirror(Z', G') → Z''
  203. Mirror(G', Z'') → G''
  204. Mirror(Z'', G'') → Z'''
  205. Mirror(G'', Z''') → G'''
  206. Mirror(Z''', G''') → W_2
  207. Mirror(G''', W_2) → P_4
  208. Mirror(O_1, I) → Q_4
  209. Mirror(I, Q_4) → I'
  210. Mirror(Q_4, I') → R_4
  211. Mirror(I', R_4) → I''
  212. Mirror(R_4, I'') → S_4
  213. Mirror(I'', S_4) → I'''
  214. Mirror(S_4, I''') → T_4
  215. Mirror(I''', T_4) → U_4
  216. OrthogonalLine(P_4, h) → i_2
  217. OrthogonalLine(W_2, h) → j_2
  218. OrthogonalLine(G''', h) → l_2
  219. OrthogonalLine(Z''', h) → m_2
  220. OrthogonalLine(G'', h) → n_2
  221. OrthogonalLine(Z'', h) → a_2
  222. OrthogonalLine(G', h) → b_2
  223. OrthogonalLine(Z', h) → i_3
  224. OrthogonalLine(Q_4, h) → j_3
  225. OrthogonalLine(I', h) → l_3
  226. OrthogonalLine(R_4, h) → m_3
  227. OrthogonalLine(I'', h) → n_3
  228. OrthogonalLine(S_4, h) → a_3
  229. OrthogonalLine(I''', h) → b_3
  230. OrthogonalLine(T_4, h) → i_4
  231. OrthogonalLine(U_4, h) → j_4
  232. Mirror(W_2, P_4) → V_4
  233. Mirror(P_4, V_4) → W_4
  234. Mirror(V_4, W_4) → Z_4
  235. Mirror(W_4, Z_4) → A_5
  236. Mirror(Z_4, A_5) → B_5
  237. Mirror(A_5, B_5) → C_5
  238. Mirror(B_5, C_5) → D_5
  239. Mirror(C_5, D_5) → E_5
  240. Mirror(D_5, E_5) → F_5
  241. Mirror(E_5, F_5) → G_5
  242. Mirror(F_5, G_5) → H_5
  243. Mirror(G_5, H_5) → I_5
  244. OrthogonalLine(V_4, h) → l_4
  245. OrthogonalLine(W_4, h) → m_4
  246. OrthogonalLine(Z_4, h) → n_4
  247. OrthogonalLine(A_5, h) → a_4
  248. OrthogonalLine(B_5, h) → b_4
  249. OrthogonalLine(C_5, h) → i_5
  250. OrthogonalLine(D_5, h) → j_5
  251. OrthogonalLine(E_5, h) → l_5
  252. OrthogonalLine(F_5, h) → m_5
  253. OrthogonalLine(G_5, h) → n_5
  254. OrthogonalLine(H_5, h) → a_5
  255. OrthogonalLine(I_5, h) → b_5
  256. Mirror(T_4, U_4) → J_5
  257. Mirror(U_4, J_5) → K_5
  258. Mirror(J_5, K_5) → L_5
  259. Mirror(K_5, L_5) → M_5
  260. Mirror(L_5, M_5) → N_5
  261. Mirror(M_5, N_5) → O_5
  262. Mirror(N_5, O_5) → P_5
  263. Mirror(O_5, P_5) → Q_5
  264. Mirror(P_5, Q_5) → R_5
  265. Mirror(Q_5, R_5) → S_5
  266. Mirror(R_5, S_5) → T_5
  267. Mirror(S_5, T_5) → U_5
  268. OrthogonalLine(J_5, h) → i_6
  269. OrthogonalLine(K_5, h) → j_6
  270. OrthogonalLine(L_5, h) → l_6
  271. OrthogonalLine(M_5, h) → m_6
  272. OrthogonalLine(N_5, h) → n_6
  273. OrthogonalLine(O_5, h) → a_6
  274. OrthogonalLine(P_5, h) → b_6
  275. OrthogonalLine(Q_5, h) → f_7
  276. OrthogonalLine(R_5, h) → g_7
  277. OrthogonalLine(S_5, h) → h_7
  278. OrthogonalLine(T_5, h) → i_7
  279. OrthogonalLine(U_5, h) → j_7
  280. Intersect(c, b_1, 2) → V_5
  281. Intersect(c, i_3, 1) → W_5
  282. Intersect(c, b_2, 1) → Z_5
  283. Intersect(c, a_2, 1) → A_6
  284. Intersect(c, n_2, 1) → B_6
  285. Intersect(c, l_2, 1) → E_6
  286. Intersect(c, j_2, 1) → F_6
  287. Intersect(c, i_2, 1) → G_6
  288. Intersect(c, m_2, 1) → C_6
  289. Intersect(c, c_1, 2) → D_6
  290. Intersect(c, j_3, 1) → H_6
  291. Intersect(c, l_3, 1) → I_6
  292. Intersect(c, m_3, 1) → J_6
