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Hyperbolic Paraboloid Model

Hyperbolic Paraboloid Model
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2020-05-22 17:06
FunctionTransformationGeometric Construction
This example demonstrates the geometric construction of a hyperbolic paraboloid. Key mathematical concepts include function graphs, transformations, and geometric construction. Using commands such as Segment, Intersect, Point, CircleArc, Line, Mirror, Rotate, Polygon, Sequence, Midpoint, Distance, and Plane, a hyperbolic paraboloid model is constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(D, A) → i
  4. Segment(C, D) → h
  5. Intersect(i, yAxis) → E
  6. Intersect(f, xAxis) → F
  7. Intersect(g, yAxis) → G
  8. Point(xAxis) → H
  9. CircleArc(G, C, I) → c
  10. CircleArc(G, I, B) → d
  11. Point(c) → J
  12. Point(c) → K
  13. Point(c) → L
  14. Point(c) → M
  15. Point(c) → N
  16. Line(J, g) → j
  17. Line(K, g) → k
  18. Line(L, g) → l
  19. Line(M, g) → m
  20. Line(N, g) → n
  21. Mirror(I_1, vgl1) → N_1
  22. Mirror(J_1, vgl1) → O_1
  23. Mirror(K_1, vgl1) → P_1
  24. Mirror(L_1, vgl1) → Q_1
  25. Mirror(M_1, vgl1) → R_1
  26. Rotate(I_1, 90°, zAxis) → S_1
  27. Rotate(J_1, 90°, zAxis) → P
  28. Rotate(L_1, 90°, zAxis) → Q
  29. Rotate(M_1, 90°, zAxis) → R
  30. Rotate(I_1, (-(90°)), zAxis) → S
  31. Line(M_1, h) → p
  32. Line(L_1, h) → q
  33. Line(K_1, h) → r
  34. Line(J_1, h) → s
  35. Line(I_1, h) → t
  36. Line(H_1, h) → a
  37. Line(R, i) → b
  38. Line(Q, i) → e
  39. Line(P, i) → f_1
  40. Line(S_1, i) → g_1
  41. Intersect(g_1, t) → T
  42. Intersect(f_1, s) → U
  43. Intersect(r, e) → V
  44. Intersect(p, b) → W
  45. Rotate(K_1, 90°, zAxis) → Z
  46. Line(Z, i) → h_1
  47. Intersect(r, h_1) → A_1
  48. Intersect(q, e) → T_1
  49. Mirror(T, vgl1) → T'
  50. Mirror(U, vgl1) → U'
  51. Mirror(A_1, vgl1) → U_1
  52. Mirror(T_1, vgl1) → V_1
  53. Mirror(W, vgl1) → W'
  54. Polygon(D, A, W', W) → vierhoek1, d_1, a_1, w', w
  55. Polygon(W, W', V_1, T_1) → vierhoek2, w_1, i_1, v_1, t_1
  56. Polygon(T_1, V_1, U_1, A_1) → vierhoek3, j_1, k_1, u_1, l_1
  57. Polygon(A_1, U_1, U', U) → vierhoek4, m_1, n_1, u', u
  58. Polygon(U, U', T', T) → vierhoek5, u_2, p_1, t', t_2
  59. Polygon(T, T', H_1) → driehoek1, q_1, t_3, r_1
  60. Segment(T, U) → b_1
  61. Segment(A_1, T_1) → e_1
  62. Segment(T_1, W) → f_2
  63. Segment(W, D) → g_2
  64. Segment(U', U_1) → j_2
  65. Segment(U_1, V_1) → k_2
  66. Segment(V_1, W') → l_2
  67. Segment(W', A) → m_2
  68. Sequence(Rotate[vierhoek1, (nn * 90°), zAxis], nn, 1, 4) → l1
  69. Sequence(Rotate[vierhoek2, (nn * 90°), zAxis], nn, 1, 4) → l2
  70. Sequence(Rotate[vierhoek3, (nn * 90°), zAxis], nn, 1, 4) → l3
