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Cube Geometry: Shortest Path Between Antipodal Points

Cube Geometry: Shortest Path Between Antipodal Points
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2015-07-14 02:20
VectorTrigonometrySolid GeometryOptimizationGeometric Construction
In collections Cube Geometry
This example demonstrates the shortest path between antipodal points on a cube's surface, illustrating spherical geometry and symmetry. Key math concepts include minimal paths, geometric constructions, and spatial relationships. Constructed using GeoGebra commands like Point, Cube, Midpoint, Mirror, OrthogonalPlane, and IntersectPath. Ideal for high school solid geometry education and spatial reasoning skills development.

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) → A_1
  6. Intersect
  7. Midpoint(A, G) → O
  8. Mirror(A_1, O) → A_2
  9. OrthogonalPlane(A_1, edgeEF) → c
  10. OrthogonalPlane(A_2, edgeEF) → d
  11. OrthogonalPlane(A_2, edgeBC) → e
  12. OrthogonalPlane(A_1, edgeFG) → f
  13. Intersect(edgeEF, c) → I
  14. Intersect(edgeAB, c) → J
  15. IntersectPath(e, c) → g
  16. Intersect(g, faceABCD) → K
  17. Segment(A_1, I) → h
  18. Segment(I, J) → i
  19. Segment(J, K) → j
  20. Segment(K, A_2) → k
  21. Intersect(edgeEF, d) → L
  22. Intersect(edgeAB, d) → M
  23. Vector(I, L) → u
  24. Vector(J, M) → v
  25. Segment(A_1, N) → l
  26. Segment(N, P) → m
  27. Segment(P, A_2) → n