  293. Intersect(c, n_3, 1) → K_6
  294. Intersect(c, a_3, 1) → L_6
  295. Intersect(c, b_3, 1) → M_6
  296. Intersect(c, i_4, 1) → N_6
  297. Intersect(a_3, b_3) → O_6
  298. Intersect(c, j_4, 1) → P_6
  299. Intersect(c, l_4, 1) → Q_6
  300. Intersect(c, m_4, 1) → R_6
  301. Intersect(c, n_4, 1) → S_6
  302. Intersect(c, a_4, 1) → T_6
  303. Intersect(c, b_4, 1) → U_6
  304. Intersect(c, i_5, 1) → V_6
  305. Intersect(c, j_5, 1) → W_6
  306. Intersect(c, l_5, 1) → Z_6
  307. Intersect(c, m_5, 1) → A_7
  308. Intersect(c, n_5, 1) → B_7
  309. Intersect(c, a_5, 1) → C_7
  310. Intersect(c, b_5, 1) → D_7
  311. Intersect(c, i_6, 1) → E_7
  312. Intersect(c, j_6, 1) → F_7
  313. Intersect(c, l_6, 1) → G_7
  314. Intersect(c, m_6, 1) → H_7
  315. Intersect(c, n_6, 1) → I_7
  316. Intersect(c, a_6, 1) → J_7
  317. Intersect(c, b_6, 1) → K_7
  318. Intersect(c, f_7, 1) → L_7
  319. Intersect(c, g_7, 1) → M_7
  320. Intersect(c, h_7, 1) → N_7
  321. Intersect(c, i_7, 1) → O_7
  322. Intersect(c, j_7, 1) → P_7
  323. Intersect(c, i_3, 2) → Q_7
  324. Intersect(c, b_2, 2) → R_7
  325. Intersect(c, a_2, 2) → S_7
  326. Intersect(c, n_2, 2) → T_7
  327. Intersect(c, m_2, 2) → U_7
  328. Intersect(c, l_2, 2) → V_7
  329. Intersect(c, j_2, 2) → W_7
  330. Intersect(c, i_2, 2) → Z_7
  331. Intersect(c, l_4, 2) → A_8
  332. Intersect(c, m_4, 2) → B_8
  333. Intersect(c, n_4, 2) → C_8
  334. Intersect(c, a_4, 2) → D_8
  335. Intersect(c, b_4, 2) → E_8
  336. Intersect(c, i_5, 2) → F_8
  337. Intersect(c, j_5, 2) → G_8
  338. Intersect(c, l_5, 2) → H_8
  339. Intersect(c, m_5, 2) → I_8
  340. Intersect(c, n_5, 2) → J_8
  341. Intersect(c, a_5, 2) → K_8
  342. Intersect(c, b_5, 2) → L_8
  343. Intersect(c, j_3, 2) → M_8
  344. Intersect(c, l_3, 2) → N_8
  345. Intersect(c, m_3, 2) → O_8
  346. Intersect(c, n_3, 2) → P_8
  347. Intersect(c, a_3, 2) → Q_8
  348. Intersect(c, b_3, 2) → R_8
  349. Intersect(c, i_4, 2) → S_8
  350. Intersect(c, j_4, 2) → T_8
  351. Intersect(c, i_6, 2) → U_8
  352. Intersect(c, j_6, 2) → V_8
  353. Intersect(c, l_6, 2) → W_8
  354. Intersect(c, m_6, 2) → Z_8
  355. Intersect(c, n_6, 2) → A_9
  356. Intersect(c, a_6, 2) → B_9
  357. Intersect(c, b_6, 2) → C_9
  358. Intersect(c, f_7, 2) → D_9
  359. Intersect(c, g_7, 2) → E_9
  360. Intersect(c, h_7, 2) → F_9
  361. Intersect(c, i_7, 2) → G_9
  362. Intersect(c, j_7, 2) → H_9
  363. Circle(V_5, G) → c_7
  364. Circle(Q_7, Z') → d_7
  365. Circle(R_7, G') → e_7
  366. Circle(S_7, Z'') → k_7
  367. Circle(T_7, Z'') → p_7
  368. Circle(U_7, Z''') → q_7
  369. Circle(V_7, G'') → r_7
  370. Circle(W_7, G''') → s_7
  371. Circle(Z_7, G''') → t_7
  372. Mirror(g_4, f) → c_8
  373. Mirror(f_4, f) → d_8
  374. Mirror(t_7, f) → e_8
  375. Mirror(s_7, f) → f_8
  376. Mirror(r_7, f) → g_8
  377. Mirror(q_7, f) → h_8
  378. Mirror(p_7, f) → k_8
  379. Mirror(k_7, f) → p_8
  380. Mirror(e_7, f) → q_8
  381. Mirror(d_7, f) → r_8
  382. Mirror(d_7, f) → s_8
  383. Mirror(c_7, f) → t_8