  71. Sequence(Rotate[vierhoek4, (nn * 90°), zAxis], nn, 1, 4) → l4
  72. Sequence(Rotate[driehoek1, (nn * 90°), zAxis], nn, 1, 4) → l6
  73. Midpoint(H, E) → W_1
  74. Line(W_1, zAxis) → n_2
  75. Distance(W_1, E) → horstraal
  76. Plane(B, D, H_1) → diag
  77. Point(g) → Z_1
  78. Mirror(Z_1, yAxis) → A_2
  79. Mirror(Z_1, O) → B_2
  80. Mirror(A_2, O) → C_2
  81. Segment(B, D) → p_2
  82. Mirror(Z_1, p_2) → D_2
  83. Mirror(A_2, p_2) → E_2
  84. Mirror(D_2, O) → F_2
  85. Mirror(E_2, O) → G_2
  86. Segment(D_2, Z_1) → q_2
  87. Segment(E_2, C_2) → r_2
  88. Segment(B_2, F_2) → s_2
  89. Segment(A_2, G_2) → a_2
  90. Plane(F_2, B_2, B_2 + (0, 0, 1)) → p_3
  91. Intersect(w_1, p_3) → H_2
  92. Intersect(j_1, p_3) → I_2
  93. Intersect(m_1, p_3) → J_2
  94. Intersect(u_2, p_3) → K_2
  95. Intersect(t_2, p_3) → L_2
  96. Mirror(K_2, diag) → M_2
  97. Mirror(J_2, diag) → N_2
  98. Mirror(I_2, diag) → O_2
  99. Mirror(H_2, diag) → P_2
  100. Segment(L_2, K_2) → b_2
  101. Segment(K_2, J_2) → c_2
  102. Segment(J_2, I_2) → d_2
  103. Segment(I_2, H_2) → e_2
  104. Segment(H_2, B_2) → f_3
  105. Segment(M_2, L_2) → g_3
  106. Segment(M_2, N_2) → h_3
  107. Segment(N_2, O_2) → i_3
  108. Segment(O_2, P_2) → j_3
  109. Segment(P_2, F_2) → k_3
  110. Sequence(Rotate[l13, (nn * 90°), zAxis], nn, 1, 3) → l14
  111. Point(d_2) → Q_2
  112. Rotate(Q_2, 90°, zAxis) → R_2
  113. Midpoint(Q_2, R_2) → S_2
  114. Mirror(Q_2, diag) → W_2
  115. Line(W_2, yAxis) → l_3
  116. Intersect(l_3, e_1) → A_3
  117. Line(A_3, xAxis) → m_3
  118. Intersect(m_3, k_2) → C_3
  119. Polygon(T_1, V_1, C_3, A_3) → vierhoek6, n_3, q_3, c_3, a_3
  120. Sequence(Rotate[vierhoek6, (nn * 90°), zAxis], nn, 1, 3) → l15
  121. Polygon(A_3, C_3, U_1, A_1) → vierhoek7, r_3, s_3, b_3, d_3
  122. Sequence(Rotate[vierhoek7, (nn * 90°), zAxis], nn, 1, 3) → l16
  123. Rotate(L_2, 90°, zAxis) → D_3
  124. Polygon(U, U', D_3, L_2) → vierhoek8, u_3, e_3, f_4, g_4
  125. Sequence(Rotate[vierhoek8, (nn * 90°), zAxis], nn, 1, 3) → l5
  126. Mirror(K_2, diag) → E_3
  127. Mirror(J_2, diag) → F_3
  128. Polygon(Q_2, U_2, Z_2, W_2, F_3, E_3, L_2, K_2, J_2) → veelhoek1, h_4, i_4, z_2, w_2, j_4, k_4, l_4, m_4, n_4
  129. Sequence(Rotate[veelhoek1, (nn * 90°), zAxis], nn, 1, 3) → l17
  130. Polygon(A_3, Q_2, J_2, A_1) → vierhoek9, p_4, q_4, r_4, s_4
  131. Sequence(Rotate[vierhoek9, (nn * 90°), zAxis], nn, 1, 3) → l18
  132. Mirror(vierhoek9, diag) → vierhoek9'
  133. Sequence(Rotate[vierhoek9', (nn * 90°), zAxis], nn, 1, 3) → l19
  134. Polygon(A_1, J_2, L_2) → driehoek2, t_4, a_4, b_4
  135. Segment(L_2, A_1) → c_1
  136. Mirror(driehoek2, diag) → driehoek2'
  137. Midpoint(Z_2, U_2) → G_3
  138. Segment(D_3, U_1) → s_1
