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Cube Geometry: Shortest Path Between Points - A Net Copy

Cube Geometry: Shortest Path Between Points - A Net Copy
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2018-06-26 13:34
PolygonSolid GeometryTransformationOptimizationGeometric Construction
① This applet demonstrates the shortest path between two points on adjacent faces of a cube, shown simultaneously on the 3D cube and its net (unfolded surface), making the correspondence between the 3D path and its 2D straight-line representation clear. ② Core mathematical concepts include surface shortest paths in solid geometry, cube nets, symmetry, and the principle that the shortest path on a surface becomes a straight line when unfolded. ③ Construction details: The cube is created using the Cube command; points are placed on faces with Point and PointIn; intersections with edges are computed via Intersect and IntersectPath; path segments are drawn with Segment; faces are constructed using Polygon and Plane; the net is generated as a copy to visualize the path. ④ Suitable for high school to early university students learning solid geometry, optimization of paths on surfaces, and the use of nets to solve 3D problems.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → A
  2. Point(xAxis) → 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, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. PointIn(faceEFGH) → I
  6. PointIn(faceABFE) → J
  7. OrthogonalPlane(J, edgeEF) → b
  8. OrthogonalPlane(I, edgeEF) → c
  9. Intersect(edgeEF, b) → K
  10. Intersect(edgeEF, c) → L
  11. OrthogonalPlane(J, edgeBF) → d
  12. IntersectPath(c, d) → e
  13. OrthogonalPlane(I, edgeFG) → f_1
  14. IntersectPath(f_1, d) → g
  15. Intersect(e, faceABFE) → M
  16. Segment(J, M) → h
  17. Segment(M, L) → i
  18. Segment(L, I) → j
  19. Vector(L, K) → u
  20. Segment(J, N) → k
  21. Segment(N, I) → l
  22. Polygon(B, A, 4, xOyPlane) → poly1, m, n, p, q, f, P
  23. Polygon(P, f, 4, xOyPlane) → poly2, r, s, t, a_1, O, Q
  24. Polygon(A, D, 4, xOyPlane) → poly3, b_1, c_1, d_1, e_1, R, S
  25. Polygon(C, B, 4, xOyPlane) → poly4, g_1, h_1, i_1, j_1, T, U
  26. Polygon(D, C, 4, xOyPlane) → poly5, k_1, l_1, m_1, n_1, V, W
  27. Circle(edgeAB, J) → q_1
  28. Intersect
  29. Intersect(xOyPlane, q_1) → C_1, D_1
  30. Circle(edgeEF, I) → p_1
  31. Plane(A, E, F) → o
  32. Intersect(o, p_1) → Z, A_1
  33. Circle(edgeAB, A_1) → r_1
  34. Intersect(xOyPlane, r_1) → B_1, E_1
  35. Segment(D_1, E_1) → s_1