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Three-Point Pursuit Problem: Spiral Curves Converging to Triangle Center

Three-Point Pursuit Problem: Spiral Curves Converging to Triangle Center
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2024-01-25 19:55
TriangleVectorParametric EquationGeometric ConstructionLocus
In collections El dominio del Tiempo (cinemática intuitiva)
This example demonstrates the classic pursuit problem where three objects starting from the vertices of an equilateral triangle each continuously chase the next one, forming spiral curves converging to the center. Core mathematical concepts include vectors, parametric equations, loci, and limits. Construction highlights: using Rotate to determine direction, UnitVector for unit direction vectors, and CurveCartesian to plot trajectories; variables M/N/P represent the pursuers, A/B/C are triangle vertices, and vM/vN/vP are velocity vectors, with real-time updates creating the inward spiral motion. Ideal for high school vector and coordinate geometry teaching and math enthusiasts exploring dynamic parametric curves.

GeoGebra Commands & Parameters

  1. Rotate(B, pi / 3, A) → C
  2. UnitVector(Vector[M, N]) → vM
  3. UnitVector(Vector[N, P]) → vN
  4. UnitVector(Vector[P, M]) → vP
  5. CurveCartesian((1 + 7 / 2, 1 + (7 / 2 * sqrt(3) / 3)) + ((7 / sqrt(3) * ℯ^((sqrt(3) * t))); (-t) - (5 * pi / 6)), t, (-2 * pi), (2 * pi)) → c1
  6. CurveCartesian((1 + 7 / 2, 1 + (7 / 2 * sqrt(3) / 3)) + ((7 / sqrt(3) * ℯ^((sqrt(3) * t))); (-t) - pi / 6), t, (-2 * pi), (2 * pi)) → c2
  7. CurveCartesian((1 + 7 / 2, 1 + (7 / 2 * sqrt(3) / 3)) + ((7 / sqrt(3) * ℯ^((sqrt(3) * t))); (-t) + pi / 2), t, (-2 * pi), (2 * pi)) → c3

AI Drawing Prompt Reference

Draw an equilateral triangle ABC, with three points M, N, and P starting from each vertex, chasing the next point at a constant speed, forming inward spiraling curves that converge to the triangle's center. First, draw the equilateral triangle ABC, then rotate B around A by 60 degrees to get C. Next, draw the unit direction vectors vM, vN, and vP for points M, N, and P. Finally, use parametric equations to draw three spiral curves c1, c2, and c3 representing the trajectories of M, N, and P.
The prompt above was compiled from the original drawing commands and is for reference only.