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Geometric Relationship Between Cuboid and 'Yangma'

Geometric Relationship Between Cuboid and 'Yangma'
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2016-05-24 17:39
VectorCoordinate GeometrySolid GeometryGeometric Construction
This example dynamically demonstrates the geometric relationship between a cuboid and its internal 'Yangma' figure, visually showcasing the composition and volume calculation of spatial geometric bodies. It involves core concepts including cuboid, Yangma volume, 3D vectors, rotation and translation transformations, volume calculation, and analytical geometry. Construction highlights utilize GeoGebra commands like PointIn (point inside object check), If (conditional logic), Rotate (3D rotation), and Translate (3D translation), along with objects t, t', P, t1, t2, A, B, C, D, Q, R, S, E, b, c, F, d, a, text1. Suitable for high school and above, for teaching and exploring solid geometry, vectors, spatial transformations, and volume calculations.

GeoGebra Commands & Parameters

  1. PointIn(Polygon[(2, 2, 4), (3, 2, 4), (3, 3, 4), (2, 3, 4)]) → P
  2. If(t' < 1, t', 1) → t1
  3. If(t' > 1, t' - 1, 0) → t2
  4. Rotate(Midpoint[P, D], ((t1 * 90))°, Segment[D, A]) → E
  5. Translate(Rotate[Rotate[Pyramid[Polygon[P, S, D, A], Q], ((t1 * 90))°, Segment[D, A]], ((t2 * 90))°, Segment[E, E + (0, 0, 1)]], Vector[(t2 * Vector[(abs(z(P) / 2 - y(P)), 0, 0)])]) → b
  6. Translate(Pyramid[Polygon[B, C, D, A], Q], Vector[(0, 0)]) → c
  7. Rotate(Midpoint[R, D], ((t1 * 90))°, Segment[C, D]) → F
  8. Translate(Rotate[Rotate[Pyramid[Polygon[R, C, D, S], Q], ((t1 * 90))°, Segment[C, D]], ((t2 * (-90)))°, Segment[F, F + (0, 0, 1)]], Vector[(t2 * Vector[(0, (-abs(x(P) - z(P) / 2)), 0)])]) → d
  9. Translate(Prism[Polygon[(x(P), y(P), 0), ((-x(P)), y(P), 0), ((-x(P)), (-y(P)), 0), (x(P), (-y(P)), 0)], P], Vector[(0, 0)]) → a

AI Drawing Prompt Reference

Draw a cuboid and construct an internal figure named 'Yangma' within it to demonstrate their geometric relationship and volume calculation. Start by defining a face of the cuboid, then use dynamic parameters t and t' to perform rotation and translation transformations to generate the initial shape of Yangma. Further refine its structure through additional rotations and translations, and finally compare Yangma with the cuboid to highlight volume calculations and spatial geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.