用立方体搭建的多面体截面与多项式几何展示

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2020-08-15 03:57

该范例展示如何通过多个立方体单元“搭建”出三维多面体，并进一步观察其面与棱对应的几何结构（便于联想到多项式/分段函数的构造思想）。核心数学概念包括：立体几何中的棱与面、由多边形形成的棱柱结构、以及用坐标/几何关系理解图形交叠与组合。作图要点：利用 GeoGebra 的 Polygon 与 Prism 命令生成底面多边形并拉伸形成棱柱，同时由 A_2,B_2,C_2,D_2 与对应的 poly1_2、a_2,b_2,c_2,d_2 构成基元，再生成 E_2,F_2,G_2,H_2 以及面对象 faceABFE_2、faceBCGF_2、faceCDHG_2、faceADHE_2、faceEFGH_2；同理在另一组坐标 A_3,B_3,C_3,D_3、poly1_3 与 e_3,f_3,g_3 等基础上建立相应的棱柱，并显式给出各条边 edgeAE_2、edgeBF_2、edgeCG_2、edgeDH_2、edgeEF_2、edgeFG_2、edgeGH_2、edgeEH_2，用于突出“搭建”结构的连接关系。适合中学高年级到大学基础几何课的课堂演示、几何建模训练与图形理解型学习用途。

