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三维简单问题1

三维简单问题1
几何画板及绘图内容加载中，请稍等……
2014-08-12 07:18
三角函数立体几何最值优化几何作图
所属专题 Graphs of power functions 2 Graphs of power functions
本范例演示了三维空间中金字塔的几何构造与性质分析。涉及的核心数学概念包括三角函数、角度计算、优化问题、质心或重心、立体几何等。作图要点使用了GeoGebra的Circle（圆）、Point（点）、Segment（线段）、Line（直线）、Angle（角度）、Polygon（多边形）、PointIn（点在对象上）、Intersect（交点）、OrthogonalLine（垂线）、Vector（向量）、Translate（平移）、Midpoint（中点）等命令和对象。适合高中及以上学生用于学习立体几何、向量与优化问题的实际应用。

GeoGebra 命令及参数

  1. Circle(O, Radius) → axiscircle
  2. Point(axiscircle) → Y
  3. Segment(O, Y) → OX
  4. Point(axiscircle) → Z
  5. Segment(O, Z) → OZ
  6. Line(A, OZ) → yaxis
  7. Point(yaxis) → edgeYPt
  8. Segment(A, edgeYPt) → edgeY
  9. Point(axiscircle) → X
  10. Angle(X, O, Z) → α
  11. Segment(O, X) → OY
  12. Line(A, OY) → CParallelCB
  13. Point(CParallelCB) → edgeXPt
  14. Polygon(edgeYPt, A, edgeXPt) → MaxBase, MaxBaseEdge2, MaxBaseEdge1, MaxBaseEdge3
  15. PointIn(MaxBase) → C
  16. Line(A, C) → MaxBaseDivider
  17. Intersect(MaxBaseEdge3, MaxBaseDivider) → MaxBaseDividerPt
  18. Polygon(MaxBaseDividerPt, A, edgeXPt) → MaxBaseLowerHalf, MaxBaseLowerHalfEdge2, MaxBaseLowerHalfEdge1, MaxBaseLowerHalfEdge3
  19. PointIn(MaxBaseLowerHalf) → B
  20. Polygon(A, B, C) → Base, AB, BC, AC
  21. PointIn(Base) → K
  22. Line(K, OX) → VerticalFromBase
  23. Point(VerticalFromBase) → VerticalFromBasePt
  24. Segment(K, VerticalFromBasePt) → MaxHeight
  25. Point(MaxHeight) → D
  26. Line(K, OZ) → KParallelZ
  27. Line(K, OY) → KParallelX
  28. Intersect(KParallelX, edgeY) → K_Z
  29. Line(C, OY) → CParallelX
  30. Intersect(CParallelX, edgeY) → C_Z
  31. Line(C, OZ) → CParallelZ
  32. Segment(A, edgeXPt) → edgeX
  33. Intersect(KParallelZ, edgeX) → K_X
  34. Circle(A_P, LengthDA / Scale) → CircleAtA_P
  35. Intersect(CParallelZ, edgeX) → C_X
  36. Circle(C_P, LengthDC / Scale) → CircleAtC_P
  37. Line(B, OY) → BParallelX
  38. Line(B, OZ) → BParallelZ
  39. Intersect(BParallelZ, edgeX) → B_X
  40. Intersect(BParallelX, edgeY) → B_Z
  41. Circle(C_Q, LengthDC / Scale) → CircleAtC_Q
  42. Circle(B_Q, LengthDB / Scale) → CircleAtB_Q
  43. OrthogonalLine(K, AC) → KPerpendicularAC
  44. Intersect(KPerpendicularAC, AC) → KonAC
  45. Point(AB) → M
  46. Line(M, OY) → MParallelX
  47. Line(M, OZ) → MParallelZ
  48. Intersect(MParallelZ, edgeX) → M_X
  49. Intersect(MParallelX, edgeY) → M_Z
  50. Point(OX) → RightAngleDistance
  51. Vector(O, RightAngleDistance) → u
  52. Translate(K, u) → DKA3
  53. Line(A, OX) → zaxis
  54. Point(zaxis) → edgeZPt
  55. Segment(A, edgeZPt) → edgeZ
  56. Segment(K, D) → KD
  57. Segment(A, D) → DA
  58. Segment(D, C) → DC
  59. Segment(D, B) → DB
  60. Segment(B_Q, C_Q) → BC_Q
  61. Intersect(CircleAtB_Q, CircleAtC_Q) → D_Q, D_{Q'}, ,
