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Geometric Modeling of Famous Domes Around the World

Geometric Modeling of Famous Domes Around the World
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2018-12-03 04:17
VectorSolid GeometryTransformationParametric EquationGeometric Construction
This GeoGebra example constructs geometric models of several world-famous domes (e.g., dome-shaped buildings), visually demonstrating their spatial structures and mathematical principles. Core mathematical concepts include the Pythagorean theorem, vector operations, geometric transformations (translation, reflection), parametric curves and surfaces, as well as basic logic and division. Construction highlights: using Vector and Translate for vector translation, Segment for line segments, Mirror for symmetry, CurveCartesian for parametric curves, Sequence for generating sequences, and Surface for 3D surfaces; it also involves numerous point, line, and surface objects (such as O, A, B, C, D, E) with coordinate definitions and relationships. This example is suitable for upper secondary to early university students, helping them understand solid geometry, analytic geometry, and parametric equations in real-world contexts, and can serve as an extension for architectural mathematics or mathematical modeling courses.

GeoGebra Commands & Parameters

  1. Vector(O, Z) → u
  2. Vector(O, X) → w
  3. Vector(O, Y) → v
  4. Translate(O, Vector[(g * u)]) → A_1
  5. Segment(O, A_1) → d
  6. Mirror(A_1, O) → E_1
  7. Segment(E_1, O) → i
  8. CurveCartesian(x(O + ((R * sin(i°)) * u) - ((R * cos(i°)) * v) + (((R * 0) * sin(i°)) * w)), y(O + ((R * sin(i°)) * u) - ((R * cos(i°)) * v) + (((R * 0) * sin(i°)) * w)), i, 0, 360) → h_1
  9. Sequence((x(A) + ((R - f) * cos(i)), y(A) + ((R - f) * sin(i)), z(A)), i, 0, 6.28319, 0.7854) → liste10
  10. Sequence((x(B) + ((R - f) * cos(i)), y(B) + ((R - f) * sin(i)), z(B)), i, 0, 6.28319, 0.7854) → liste11
  11. Sequence(Segment[Element[liste10, i], Element[liste11, i]], i, 1, Length[liste11]) → liste12
  12. Sequence(Segment[Element[liste10, i], Element[liste10, i + 1]], i, 1, Length[liste10]) → liste13
  13. Sequence(Segment[Element[liste11, i], Element[liste11, i + 1]], i, 1, Length[liste11]) → liste14
  14. Sequence(Translate[liste13, Vector[(Vector[A, B] * i)]], i, 0, 1, 0.1) → liste15
  15. CurveCartesian(x(C) + (R / tt * sin(i°)), y(C) - (R / tt * cos(i°)), z(C) + ((R * 0) * sin(i°)), i, 0, 360) → e
  16. Sequence(Translate[Dilate[e, i, D], Vector[((-i) * w / hh)]], i, 0, 0.96, 0.03) → liste16
  17. Sequence(Translate[Dilate[e, i, D], Vector[((-i) * w / hh)]], i, 0, 0.56, 0.03) → liste17
  18. Segment(H, G) → l
  19. Sequence(CurveCartesian[x(O) + ((R * cos(j + 0.05)) * sin(i)^(r)), y(O) + ((R * sin(j + 0.05)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste3
  20. Sequence(CurveCartesian[x(O) + ((R * cos(j)) * sin(i)^(r)), y(O) + ((R * sin(j)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste2
  21. Sequence(CurveCartesian[x(O) + ((R * cos(j - 0.05)) * sin(i)^(r)), y(O) + ((R * sin(j - 0.05)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste4
  22. Sequence(CurveCartesian[x(O) + ((R * cos(j + 0.1)) * sin(i)^(r)), y(O) + ((R * sin(j + 0.1)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste5
  23. Sequence(CurveCartesian[x(O) + ((R * cos(j - 0.1)) * sin(i)^(r)), y(O) + ((R * sin(j - 0.1)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste6
  24. Sequence(CurveCartesian[x(O) + ((R * cos(j + 0.12)) * sin(i)^(r)), y(O) + ((R * sin(j + 0.12)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste7
  25. Sequence(CurveCartesian[x(O) + ((R * cos(j - 0.12)) * sin(i)^(r)), y(O) + ((R * sin(j - 0.12)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / n) → liste8
  26. Sequence(CurveCartesian[x(O) + ((R * cos(j + π)) * sin(i)^(r)), y(O) + ((R * sin(j + π)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), i, tt_1, t - 0.11], j, 0, 6.28319, π / ((4 * n))) → liste9
  27. Sequence(CurveCartesian[x(O) + ((R * cos(j°)) * sin(i)^(r)), y(O) + ((R * sin(j°)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), j, 0, 360], i, tt_1, t, π / ((10 * n))) → liste1
  28. Surface(x(O) + ((R * cos(j°)) * sin(i)^(r)), y(O) + ((R * sin(j°)) * sin(i)^(r)), z(O) + ((R * h) * cos(i)), j, 0, 360, i, tt_1, t) → b
  29. Segment(F, D + (0, 0, 0.1)) → j