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卡瓦利里原理 by Mike May

卡瓦利里原理 by Mike May
几何画板及绘图内容加载中，请稍等……
2015-06-22 04:51
立体几何变换几何作图
所属专题 Connolly Geometry Volume
这个范例通过直观的方式解释了卡瓦利里原理，即两个立体图形如果在每个平行截面上的面积相等，则这两个立体图形的体积相等。核心概念包括立体几何和面积相等。通过使用GeoGebra命令如Point, Intersect, Segment等进行图形构造。适合高中立体几何教学。

GeoGebra 命令及参数

  1. Point(xAxis) → B
  2. Point(yAxis) → U_1
  3. Intersect(xAxis, yAxis) → A
  4. Segment(U_1, A) → b_3
  5. Point(b_3) → M_2
  6. Line(A, B) → a
  7. Vector(A, C) → u
  8. Ray(A, C) → b
  9. Line(D, a) → c
  10. Point(c) → G
  11. Circle(G, F) → d
  12. Circle(F, G) → e
  13. OrthogonalLine(G, c) → f
  14. OrthogonalLine(F, c) → g
  15. Intersect(d, f, 2) → H
  16. Intersect(e, g, 2) → I
  17. Intersect(a, f) → J
  18. Intersect(a, g) → K
  19. Translate(J, u4) → J'
  20. Translate(K, u4) → K'
  21. Segment(H, I) → l
  22. Segment(I, F) → m
  23. Segment(H, G) → n
  24. Segment(G, F) → p
  25. Polygon(J, K, K', J') → poly1, j, k, k', j'
  26. Polygon(G, F, I, H) → poly2, g_1, f_1, i, h
  27. Point(c) → L
  28. Point(c) → M
  29. Polygon(L, M, 3) → poly3, q, r, s, N
  30. Dilate(poly3, sqrt(poly2 / poly3), L) → poly3'
  31. Point(c) → O
  32. Point(c) → P
  33. Polygon(O, P, Q, R, S, T, U) → poly4, o, p_1, q_1, r_1, s_1, t, u_1
  34. OrthogonalLine(T, c) → a_1
  35. OrthogonalLine(U, o) → b_1
  36. OrthogonalLine(S, o) → c_1
  37. OrthogonalLine(R, o) → d_1
  38. OrthogonalLine(Q, c) → e_1
  39. Intersect(a_1, b_1) → C_1
  40. OrthogonalLine(P, c) → n_1
  41. OrthogonalLine(O, c) → t_1
  42. Dilate(O, sqrt(poly2 / poly4), O) → O'
  43. Dilate(P, sqrt(poly2 / poly4), O) → P'
  44. Dilate(Q, sqrt(poly2 / poly4), O) → Q'
  45. Dilate(R, sqrt(poly2 / poly4), O) → R'
  46. Dilate(S, sqrt(poly2 / poly4), O) → S'
  47. Dilate(T, sqrt(poly2 / poly4), O) → T'
  48. Dilate(U, sqrt(poly2 / poly4), O) → U'
  49. Polygon(O', P', Q', R', S', T', U') → poly4', o', p', q', r', s', t', u'
  50. OrthogonalLine(O, c) → h_1
  51. OrthogonalLine(P', c) → i_1
  52. OrthogonalLine(Q', c) → j_1
  53. OrthogonalLine(R', c) → k_1
  54. OrthogonalLine(S', c) → l_1
  55. OrthogonalLine(T', c) → m_1
  56. OrthogonalLine(U', c) → a_2
  57. Intersect(h_1, a) → V
  58. Intersect(i_1, a) → W
  59. Intersect(j_1, a) → Z
  60. Intersect(k_1, a) → A_1
  61. Intersect(l_1, a) → B_1
  62. Intersect(m_1, a) → D_1
  63. Intersect(a_2, a) → E_1
  64. Intersect(c, a_2) → F_1
  65. Segment(U', F_1) → b_2
  66. Intersect(m_1, c) → G_1
  67. Intersect(l_1, c) → H_1
  68. Intersect(k_1, c) → I_1
  69. Intersect(j_1, c) → J_1
  70. Segment(U', F_1) → c_2
  71. Segment(T', G_1) → d_2
  72. Segment(S', H_1) → e_2
  73. Segment(R', I_1) → f_2
  74. Segment(Q', J_1) → g_2
  75. Translate(E_1, ub) → K_1
  76. Translate(D_1, ud) → L_1
  77. Translate(B_1, ue) → M_1
  78. Translate(A_1, uf) → N_1
  79. Translate(Z, ug) → Z'
  80. Polygon(K_1, V, W, Z', N_1, M_1, L_1) → poly5, k_2, v, w, z', n_2, m_2, l_2
  81. Dilate(N, sqrt(poly2 / poly3), L) → N'