  139. Sequence(Rotate[l7, (nn * 90°), zAxis], nn, 2, 3) → l8
  140. Mirror(G_3, zAxis) → H_3
  141. Segment(G_3, H_3) → h_2
  142. Mirror(U_2, zAxis) → I_3
  143. Segment(I_3, U_2) → i_2
  144. Mirror(Z_2, J_3) → K_3
  145. Rotate(K_3, 90°, i_2) → L_3
  146. CircleArc(J_3, K_3, L_3) → c_4
  147. CircleArc(J_3, L_3, Z_2) → d_4
  148. Point(c_4) → M_3
  149. Point(c_4) → N_3
  150. Point(c_4) → O_3
  151. Point(c_4) → P_3
  152. Point(c_4) → Q_3
  153. Rotate(S_3, 90°, zAxis) → Z_3
  154. Rotate(T_3, 90°, zAxis) → A_4
  155. Rotate(U_3, 90°, zAxis) → B_4
  156. Rotate(V_3, 90°, zAxis) → C_4
  157. Rotate(W_3, 90°, zAxis) → D_4
  158. Polygon(R_3, S_3, Z_3) → driehoek3, z_3, e_4, f_5
  159. Polygon(W_3, D_4, C_4) → driehoek4, t_5, w_3, a_5
  160. Polygon(W_3, V_3, C_4) → driehoek5, b_5, c_5, v_3
  161. Polygon(W_3, I_3, K_3) → driehoek6, d_5, e_5, f_6
  162. Polygon(W_3, I_3, D_4) → driehoek7, g_6, h_6, i_6
  163. Polygon(U_3, V_3, C_4) → driehoek8, j_6, k_6, l_6
  164. Polygon(U_3, B_4, C_4) → driehoek9, m_6, n_6, p_6
  165. Polygon(T_3, U_3, B_4) → driehoek10, q_6, r_6, s_6
  166. Polygon(T_3, B_4, A_4) → driehoek11, t_6, a_6, b_6
  167. Polygon(T_3, S_3, A_4) → driehoek12, c_6, d_6, e_6
  168. Polygon(S_3, Z_3, A_4) → driehoek13, f_7, g_7, h_7
  169. Sequence(Rotate[l9, (nn * 90°), zAxis], nn, 1, 3) → l10
  170. Segment(I_3, D_4) → g_5
  171. Segment(D_4, C_4) → h_5
  172. Segment(C_4, B_4) → i_5
  173. Segment(B_4, A_4) → j_5
  174. Segment(A_4, Z_3) → k_5
  175. Segment(Z_3, R_3) → l_5
  176. Sequence(Rotate[l11, (nn * 90°), zAxis], nn, 1, 3) → l12
  177. Intersect(f, i) → E_4
  178. Intersect(g, h) → F_4
  179. Midpoint(r_2) → G_4
  180. Midpoint(a_2) → H_4
  181. Midpoint(s_2) → I_4
  182. Midpoint(q_2) → J_4
  183. Intersect(i, h) → K_4
  184. Intersect(f, g) → L_4
  185. Intersect(q_2, r_2) → M_4
  186. Intersect(s_2, a_2) → N_4
  187. Intersect(r_2, s_2) → O_4
  188. Intersect(q_2, a_2) → P_4
  189. Segment(E_4, G_4) → m_5
  190. Segment(K_4, I_4) → n_5
  191. Segment(H_4, F_4) → p_5
  192. Segment(J_4, L_4) → q_5
  193. Segment(O_4, Q_4) → r_5
  194. Segment(Q_4, N_4) → s_5
  195. Segment(Q_4, P_4) → i_7
  196. Segment(Q_4, M_4) → j_7
  197. Segment(A_1, U) → k_7
  198. Segment(U, L_2) → l_7
  199. Polygon(A_1, J_2, K_2, U) → vierhoek10, m_7, n_7, p_7, u_4
  200. Polygon(U, K_2, L_2) → driehoek14, q_7, u_5, r_7
  201. Mirror(driehoek14, diag) → driehoek14'
  202. Mirror(vierhoek10, diag) → vierhoek10'
  203. Sequence(Rotate[driehoek14, (nn * 90°), zAxis], nn, 1, 3) → l20
  204. Sequence(Rotate[driehoek14', (nn * 90°), zAxis], nn, 1, 3) → l21
  205. Sequence(Rotate[vierhoek10, (nn * 90°), zAxis], nn, 1, 3) → l22
  206. Sequence(Rotate[vierhoek10', (nn * 90°), zAxis], nn, 1, 3) → l23