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平面间夹角

平面间夹角
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2019-01-30 09:02
向量立体几何几何作图
①本范例演示了两个平面之间夹角的计算与可视化；②涉及的核心数学概念包括向量叉积、空间几何、角度计算；③作图要点提炼自构造摘要，使用了Polygon（多边形）、Vector（向量）、Line（直线）、Angle（角度）等GeoGebra命令与对象；④适合高中及以上学段学习立体几何、向量与空间角的应用，可用于教学演示与几何探究。

GeoGebra 命令及参数

  1. Polygon(Control1, Control2, 4) → poly2, i, j, k, l, R000, S000
  2. PointIn(poly2) → ControlR
  3. Polygon(Control1, Control2, 4) → controlpoly, a_1, b_1, d, e, Control3, Control4
  4. Vector(O000, K1) → k1
  5. Vector(O000, J1) → j1
  6. Vector(O000, I1) → i1
  7. Segment(A, B) → L01
  8. Segment(B, C) → L03
  9. Segment(C, D) → L05
  10. Segment(D, A) → L07
  11. Segment(A, E) → L02
  12. Segment(B, F) → L04
  13. Segment(F, G) → L10
  14. Segment(G, H) → L11
  15. Segment(E, H) → L12
  16. Segment(E, F) → L09
  17. Segment(C, G) → L06
  18. Segment(D, H) → L08
  19. Textfield(TTT10) → inputBox1
  20. Textfield(TTT20) → inputBox2
  21. TextToUnicode(TTT10) → TTT11
  22. Object(UnicodeToLetter[Element[TTT11, 1]]) → PP11
  23. Object(UnicodeToLetter[Element[TTT11, 2]]) → PP12
  24. TextToUnicode(TTT20) → TTT21
  25. Object(UnicodeToLetter[Element[TTT21, 1]]) → PP21
  26. Object(UnicodeToLetter[Element[TTT21, 2]]) → PP22
  27. Object(UnicodeToLetter[Element[TTT21, 3]]) → PP23
  28. Object(UnicodeToLetter[Element[TTT21, 4]]) → PP24
  29. Segment(PP11, PP12) → LL11_1
  30. Polygon(PP21, PP22, PP23, PP24) → plane11, pp21, pp22, pp23, pp24
  31. Intersect(pp22, pp23) → test13
  32. Intersect(pp21, pp24) → test14
  33. Intersect(pp21, pp23) → test11
  34. Intersect(pp22, pp24) → test12
  35. Polygon(PP21, PP22, PP23) → plane12, pp23_1, pp21_1, pp22_1
  36. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT10], 1]])) → FFF11
  37. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT10], 2]])) → FFF12
  38. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT20], 1]])) → FFF21
  39. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT20], 2]])) → FFF22
  40. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT20], 3]])) → FFF23
  41. Element((Invert[{{Element[FFF22, 1, 1] - Element[FFF21, 1, 1], Element[FFF23, 1, 1] - Element[FFF21, 1, 1], Element[FFF11, 1, 1] - Element[FFF12, 1, 1]}, {Element[FFF22, 2, 1] - Element[FFF21, 2, 1], Element[FFF23, 2, 1] - Element[FFF21, 2, 1], Element[FFF11, 2, 1] - Element[FFF12, 2, 1]}, {Element[FFF22, 3, 1] - Element[FFF21, 3, 1], Element[FFF23, 3, 1] - Element[FFF21, 3, 1], Element[FFF11, 3, 1] - Element[FFF12, 3, 1]}}] * (FFF11 - FFF21)), 3, 1) → r
  42. Segment(PPP1, PPP2) → LL12_1
  43. Segment(PPP1, PPP3) → LL13
  44. Segment(PPP3, PP12) → LL11
  45. Segment(PP11, PPP2) → LL12
  46. Line(PPPP11, LL12_1) → lll11
  47. Line(PPPP12, LL12) → lll12
  48. Intersect(lll12, lll11) → I000
  49. Segment(I000, PPPP11) → lllll12
