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Cuboid Shadow and Section Construction

Cuboid Shadow and Section Construction
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2019-06-17 20:22
PolygonVectorSolid GeometryTransformationGeometric Construction
This example demonstrates the shadow cast by a cuboid under a light source, visualized by intersecting an inclined plane with the cuboid to form a polygonal shadow region. The core mathematical concepts involve solid geometry, including cuboids, plane-polyhedron intersections, cross-section polygons, and the spatial relationships among points, lines, and planes using vectors. Construction details: using GeoGebra commands such as Point, Cube, Intersect, Plane, PointIn, Polygon, and Vector, the cuboid's vertices and edges are created, an inclined plane is defined, points inside the shadow region are determined, and the shadow polygon is drawn. Suitable for high school or early college students to explore 3D sections and projections, and as a visual aid in teaching solid geometry.

GeoGebra Commands & Parameters

  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