  62. Segment(C_Q, D_Q) → DC_Q
  63. Segment(D_Q, B_Q) → DB_Q
  64. Segment(D_S, M_S) → DM_S
  65. Segment(D_S, K_S) → DK_S
  66. Segment(C_R, A_R) → AC_R
  67. Segment(A_R, B_R) → AB_R
  68. Segment(B_R, C_R) → BC_R
  69. Segment(K_R, A_R) → KA_R
  70. Segment(K_R, B_R) → KB_R
  71. Segment(K_R, C_R) → KC_R
  72. Segment(K, M) → MK
  73. Segment(D, M) → DM
  74. Segment(K_R, M_R) → KM_R
  75. Circle(K, KVertical) → CircleAtK
  76. Intersect(CircleAtK, MK) → KHorizontal1,
  77. Line(KHorizontal1, KD) → KPerpendicularMK
  78. Line(KVertical, MK) → KParallelMK
  79. Intersect(KParallelMK, KPerpendicularMK) → KAngle1
  80. Segment(KHorizontal1, KAngle1) → KAngleVertical1
  81. Segment(KAngle1, KVertical) → KAngleHorizontal1
  82. Line(KVertical, AC) → KParallelAC
  83. Intersect(CircleAtK, AC) → KHorizontal2, KHorizontal3
  84. Line(KHorizontal3, KD) → KParallelDK2
  85. Line(KHorizontal2, KD) → KParallelDK1
  86. Intersect(KParallelDK2, KParallelAC) → KAngle3
  87. Intersect(KParallelDK1, KParallelAC) → KAngle2
  88. Segment(KHorizontal3, KAngle3) → KAngleVertical3
  89. Segment(KAngle3, KVertical) → KAngleHorizontal3
  90. Segment(KVertical, KAngle2) → KAngleHorizontal2
  91. Segment(KAngle2, KHorizontal2) → KAngleVertical2
  92. Intersect(CircleAtK, BC) → KHorizontal4, KHorizontal5
  93. Line(KHorizontal4, KD) → KParallelDK3
  94. Line(KHorizontal5, KD) → KParallelDK4
  95. Line(KVertical, BC) → KParallelBC
  96. Intersect(KParallelDK3, KParallelBC) → KAngle4
  97. Intersect(KParallelDK4, KParallelBC) → KAngle5
  98. Segment(KVertical, KAngle4) → KAngleHorizontal4
  99. Segment(KHorizontal4, KAngle4) → KAngleVertical4
  100. Segment(KVertical, KAngle5) → KAngleHorizontal5
  101. Segment(KAngle5, KHorizontal5) → KAngleVertical5
  102. Segment(A_{R2}, A_{R3}) → A_RAngleHorizontal
  103. Segment(A_{R1}, A_{R3}) → A_RAngleVertical
  104. Segment(B_{R3}, B_{R2}) → B_RAngleHorizontal
  105. Segment(B_{R1}, B_{R2}) → B_RAngleVertical
  106. Circle(B, BAnglePoint1) → CircleAtB
  107. Intersect(CircleAtB, BC) → BAnglePoint2,
  108. Line(BAnglePoint2, edgeX) → BParallelX1
  109. Line(BAnglePoint1, BC) → BParallelZ1
  110. Intersect(BParallelZ1, BParallelX1) → BAnglePoint3
  111. Segment(BAnglePoint1, BAnglePoint3) → BAngleLine1
  112. Segment(BAnglePoint3, BAnglePoint2) → BAngleLine2
  113. Circle(C, 0.5) → CircleAtC
  114. Intersect(CircleAtC, AC) → CAnglePoint2,
  115. Line(CAnglePoint2, BC) → CParallelBC
  116. Circle(C_R, 0.5) → CircleAtC_T
  117. Intersect(CircleAtC_T, AC_R) → C_{R1},
  118. Intersect(CircleAtC_T, BC_R, 1) → C_{R2}
  119. Line(C_{R1}, BC_R) → C_TParallelBC
  120. Line(C_{R2}, AC_R) → C_TParallelAC
  121. Intersect(C_TParallelBC, C_TParallelAC) → C_{R3}
  122. Segment(C_{R3}, C_{R1}) → C_RAngle1
