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四面体棱柱的构造与几何探索

四面体棱柱的构造与几何探索
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2018-06-01 04:35
多边形立体几何变换几何作图
该范例演示了如何基于四面体（八面体的一部分）构建棱柱，并展示其面、棱与顶点的空间关系。核心数学概念包括立体几何中的棱柱定义、四面体与八面体的关联、以及通过旋转和点操作生成三维图形。作图要点：使用GeoGebra的Point、Rotate、Octahedron和Prism命令，创建了多个顶点（A、B、C、D、E等）和面（如faceAA'B、faceABE等），并通过旋转生成对称结构，最终形成棱柱及其附加面。适合高中或大学低年级学生用于理解三维几何体的构造方法，也可作为教师演示立体几何中棱柱与多面体关系的教学素材。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Rotate(A, (2 * π / 3)) → A'
  3. Point(Circle[Midpoint[A, A'], (Distance[A, A'] * sqrt(3) / 2), Segment[A, A']]) → B
  4. Octahedron(A, A', B) → a, C, faceBDE, faceCDE, edgeAB, edgeAA', edgeA'B, edgeAE, edgeBE, edgeAC, edgeCE, edgeA'C, D, edgeA'D, edgeCD, edgeBD, edgeDE, E, faceAA'B, faceABE, faceACE, faceAA'C, faceA'CD, faceA'BD
  5. Prism(faceA'BD, g) → b, F, edgeDH, edgeFG, edgeGH, edgeFH, G, H, faceA'BFG, faceA'DHG, faceBDHF, faceFGH, edgeBF, edgeA'G
  6. Prism(faceA'CD, g) → c, I, edgeCK, edgeIJ, edgeJK, edgeIK, J, K, faceA'DIJ, faceA'CKJ, faceCDIK, faceIJK, edgeDI, edgeA'J
  7. Prism(faceAA'B, g) → d, L, edgeA'N, edgeLM, edgeMN, edgeLN, M, N, faceABLM, faceAA'NM, faceA'BLN, faceLMN, edgeBL, edgeAM
  8. Prism(faceAA'C, g) → e, O, edgeCQ, edgeOP, edgePQ, edgeOQ, P, Q, faceAA'OP, faceACQP, faceA'CQO, faceOPQ, edgeA'O, edgeAP
  9. Prism(faceABE, g) → f, R, edgeBT, edgeRS, edgeST, edgeRT, S, T, faceAERS, faceABTS, faceBERT, faceRST, edgeER, edgeAS
  10. Prism(faceACE, g) → h, U, edgeEW, edgeUV, edgeVW, edgeUW, V, W, faceACUV, faceAEWV, faceCEWU, faceUVW, edgeCU, edgeAV
  11. Prism(faceACE, g) → i, Z, edge2, edge3, edge5, edge4, A_1, B_1, face2, face3, face4, face1, edgeCZ, edge1
  12. Prism(faceBDE, g) → j, C_1, edge7, edge9, edge11, edge10, D_1, E_1, face7, face6, face8, face5, edge8, edge6
  13. Prism(faceCDE, g) → k, F_1, edge12, edge15, edge17, edge16, G_1, H_1, face12, face10, face11, face9, edge14, edge13