  50. Segment(I000, PPPP12) → lllll11
  51. Line(PPPP21, LL13) → lll21
  52. Line(PPPP22, LL11) → lll22
  53. Intersect(lll22, lll21) → PPPPP2
  54. Segment(PPPP22, PPPPP2) → lllll21
  55. Segment(PPPP21, PPPPP2) → lllll22
  56. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT20], 4]])) → FFF24
  57. If(FFFFF1 ≟ 0, FFFF12, FFFF13) → FFFFFF2
  58. If(FFFFFFF2 > 90°, FFFFFFF2 - 90°, 90° - FFFFFFF2) → FFFFFFF3
  59. Polygon(Control1, Control2, 4) → poly1, c, f, g, h, P000, Q000
  60. Textfield(TTT30) → inputBox3
  61. TextToUnicode(TTT30) → TTT31
  62. Object(UnicodeToLetter[Element[TTT31, 1]]) → PP31
  63. Object(UnicodeToLetter[Element[TTT31, 2]]) → PP32
  64. Object(UnicodeToLetter[Element[TTT31, 3]]) → PP33
  65. Object(UnicodeToLetter[Element[TTT31, 4]]) → PP34
  66. Object(UnicodeToLetter[Element[TTT41, 1]]) → PPP41
  67. Object(UnicodeToLetter[Element[TTT41, 2]]) → PPP42
  68. Object(UnicodeToLetter[Element[TTT51, 1]]) → PPP51
  69. Object(UnicodeToLetter[Element[TTT51, 2]]) → PPP52
  70. Intersection(TTT21, TTT31) → TTT61
  71. Object(UnicodeToLetter[Element[TTT61, 1]]) → PPP61
  72. Object(UnicodeToLetter[Element[TTT61, 2]]) → PPP62
  73. Object("F" + (UnicodeToLetter[Element[TTT41, 1]])) → FFF41
  74. Object("F" + (UnicodeToLetter[Element[TTT41, 2]])) → FFF42
  75. Object("F" + (UnicodeToLetter[Element[TTT51, 1]])) → FFF51
  76. Object("F" + (UnicodeToLetter[Element[TTT51, 2]])) → FFF52
  77. Object("F" + (UnicodeToLetter[Element[TTT61, 1]])) → FFF61
  78. Object("F" + (UnicodeToLetter[Element[TTT61, 2]])) → FFF62
  79. Polygon(PP31, PP32, PP33, PP34) → poly3, pp31, pp32, pp33, pp34
  80. Segment(PPP61, PPP62) → line6162
  81. Length(TTT21) → length21
  82. Length(TTT31) → length31
  83. If(length21 ≤ length31, TTT21, TTT31) → TTT71
  84. Object("F" + (UnicodeToLetter[Element[TTT71, 1]])) → FFF71
  85. Object("F" + (UnicodeToLetter[Element[TTT71, 2]])) → FFF72
  86. Polygon(PP31, PP32, PP33) → poly4, m, n, p
  87. Angle(PPP41, HHH41, PPP51) → line41
  88. Angle(PPP42, HHH42, PPP52) → line52
  89. Angle(PPP42, HHH42, PPP51) → line42
  90. Angle(PPP41, HHH41, PPP52) → line51
  91. Angle(PPP41, HHH41, PPP51) → lline41
  92. Angle(PPP42, HHH42, PPP51) → lline42
  93. Angle(PPP41, HHH41, PPP52) → lline51
  94. Unique(RemoveUndefined[{GGG41, GGG42, GGG51, GGG52}]) → aaaaaaaaaaaaaaaaaaaaaaaaaa_3
  95. Frequency(RemoveUndefined[{GGG41, GGG42, GGG51, GGG52}]) → aaaaaaaaaaaaaaaaaaaaaaaaaa_2
  96. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT30], 1]])) → FFF31
  97. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT30], 2]])) → FFF32
  98. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT30], 3]])) → FFF33
  99. Object("F" + (UnicodeToLetter[Element[TextToUnicode[TTT30], 4]])) → FFF34
