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Spiral Ramp 3D Modeling and Geometric Construction

Spiral Ramp 3D Modeling and Geometric Construction
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2018-01-13 17:24
Solid GeometryParametric EquationGeometric ConstructionGraphing
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
① This example demonstrates a spiral-shaped ascending ramp (Spiralförmiger Aufgang) in 3D, constructed using parametric curves and surfaces to create a helical staircase-like solid structure. ② Core mathematical concepts include spatial curves (CurveCartesian), parametric surfaces (Surface), rays (Ray), and the integration of geometric elements such as centroid, orthocenter, and tangents in three-dimensional space, along with circular/pie chart visualizations. ③ Construction highlights: GeoGebra commands Ray, Surface, and CurveCartesian are used to build the spiral path and surfaces; objects include parameters b, h, r, H for dimensions, Stange (rod), Boden (ground), Glas_1/Glas_2 (glass), and angle variables γ_1 to γ_4; Text1-3 provide annotations. ④ Suitable for high school to university levels in solid geometry, parametric equations, or 3D modeling instruction, and useful for mathematical experiments or artistic spiral structure design.

GeoGebra Commands & Parameters

  1. Ray((0, 0, 0), (0, 0, 1)) → Stange
  2. Surface(((((a * r) * (t + 1)) + (s * b)) * cos(t)), ((((a * r) * (t + 1)) + (s * b)) * sin(t)), (h / ((2 * π)) * t), t, 0, (2 * π), s, 0, 1) → Boden
  3. Surface((((a * r) * (t + 1)) * cos(t)), (((a * r) * (t + 1)) * sin(t)), (h / ((2 * π)) * t) + (s * H), t, 0, (2 * π), s, 0, 1) → Glas_1
  4. Surface(((((a * r) * (t + 1)) + b) * cos(t)), ((((a * r) * (t + 1)) + b) * sin(t)), (h / ((2 * π)) * t) + (s * H), t, 0, (2 * π), s, 0, 1) → Glas_2
  5. CurveCartesian((((a * r) * (t + 1)) * cos(t)), (((a * r) * (t + 1)) * sin(t)), (h / ((2 * π)) * t), t, 0, (2 * π)) → γ_1
  6. CurveCartesian(((((a * r) * (t + 1)) + b) * cos(t)), ((((a * r) * (t + 1)) + b) * sin(t)), (h / ((2 * π)) * t), t, 0, (2 * π)) → γ_2
  7. CurveCartesian((((a * r) * (t + 1)) * cos(t)), (((a * r) * (t + 1)) * sin(t)), H + (h / ((2 * π)) * t), t, 0, (2 * π)) → γ_3
  8. CurveCartesian(((((a * r) * (t + 1)) + b) * cos(t)), ((((a * r) * (t + 1)) + b) * sin(t)), 1 + (h / ((2 * π)) * t), t, 0, (2 * π)) → γ_4