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True Length of a Segment Visualization

True Length of a Segment Visualization
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2020-05-30 03:22
AngleVectorSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates how to find the true length of a spatial segment by projecting it onto a plane and using orthogonal vectors. The core mathematical concepts involve spatial line segments, orthogonal projection, unit orthogonal vectors, rotation, and length measurement. The construction uses commands such as Segment, Polygon, UnitOrthogonalVector, Angle, and CurveCartesian, along with objects like points A, B, A', B', B'', lines s, s', s'', and auxiliary elements d, e, f, g, C, D, h, i, j, k, polygon Vieleck1, and rotation operations. It is suitable for high school or early college students learning solid geometry and vector projection, helping them understand the difference between true length and projected length of a segment in 3D space, and how to model such problems using GeoGebra.

GeoGebra Commands & Parameters

  1. Segment(A, B) → s
  2. Segment(A', B') → s'
  3. Segment(A'', B'') → s''
  4. Segment(A, A') → d
  5. Segment(B, B') → e
  6. Segment(A, A'') → f
  7. Segment(B, B'') → g
  8. Segment(A', C) → h
  9. Segment(C, A'') → i
  10. Segment(B', D) → j
  11. Segment(D, B'') → k
  12. Polygon(A', B', B, A) → Vieleck1, a', b', b, a_1
  13. UnitOrthogonalVector(Vieleck1) → u
  14. Segment(B', B_0) → a
  15. Segment(A', A_0) → c
  16. Segment(A_0, B_0) → s_0
  17. Angle(A', B', B_0) → α
  18. Angle(A_0, A', B') → β
  19. CurveCartesian(x(B') + ((x(B_0) - x(B')) * sin(t)), y(B') + ((y(B_0) - y(B')) * sin(t)), (z(B) * cos(t)), t, 0, 1.5708) → b_2
  20. CurveCartesian(x(A') + ((x(A_0) - x(A')) * sin(t)), y(A') + ((y(A_0) - y(A')) * sin(t)), (z(A) * cos(t)), t, 0, 1.5708) → b_1
  21. Segment(E, F) → l

AI Drawing Prompt Reference

Draw the true length of a spatial segment AB by projecting it onto a plane and using orthogonal vectors. Start by creating segment AB, then find the orthogonal points A', B', and A''. Draw auxiliary segments d, e, f, and g. Connect A' to C, C to A'', B' to D, and D to B''. Draw quadrilateral A'B'BA and find its orthogonal vector u. Draw segments B' to B_0 and A' to A_0, and segment A_0 to B_0, s_0, and calculate angles α and β. Finally, draw the rotation curves b_1 and b_2 for B' and A', and segment EF.
The prompt above was compiled from the original drawing commands and is for reference only.