  100. Determinant({{Element[FFF31, 2, 1] - Element[FFF32, 2, 1], Element[FFF31, 3, 1] - Element[FFF32, 3, 1]}, {Element[FFF32, 2, 1] - Element[FFF33, 2, 1], Element[FFF32, 3, 1] - Element[FFF33, 3, 1]}}) → aangle21
  101. Determinant({{Element[FFF21, 2, 1] - Element[FFF22, 2, 1], Element[FFF21, 3, 1] - Element[FFF22, 3, 1]}, {Element[FFF22, 2, 1] - Element[FFF23, 2, 1], Element[FFF22, 3, 1] - Element[FFF23, 3, 1]}}) → aangle11
  102. Determinant({{Element[FFF21, 1, 1] - Element[FFF22, 1, 1], Element[FFF21, 2, 1] - Element[FFF22, 2, 1]}, {Element[FFF22, 1, 1] - Element[FFF23, 1, 1], Element[FFF22, 2, 1] - Element[FFF23, 2, 1]}}) → aangle13
  103. Determinant({{Element[FFF31, 1, 1] - Element[FFF32, 1, 1], Element[FFF31, 2, 1] - Element[FFF32, 2, 1]}, {Element[FFF32, 1, 1] - Element[FFF33, 1, 1], Element[FFF32, 2, 1] - Element[FFF33, 2, 1]}}) → aangle23
  104. Segment(PPP41, HHH41) → angle411
  105. Segment(HHH41, PPP51) → angle412
  106. Segment(PPP42, HHH42) → angle421
  107. Segment(HHH42, PPP51) → angle422
  108. Segment(PPP41, HHH41) → angle511
  109. Segment(HHH41, PPP52) → angle512
  110. Segment(PPP42, HHH42) → angle521
  111. Segment(HHH42, PPP52) → angle522
  112. Segment(PPP41, HHH41) → anglea411
  113. Segment(HHH41, PPP51) → anglea412
  114. Segment(PPP42, HHH42) → anglea421
  115. Segment(HHH42, PPP51) → anglea422
  116. Segment(PPP41, HHH41) → anglea511
  117. Segment(HHH41, PPP52) → anglea512
  118. If(length21 ≤ length31, TTT31, TTT21) → TTT81
  119. Object("F" + (UnicodeToLetter[Element[TTT71, 3]])) → FFF73
  120. Object("F" + (UnicodeToLetter[Element[TTT71, 4]])) → FFF74
  121. Object("F" + (UnicodeToLetter[Element[TTT81, 1]])) → FFF81
  122. Object("F" + (UnicodeToLetter[Element[TTT81, 2]])) → FFF82
  123. Object("F" + (UnicodeToLetter[Element[TTT81, 3]])) → FFF83
  124. Object("F" + (UnicodeToLetter[Element[TTT81, 4]])) → FFF84
  125. Element((Invert[{{Element[FFF82, 1, 1] - Element[FFF81, 1, 1], Element[FFF83, 1, 1] - Element[FFF81, 1, 1], Element[FFF71, 1, 1] - Element[FFF72, 1, 1]}, {Element[FFF82, 2, 1] - Element[FFF81, 2, 1], Element[FFF83, 2, 1] - Element[FFF81, 2, 1], Element[FFF71, 2, 1] - Element[FFF72, 2, 1]}, {Element[FFF82, 3, 1] - Element[FFF81, 3, 1], Element[FFF83, 3, 1] - Element[FFF81, 3, 1], Element[FFF71, 3, 1] - Element[FFF72, 3, 1]}}] * (FFF71 - FFF81)), 3, 1) → r712
  126. Element((Invert[{{Element[FFF82, 1, 1] - Element[FFF81, 1, 1], Element[FFF83, 1, 1] - Element[FFF81, 1, 1], Element[FFF72, 1, 1] - Element[FFF73, 1, 1]}, {Element[FFF82, 2, 1] - Element[FFF81, 2, 1], Element[FFF83, 2, 1] - Element[FFF81, 2, 1], Element[FFF72, 2, 1] - Element[FFF73, 2, 1]}, {Element[FFF82, 3, 1] - Element[FFF81, 3, 1], Element[FFF83, 3, 1] - Element[FFF81, 3, 1], Element[FFF72, 3, 1] - Element[FFF73, 3, 1]}}] * (FFF72 - FFF81)), 3, 1) → r723
