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Cross Section of a Rectangular Pyramid by Bill MacIntosh

Cross Section of a Rectangular Pyramid by Bill MacIntosh
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2014-12-17 00:51
PolygonVectorTrigonometrySolid GeometryTransformation
This GeoGebra example dynamically demonstrates the cross-section of a rectangular pyramid as a cutting plane moves through it. It illustrates how the shape of the section changes with the plane's orientation and position. Core mathematical concepts include solid geometry, cross-sections, similarity transformations, vector operations, and trigonometric functions used for spatial projections. The construction uses commands such as Vector, Dilate, Polygon, Translate, PointIn, Segment, and Sequence to build the pyramid and generate the section polygon, with sliders controlling the cutting plane. Suitable for high school or early college students learning about 3D geometry and cross-sections, and useful for interactive classroom demonstrations.

GeoGebra Commands & Parameters

  1. Vector(O, Z) → w
  2. Vector(O, Y) → v
  3. Vector(O, X) → u
  4. Dilate(D, 0.6, O') → D'_3
  5. Dilate(C, 0.6, O') → C'_3
  6. Dilate(B, 0.6, O') → B'_3
  7. Dilate(A, 0.6, O') → A'_3
  8. Polygon(A, B, C, D) → poly1, a, b_1, c_1, d_1
  9. Translate(A, Vector[(hauteur * u)]) → A'
  10. Translate(B, Vector[(hauteur * u)]) → B'
  11. Translate(C, Vector[(hauteur * u)]) → C'
  12. Translate(D, Vector[(hauteur * u)]) → D'
  13. Polygon(A', B', C', D') → poly1', a', b', c', d'
  14. PointIn(poly1') → S
  15. Translate(B, Vector[(((1 - hauteur3) * hauteur) * u)]) → B'_2
  16. Translate(C, Vector[(((1 - hauteur3) * hauteur) * u)]) → C'_2
  17. Translate(D, Vector[(((1 - hauteur3) * hauteur) * u)]) → D'_2
  18. Translate(A, Vector[(((1 - hauteur3) * hauteur) * u)]) → A'_2
  19. Polygon(A'_2, B'_2, C'_2, D'_2) → poly1'', a'_2, b'_2, c'_2, d'_2
  20. Translate(A, Vector[((1 - hauteur3) * u)]) → A'_1
  21. Translate(B, Vector[((1 - hauteur3) * u)]) → B'_1
  22. Translate(C, Vector[((1 - hauteur3) * u)]) → C'_1
  23. Translate(D, Vector[((1 - hauteur3) * u)]) → D'_1
  24. Translate(S, Vector[((-hauteur) * u)]) → S'
  25. Segment(S', S) → j
  26. Sequence(O' + ((b * v) * cos(((k * 360 / Sommets))° + i°)) + ((b * w) * sin(((k * 360 / Sommets))° + i°)), k, 0, Sommets) → liste1
  27. Sequence(Segment[Element[liste1, l], S], l, 0, Sommets) → liste4
  28. Polygon(liste1) → poly2
  29. Dilate(poly2, hauteur3, S) → poly2'
  30. Sequence(Polygon[Element[liste1, k], Element[liste1, k + 1], S], k, 0, Sommets) → liste9

AI Drawing Prompt Reference

Draw a rectangular pyramid with a base rectangle and an apex translated vertically above it. Inside the pyramid, place a movable cutting plane that intersects the pyramid to produce a cross-section polygon whose shape changes with the plane's position and orientation. Also, include a smaller similar base near the top, connected by segments to the apex, forming an inner framework. Use sliders to control the plane's height and tilt, dynamically showing how the section shape varies.
The prompt above was compiled from the original drawing commands and is for reference only.