GeoGebra 命令及参数

Polygon(A_2, B_2, C_2, D_2) → poly1_2, a_2, b_2, c_2, d_2
Prism(poly1_2, E_2) → e_2, F_2, edgeBF_2, edgeCG_2, edgeDH_2, edgeEF_2, edgeFG_2, edgeGH_2, edgeEH_2, G_2, H_2, faceABFE_2, faceBCGF_2, faceCDHG_2, faceADHE_2, faceEFGH_2, edgeAE_2
Polygon(A_3, B_3, C_3, D_3) → poly1_3, a_3, b_3, c_3, d_3
Prism(poly1_3, E_3) → e_3, F_3, edgeBF_3, edgeCG_3, edgeDH_3, edgeEF_3, edgeFG_3, edgeGH_3, edgeEH_3, G_3, H_3, faceABFE_3, faceBCGF_3, faceCDHG_3, faceADHE_3, faceEFGH_3, edgeAE_3
Polygon(A_4, B_4, C_4, D_4) → poly1_4, a_4, b_4, c_4, d_4
Prism(poly1_4, E_4) → e_4, F_4, edgeBF_4, edgeCG_4, edgeDH_4, edgeEF_4, edgeFG_4, edgeGH_4, edgeEH_4, G_4, H_4, faceABFE_4, faceBCGF_4, faceCDHG_4, faceADHE_4, faceEFGH_4, edgeAE_4
Polygon(A_5, B_5, C_5, D_5) → poly1_5, a_5, b_5, c_5, d_5
Prism(poly1_5, E_5) → e_5, F_5, edgeBF_5, edgeCG_5, edgeDH_5, edgeEF_5, edgeFG_5, edgeGH_5, edgeEH_5, G_5, H_5, faceABFE_5, faceBCGF_5, faceCDHG_5, faceADHE_5, faceEFGH_5, edgeAE_5
Polygon(A_6, B_6, C_6, D_6) → poly1_6, a_6, b_6, c_6, d_6
Prism(poly1_6, E_6) → e_6, F_6, edgeBF_6, edgeCG_6, edgeDH_6, edgeEF_6, edgeFG_6, edgeGH_6, edgeEH_6, G_6, H_6, faceABFE_6, faceBCGF_6, faceCDHG_6, faceADHE_6, faceEFGH_6, edgeAE_6
Polygon(A_7, B_7, C_7, D_7) → poly1_7, a_7, b_7, c_7, d_7
Prism(poly1_7, E_7) → e_7, F_7, edgeBF_7, edgeCG_7, edgeDH_7, edgeEF_7, edgeFG_7, edgeGH_7, edgeEH_7, G_7, H_7, faceABFE_7, faceBCGF_7, faceCDHG_7, faceADHE_7, faceEFGH_7, edgeAE_7
Polygon(A_8, B_8, C_8, D_8) → poly1_8, a_8, b_8, c_8, d_8
Prism(poly1_8, E_8) → e_8, F_8, edgeBF_8, edgeCG_8, edgeDH_8, edgeEF_8, edgeFG_8, edgeGH_8, edgeEH_8, G_8, H_8, faceABFE_8, faceBCGF_8, faceCDHG_8, faceADHE_8, faceEFGH_8, edgeAE_8
Polygon(A_9, B_9, C_9, D_9) → poly1_9, a_9, b_9, c_9, d_9
Prism(poly1_9, E_9) → e_9, F_9, edgeBF_9, edgeCG_9, edgeDH_9, edgeEF_9, edgeFG_9, edgeGH_9, edgeEH_9, G_9, H_9, faceABFE_9, faceBCGF_9, faceCDHG_9, faceADHE_9, faceEFGH_9, edgeAE_9
Polygon(A_{10}, B_{10}, C_{10}, D_{10}) → poly1_{10}, a_{10}, b_{10}, c_{10}, d_{10}
Prism(poly1_{10}, E_{10}) → e_{10}, F_{10}, edgeBF_{10}, edgeCG_{10}, edgeDH_{10}, edgeEF_{10}, edgeFG_{10}, edgeGH_{10}, edgeEH_{10}, G_{10}, H_{10}, faceABFE_{10}, faceBCGF_{10}, faceCDHG_{10}, faceADHE_{10}, faceEFGH_{10}, edgeAE_{10}
Polygon(A_{11}, B_{11}, C_{11}, D_{11}) → poly1_{11}, a_{11}, b_{11}, c_{11}, d_{11}
Prism(poly1_{11}, E_{11}) → e_{11}, F_{11}, edgeBF_{11}, edgeCG_{11}, edgeDH_{11}, edgeEF_{11}, edgeFG_{11}, edgeGH_{11}, edgeEH_{11}, G_{11}, H_{11}, faceABFE_{11}, faceBCGF_{11}, faceCDHG_{11}, faceADHE_{11}, faceEFGH_{11}, edgeAE_{11}
Polygon(A_1, B_1, C_1, D_1) → poly1_1, a_1, b_1, c_1, d_1
Prism(poly1_1, E_1) → e_1, F_1, edgeBF_1, edgeCG_1, edgeDH_1, edgeEF_1, edgeFG_1, edgeGH_1, edgeEH_1, G_1, H_1, faceABFE_1, faceBCGF_1, faceCDHG_1, faceADHE_1, faceEFGH_1, edgeAE_1
Polygon(A_{12}, B_{12}, C_{12}, D_{12}) → poly1_{12}, a_{12}, b_{12}, c_{12}, d_{12}
Prism(poly1_{12}, E_{12}) → e_{12}, F_{12}, edgeBF_{12}, edgeCG_{12}, edgeDH_{12}, edgeEF_{12}, edgeFG_{12}, edgeGH_{12}, edgeEH_{12}, G_{12}, H_{12}, faceABFE_{12}, faceBCGF_{12}, faceCDHG_{12}, faceADHE_{12}, faceEFGH_{12}, edgeAE_{12}
Polygon(E, B, C, D) → poly1, a, b, c, d
Polygon(A_{13}, B_{13}, C_{13}, D_{13}) → poly1_{13}, a_{13}, b_{13}, c_{13}, d_{13}
Prism(poly1_{13}, E_{13}) → e_{13}, F_{13}, edgeBF_{13}, edgeCG_{13}, edgeDH_{13}, edgeEF_{13}, edgeFG_{13}, edgeGH_{13}, edgeEH_{13}, G_{13}, H_{13}, faceABFE_{13}, faceBCGF_{13}, faceCDHG_{13}, faceADHE_{13}, faceEFGH_{13}, edgeAE_{13}
Polygon(A_{14}, B_{14}, C_{14}, D_{14}) → poly1_{14}, a_{14}, b_{14}, c_{14}, d_{14}
Prism(poly1_{14}, E_{14}) → e_{14}, F_{14}, edgeBF_{14}, edgeCG_{14}, edgeDH_{14}, edgeEF_{14}, edgeFG_{14}, edgeGH_{14}, edgeEH_{14}, G_{14}, H_{14}, faceABFE_{14}, faceBCGF_{14}, faceCDHG_{14}, faceADHE_{14}, faceEFGH_{14}, edgeAE_{14}
Polygon(A_{15}, B_{15}, C_{15}, D_{15}) → poly1_{15}, a_{15}, b_{15}, c_{15}, d_{15}
Prism(poly1_{15}, E_{15}) → e_{15}, F_{15}, edgeBF_{15}, edgeCG_{15}, edgeDH_{15}, edgeEF_{15}, edgeFG_{15}, edgeGH_{15}, edgeEH_{15}, G_{15}, H_{15}, faceABFE_{15}, faceBCGF_{15}, faceCDHG_{15}, faceADHE_{15}, faceEFGH_{15}, edgeAE_{15}
Polygon(A_{16}, B_{16}, C_{16}, D_{16}) → poly1_{16}, a_{16}, b_{16}, c_{16}, d_{16}
Prism(poly1_{16}, E_{16}) → e_{16}, F_{16}, edgeBF_{16}, edgeCG_{16}, edgeDH_{16}, edgeEF_{16}, edgeFG_{16}, edgeGH_{16}, edgeEH_{16}, G_{16}, H_{16}, faceABFE_{16}, faceBCGF_{16}, faceCDHG_{16}, faceADHE_{16}, faceEFGH_{16}, edgeAE_{16}
Polygon(E_{25}, B_{25}, C_{25}, D_{25}) → poly1_{25}, a_{25}, b_{25}, c_{25}, d_{25}
Polygon(E_{38}, B_{38}, C_{38}, D_{38}) → poly1_{38}, a_{38}, b_{38}, c_{38}, d_{38}

AI 绘图提示语参考

在三维空间中绘制一个“由立方体式单元拼接的棱柱结构”。先画多个四边形底面，分别在每组底面上取四个顶点并构成四边形面。随后把每个四边形沿同一方向拉伸成棱柱，生成对应的上层四个顶点以及连接上下层的边与侧面。对每个棱柱突出显示其四个侧面：连接底面相邻顶点所形成的面（如前后两侧各两块侧面），并同时强调底面到顶面的竖向边与顶层轮廓边。整体呈现若干个重复的棱柱块相互连接，强调面、边的邻接关系与“搭积木”的视觉效果。

以上提示语为根据原始绘图命令整理而来，仅供参考。