  127. Element((Invert[{{Element[FFF82, 1, 1] - Element[FFF81, 1, 1], Element[FFF83, 1, 1] - Element[FFF81, 1, 1], Element[FFF73, 1, 1] - Element[FFF74, 1, 1]}, {Element[FFF82, 2, 1] - Element[FFF81, 2, 1], Element[FFF83, 2, 1] - Element[FFF81, 2, 1], Element[FFF73, 2, 1] - Element[FFF74, 2, 1]}, {Element[FFF82, 3, 1] - Element[FFF81, 3, 1], Element[FFF83, 3, 1] - Element[FFF81, 3, 1], Element[FFF73, 3, 1] - Element[FFF74, 3, 1]}}] * (FFF73 - FFF81)), 3, 1) → r734
  128. Element((Invert[{{Element[FFF82, 1, 1] - Element[FFF81, 1, 1], Element[FFF83, 1, 1] - Element[FFF81, 1, 1], Element[FFF74, 1, 1] - Element[FFF71, 1, 1]}, {Element[FFF82, 2, 1] - Element[FFF81, 2, 1], Element[FFF83, 2, 1] - Element[FFF81, 2, 1], Element[FFF74, 2, 1] - Element[FFF71, 2, 1]}, {Element[FFF82, 3, 1] - Element[FFF81, 3, 1], Element[FFF83, 3, 1] - Element[FFF81, 3, 1], Element[FFF74, 3, 1] - Element[FFF71, 3, 1]}}] * (FFF74 - FFF81)), 3, 1) → r741
  129. Segment(B, G) → LLL111
  130. Segment(C, F) → LLL112
  131. Segment(A, H) → LLL121
  132. Segment(D, E) → LLL122
  133. Intersect(LLL111, LLL112) → LLL131
  134. Intersect(LLL121, LLL122) → LLL132
  135. Angle(B, LLL131, C) → LLL141
  136. Angle(F, LLL131, G) → LLL142
  137. Angle(B, LLL131, F) → LLL143
  138. Angle(C, LLL131, G) → LLL144
  139. Angle(A, LLL132, D) → LLL151
  140. Angle(E, LLL132, H) → LLL152
  141. Angle(A, LLL132, E) → LLL153
  142. Angle(D, LLL132, H) → LLL154
  143. Segment(A, F) → LLL211
  144. Segment(B, E) → LLL212
  145. Segment(D, G) → LLL221
  146. Segment(C, H) → LLL222
  147. Intersect(LLL211, LLL212) → LLL231
  148. Intersect(LLL221, LLL222) → LLL232
  149. Angle(A, LLL231, B) → LLL241
  150. Angle(E, LLL231, F) → LLL242
  151. Angle(E, LLL231, A) → LLL243
  152. Angle(B, LLL231, F) → LLL244
  153. Angle(D, LLL232, C) → LLL251
  154. Angle(G, LLL232, H) → LLL252
  155. Angle(D, LLL232, H) → LLL253
  156. Angle(C, LLL232, G) → LLL254
  157. Segment(A, C) → LLL311
  158. Segment(B, D) → LLL312
  159. Segment(E, G) → LLL321
  160. Segment(F, H) → LLL322
  161. Intersect(LLL311, LLL312) → LLL331
  162. Intersect(LLL321, LLL322) → LLL332
  163. Angle(A, LLL331, B) → LLL341
  164. Angle(C, LLL331, D) → LLL342
  165. Angle(A, LLL331, D) → LLL343
  166. Angle(B, LLL331, C) → LLL344
  167. Angle(E, LLL332, F) → LLL351
  168. Angle(G, LLL332, H) → LLL352
  169. Angle(E, LLL332, H) → LLL353
  170. Angle(F, LLL332, G) → LLL354
  171. Intersect(pp31, pp33) → test21
  172. Intersect(pp32, pp34) → test22
  173. Intersect(pp32, pp33) → test23
  174. Intersect(pp31, pp34) → test24
  175. Segment(LLL131, LLL132) → LLL6
  176. Segment(LLL231, LLL232) → LLL62
  177. Segment(LLL331, LLL332) → LLL63

AI 绘图提示语参考

绘制两个相交的平面，其中一个平面由点F11和F12定义，另一个平面由点F21、F22和F23定义。使用向量交叉积计算两个平面之间的夹角，并通过点I000找到两个平面的交线。首先创建四个点F11、F12、F21、F22、F23，然后创建两个平面，接着计算平面夹角，最后找到交线并标记交点I000。
以上提示语为根据原始绘图命令整理而来，仅供参考。