  123. Segment(C_{R3}, C_{R2}) → C_RAngle2
  124. Intersect(CircleAtC, BC) → CAnglePoint1,
  125. Line(CAnglePoint1, AC) → CParallelAC
  126. Intersect(CParallelAC, CParallelBC) → CAnglePoint3
  127. Segment(CAnglePoint2, CAnglePoint3) → CAngleLine1
  128. Segment(CAnglePoint3, CAnglePoint1) → CAngleLine2
  129. Circle(A, 0.5) → CircleAtA
  130. Intersect(CircleAtA, AC) → AAnglePoint1,
  131. Intersect(CircleAtA, AB) → AAnglePoint2,
  132. Line(AAnglePoint1, AB) → AParallelAB
  133. Line(AAnglePoint2, AC) → AParallelAC
  134. Intersect(AParallelAC, AParallelAB) → AAnglePoint3
  135. Segment(AAnglePoint2, AAnglePoint3) → AAngleLine1
  136. Segment(AAnglePoint3, AAnglePoint1) → AAngleLine2
  137. Circle(M_R, 0.5) → CircleAtM_T
  138. Intersect(CircleAtM_T, AB_R) → M_RAnglePoint1, M_RAnglePoint2
  139. Intersect(CircleAtM_T, KM_R) → M_RAnglePoint3,
  140. Line(M_RAnglePoint1, KM_R) → M_TParallelMK2
  141. Line(M_RAnglePoint3, AB_R) → M_TParallelAB
  142. Line(M_RAnglePoint2, KM_R) → M_TParallelMK1
  143. Intersect(M_TParallelMK2, M_TParallelAB) → M_RAnglePoint4
  144. Intersect(M_TParallelMK1, M_TParallelAB) → M_RAnglePoint5
  145. Segment(M_RAnglePoint2, M_RAnglePoint5) → M_RAngleLine1
  146. Segment(M_RAnglePoint5, M_RAnglePoint3) → M_RAngleLine2
  147. Segment(M_RAnglePoint1, M_RAnglePoint4) → M_RAngleLine3
  148. Segment(M_RAnglePoint4, M_RAnglePoint3) → M_RAngleLine4
  149. Intersect(CircleAtC_P, CircleAtA_P) → D_P, D_{P'}, ,
  150. Segment(D_P, A_P) → DA_P
  151. Segment(D_P, C_P) → DC_P
  152. Segment(A_P, C_P) → AC_P
  153. Midpoint(A, C) → MidPointAC
  154. Midpoint(A_R, C_R) → MidPointAC_R
  155. Midpoint(A, B) → MidPointAB
  156. Midpoint(A_R, B_R) → MidPointAB_R
  157. Midpoint(B, C) → MidPointBC
  158. Midpoint(B_R, C_R) → MidPointBC_R
  159. Midpoint(D_Q, C_Q) → MidPointDC_Q
  160. Midpoint(D, C) → MidPointDC
  161. Midpoint(D, B) → MidPointDB
  162. Segment(M_S, K_S) → MK_S
  163. Midpoint(B_Q, C_Q) → MidPointBC_Q
  164. Midpoint(B_Q, D_Q) → MidPointDB_Q
  165. Midpoint(M, K) → MidPointMK
  166. Midpoint(D, K) → MidPointDK
  167. Midpoint(M, D) → MidPointDM
  168. Midpoint(M_S, K_S) → MidPointMK_S
  169. Midpoint(D_S, M_S) → MidPointDM_S
  170. Midpoint(D_S, K_S) → MidPointDK_S
  171. Midpoint(A, D) → MidPointDA
  172. Midpoint(A_P, C_P) → MidPointAC_P
  173. Midpoint(D_P, A_P) → MidPointAD_P
  174. Midpoint(D_P, C_P) → MidPointCD_P
  175. Angle(D_S, K_S, M_S) → DKM_S
  176. Angle(D_P, A_P, C_P) → DAC_P
  177. Angle(A_P, C_P, D_P) → DCA_P
  178. Angle(D_Q, C_Q, B_Q) → DCB_Q
  179. Angle(C_Q, B_Q, D_Q) → DBC_Q

AI 绘图提示语参考

在三维空间中绘制一个金字塔，首先画一个圆并标记圆上的点，连接圆心与这些点形成边，并计算角度。接着，以圆心为顶点，连接不同点形成金字塔的底面，并找到底面的重心。然后，从重心向底面的一边作垂线，确定金字塔的高度。最后，在金字塔的各顶点处绘制圆，并连接相应的点形成侧面，完成金字塔的绘制。
以上提示语为根据原始绘图命令整理而来，仅供参考。