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Dynamic Construction of the Geometric Median in 3D

Dynamic Construction of the Geometric Median in 3D
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2019-03-02 20:40
TrigonometrySolid GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates the dynamic construction of the Fermat point within a polyhedron with n vertices on a spherical surface using the Weiszfeld iteration method. Key mathematical concepts involved include geometric median, Weiszfeld iterations, and spherical geometry. The construction utilizes GeoGebra commands such as Sphere, Textfield, UnicodeToLetter, Sequence, PointIn, Zip, and Sum, along with objects like A, R, b, F, bue1, a, r, InputBox1, button1, button18, n, u34, Lä, j, Points0, bue_i, button2, button3, TotalSum0, bue3, button4, Atxt, InputBox2, button5, text1, button6, text3, B, button7, button8, button9, Fo, TotalSumA0, text2b, Cmo, c, text5, text4, text6, text2. It is suitable for high school students and educators for learning and teaching demonstrations.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), R) → b
  2. Textfield(r) → InputBox1
  3. UnicodeToLetter(34) → u34
  4. Sequence((x(Corner[1]) + (random() * x(Corner[2] - Corner[1])), y(Corner[1]) + (random() * y(Corner[4] - Corner[1]))), i, 1, n) → Points0
  5. Textfield(n) → InputBox2
  6. PointIn(b) → P_{1}
  7. PointIn(b) → P_{2}
  8. PointIn(b) → P_{3}
  9. PointIn(b) → P_{4}
  10. PointIn(b) → P_{5}
  11. Zip(Length[a - F], a, lP) → lseg
  12. Sum(Zip[Length[a - F], a, lP]) → TotalSum
  13. Sum(Zip[Length[a - A], a, lP]) → TotalSum_A
  14. Zip(Segment[A, a], a, lP) → lA

AI Drawing Prompt Reference

Dynamically construct the Fermat point of a tetrahedron on a sphere. First, draw a sphere with radius R. Then, randomly generate n points as the tetrahedron's vertices. Next, randomly select five points P1 to P5 on the sphere and use the Weiszfeld iteration method to find their geometric median F. Finally, connect F to each vertex and calculate the sum of all segment lengths TotalSum and the sum of lengths to point A TotalSum_A.
The prompt above was compiled from the original drawing commands and is for reference only.