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Tulip Surface Exploration with Geometry and Probability

Tulip Surface Exploration with Geometry and Probability
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2020-08-28 15:07
TriangleProbability & StatisticsTransformationParametric EquationGeometric Construction
In collections Surfaces and flowers
This GeoGebra example constructs a tulip‑shaped surface by combining a parametric curve, a surface of revolution, and geometric transformations. It illustrates concepts such as equilateral triangles, conditional probability, continuity, square symmetry, and reflection. The construction uses the Sequence command to generate parameter lists, CurveCartesian to define the base curve, Surface to create the 3D shape, Rotate for rotational symmetry, and Segment for outline edges. It is intended for high‑school and undergraduate students studying visualization of surfaces, parametric equations, and the interplay between geometry and probability.

GeoGebra Commands & Parameters

  1. Sequence(Surface[(e * sin(v)), (((1 + (c * (-1)^(i))) * cos((i * π / n) + (u * g(v)))) * f(v)), (((1 + (c * (-1)^(i))) * sin((i * π / n) + (u * g(v)))) * f(v)), u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → liste1
  2. Sequence(Surface[(e * sin(v)), (((1 + (c * (-1)^(i))) * cos((i * π / n) + (u * g(v)))) * f(v)), (((1 + (c * (-1)^(i))) * sin((i * π / n) + (u * g(v)))) * f(v)), u, ((-1.01 * π)) / n, (1.01 * π / n), v, -1.5708, 1.5708], i, 1, (2 * n)) → l1
  3. CurveCartesian(sin(t) - 3, sin(t)^(3) + 1, (1 / 2 * cos(t)^(3)), t, (-π), π) → a
  4. Surface((2 * sin(t)) - 3, (2 / 1.25 * sin(t)^(3)) + 1.5, ((v * 1 / 1.5) * cos(t)^(3)), t, (-π), π, v, -1, 1) → i
  5. Rotate(i, 80°, xAxis) → i'
  6. Surface((3 * sin(t)) - 2.5, (-2 / 1.45 * sin(t)^(3)) - 1.35, ((v * 1 / 1.5) * cos(t)^(3)), t, (-π), π, v, -1, 1) → b
  7. Segment(O, O - (4.5, 0, 0)) → h
  8. Rotate(i, 260°, xAxis) → i'_1
  9. Surface((e * sin(v)) + 0.1, (((c + 1) * cos(u)) * cos(v)), (((c + 1) * sin(u)) * cos(v)), u, -3.14159, π, v, -1.5708, -0.75) → j
  10. Surface((e * sin(v)) - 0.051, (((c + 1) * cos(u)) * cos(v)), (((c + 1) * sin(u)) * cos(v)), u, -3.14159, π, v, -1.5708, -0.75) → e_1

AI Drawing Prompt Reference

Create a tulip-shaped surface with equilateral triangles, probability, transformations, and geometric constructions. Start by drawing a parametric curve, then rotate it around the x-axis to form a 3D shape. Use the sequence command to generate multiple rotationally symmetric surfaces, and finally add outline edges.
The prompt above was compiled from the original drawing commands and is for reference only.