  82. Dilate(M, sqrt(poly2 / poly3), L) → M'
  83. OrthogonalLine(L, a) → h_2
  84. OrthogonalLine(M', a) → i_2
  85. OrthogonalLine(N', a) → j_2
  86. Intersect(j_2, a) → O_1
  87. Intersect(i_2, a) → P_1
  88. Intersect(h_2, a) → Q_1
  89. Intersect(c, j_2) → R_1
  90. Segment(N', R_1) → p_2
  91. Translate(O_1, up) → S_1
  92. Polygon(Q_1, P_1, S_1) → poly6, s_2, q_2, p_3
  93. Translate(A, u6) → A'
  94. Translate(E, u6) → E'
  95. Point(a) → T_1
  96. Vector(A, T_1) → v_1
  97. Translate(E, v_1) → E'_1
  98. Translate(E', v_1) → E''
  99. Translate(A', v_1) → A''
  100. Midpoint(E'_1, E'') → V_1
  101. Translate(U_1, u6) → W_1
  102. Midpoint(W_1, U_1) → Z_1
  103. Line(Z_1, a) → r_2
  104. Point(r_2) → A_2
  105. Point(r_2) → B_2
  106. Point(r_2) → C_2
  107. Polygon(E, E'_1, E'', E') → poly7, e_3, e'_1, e'', e'
  108. Midpoint(A, A') → D_2
  109. Segment(Z_1, D_2) → t_2
  110. Midpoint(E, E') → E_2
  111. Segment(Z_1, E_2) → a_3
  112. Dilate(Q_1, ratio, A_2) → F_2
  113. Dilate(P_1, ratio, A_2) → G_2
  114. Dilate(S_1, ratio, A_2) → H_2
  115. Dilate(J, ratio, B_2) → J'_1
  116. Dilate(J', ratio, B_2) → J''
  117. Dilate(K', ratio, B_2) → K''
  118. Dilate(K, ratio, B_2) → K'_1
  119. Dilate(V, ratio, C_2) → V'
  120. Dilate(K_1, ratio, C_2) → I_2
  121. Dilate(L_1, ratio, C_2) → J_2
  122. Dilate(M_1, ratio, C_2) → K_2
  123. Dilate(N_1, ratio, C_2) → L_2
  124. Dilate(Z', ratio, C_2) → Z''
  125. Dilate(W, ratio, C_2) → W'
  126. Polygon(F_2, G_2, H_2) → poly8, h_3, f_3, g_3
  127. Polygon(J'_1, K'_1, K'', J'') → poly9, j'_1, k'_1, k'', j''
  128. Polygon(I_2, V', W', Z'', L_2, K_2, J_2) → poly10, i_3, v', w', z'', l_3, k_3, j_3
  129. Segment(Q_1, A_2) → c_3
  130. Segment(S_1, A_2) → d_3
  131. Segment(P_1, A_2) → m_3
  132. Segment(J, B_2) → n_3
  133. Segment(J', B_2) → q_3
  134. Segment(K', B_2) → r_3
  135. Segment(K, B_2) → s_3
  136. Segment(V, C_2) → t_3
  137. Segment(K_1, C_2) → a_4
  138. Segment(L_1, C_2) → b_4
  139. Segment(M_1, C_2) → c_4
  140. Segment(N_1, C_2) → d_4
  141. Segment(Z', C_2) → e_4
  142. Segment(W, C_2) → f_4
  143. Translate(U_1, v_1) → N_2
  144. Translate(W_1, v_1) → O_2
  145. Polygon(U_1, N_2, O_2, W_1) → poly11, u_2, n_4, o_2, w_1
  146. Polygon(A, T_1, A'', A') → poly12, a_5, t_4, a'', a'
  147. Point(e'') → P_2
  148. Point(e'') → Q_2
  149. Point(e'') → R_2
  150. Vector(S_2, T_2) → v_2
  151. Translate(P_2, v_2) → U_2
  152. Translate(Q_2, v_2) → V_2
  153. Translate(R_2, v_2) → W_2
  154. Vector(A, M_2) → w_2
  155. Translate(Z_2, w_2) → B_3
  156. Translate(A_3, w_2) → C_3
  157. Vector(M_2, U_1) → u_3
  158. Translate(B_3, u_3) → D_3
  159. Translate(C_3, u_3) → E_3
  160. Polygon(Z_2, A_3, A'', A') → poly13, z_2, a_6, a''_1, a'_1
  161. Polygon(B_3, C_3, E'', E') → poly14, b_5, c_5, e''_1, e'_2
  162. Polygon(D_3, E_3, O_2, W_1) → poly15, d_5, e_5, o_1, w_3
  163. Midpoint(b_3) → F_3
  164. Vector(G_3, H_3) → v_3
  165. Translate(F_3, v_3) → I_3