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The Flight of Insects: Conchospirals on a Cone

The Flight of Insects: Conchospirals on a Cone
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2016-01-17 11:25
SequenceSolid GeometryParametric EquationSlider AnimationLocus
In collections Logarithmic Spiral and Fibonacci Numbers
This GeoGebra example demonstrates the trajectory of insects flying towards a light source at night, modeled as a conchospiral winding on a cone. The core mathematical concepts include parametric curves, exponential functions, solid geometry (specifically the geometry of a cone), and optimization problems. To construct this model, GeoGebra commands such as `Cone` are used to define the spatial surface, while `CurveCartesian` and `Sequence` generate the spiral path based on parameters like theta, alpha, and beta. The `Intersect` command is applied to find key points along the trajectory. This interactive tool is suitable for high school or university students studying calculus, coordinate geometry, and mathematical modeling, providing a visual representation of complex spatial curves.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → A
  2. Cone(C, A, rr) → c, d, e
  3. CurveCartesian(((r * ℯ^(((-m) * t))) * cos(t)), ((r * ℯ^(((-m) * t))) * sin(t)), (z_0 * ℯ^(((-m) * t))), t, 0, θ) → a
  4. Sequence(Sequence[(((l * sin(k)) * cos(t)), ((l * sin(k)) * sin(t)), (l * cos(k))), k, 0, π, π / n], t, 0, (2 * π), π / n) → list1
  5. Sequence(Sequence[Segment[A, (((l * sin(k)) * cos(t)), ((l * sin(k)) * sin(t)), (l * cos(k)))], k, 0, π, π / n], t, 0, (2 * π), π / n) → list2

AI Drawing Prompt Reference

Create a 3D drawing of a cone and a spiral trajectory. Define the cone by its apex, base center, and radius, then draw a parametric curve on it whose horizontal radius and height decay exponentially as the angle increases, using cosine and sine for the horizontal coordinates. Add a mesh of line segments from the apex to points generated by spherical coordinates with regular angular steps, so the spiral path and cone surface are visually supported. Include adjustable parameters for decay, radius, height, and angle range.
The prompt above was compiled from the original drawing commands and is for reference only.