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长方体阴影与截面构造

长方体阴影与截面构造
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2019-06-17 20:22
多边形向量立体几何变换几何作图
该范例演示了一个三维长方体在光照下产生的阴影效果，通过构造一个倾斜平面与长方体相交，形成阴影区域的多边形截面。核心数学概念包括立体几何中的长方体、平面与多面体的交线、截面多边形，以及向量和点、线、面的位置关系。作图要点：使用GeoGebra的Point、Cube命令创建长方体的顶点和棱，利用Intersect和Plane命令构造一个倾斜平面，再通过PointIn和Polygon命令确定阴影区域内的点并绘制阴影多边形，最后用Vector命令辅助表示方向。适合高中或大学低年级学生用于理解三维空间中的截面与投影，也可作为立体几何教学中的可视化演示工具。

GeoGebra 命令及参数

  1. Point(zAxis) → A
  2. Point(zAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD_1, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(Circle[C, Distance[D, C], Segment[D, C]]) → I
  6. Cube(D, C, I) → b, J, faceIJNM, faceKLMN, edgeDJ, edgeCD, edgeCI, edgeIJ_1, edgeDK, edgeJN, edgeKN, edgeKL, K, edgeCL_1, edgeLM, edgeIM, edgeMN, L, M, N, faceCDJI, faceDJNK, faceCDKL, faceCIML
  7. Point(Circle[I, Distance[J, I], Segment[J, I]]) → O
  8. Cube(J, I, O) → c, P, faceOPTS, faceQRST, edgeJP, edgeIJ, edgeIO, edgeOP, edgeJQ, edgePT, edgeQT, edgeQR, Q, edgeIR, edgeRS, edgeOS, edgeST, R, S, T, faceIJPO, faceJPTQ, faceIJQR, faceIOSR
  9. Point(Circle[L, Distance[C, L], Segment[C, L]]) → U
  10. Cube(C, L, U) → d, V, face3, face4, edgeCV, edgeCL, edgeLU, edgeUV, edgeCW, edge2, edge3, edgeWZ, W, edgeLZ, edge4, edge1, edge5, Z, A_1, B_1, faceCLUV, face1, faceCLZW, face2
  11. Intersect(yAxis, zAxis) → C_1
  12. Point(yAxis) → D_1
  13. Point(xAxis) → E_1
  14. Plane(C_1, D_1, E_1) → e
  15. Intersect(zAxis, xAxis) → F_1
  16. Point(yAxis) → G_1
  17. PointIn(e) → H_1
  18. Point(xAxis) → I_1
  19. Point(xAxis) → K_1
  20. PointIn(e) → L_1
  21. PointIn(e) → M_1
  22. Point(xAxis) → N_1
  23. PointIn(e) → O_1
  24. Polygon(N_1, K_1, L_1, O_1) → q3, n_1, k_2, l_2, o_1
  25. Intersect(n_1, k_2) → J_1
  26. Point(xAxis) → P_1
  27. Polygon(J_1, P_1, Q_1, R_1) → q1, j_1, p_1, q_1, r_1
  28. Point(yAxis) → S_1
  29. Intersect(xAxis, yAxis) → T_1
  30. Polygon(Q_1, S_1, T_1, P_1) → q2, q_2, s_1, t_1, p_2
  31. Vector(U_1, V_1) → u