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Polygonal Archimedean Spiral

Polygonal Archimedean Spiral
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2016-05-26 09:57
VectorSequenceCoordinate GeometryParametric Equation
This example demonstrates the construction and properties of a polygonal Archimedean spiral. Key math concepts include vector, sequence, coordinate geometry, and geometric construction. It uses commands like Vector, Sequence, PolyLine, Intersect, Textfield, Element in GeoGebra. Suitable for high school and above, exploring parametric equations and geometric transformations in complex shapes.

GeoGebra Commands & Parameters

  1. Vector((cos(((-π)) / N), sin(((-π)) / N)), (cos(π / N), sin(π / N))) → u
  2. Sequence(Vector[Rotate[u / Length[u], ((i * 360) * ° / N)]], i, 0, N - 1) → list1
  3. Sequence(Vector[(i * Element[list1, f(i)])], i, 1, mx + N) → list3
  4. Sequence(Vector[Sum[list3, i]], i, 1, mx + N) → list4
  5. Sequence((0, 0) + Element[list4, i], i, 1, mx + N) → list5
  6. Sequence(Element[list5, i] + Element[list1, f(i + 1)], i, 1, mx) → list6
  7. Sequence(Polygon[Element[list5, i], Element[list5, i + 1], Element[list5, i + 1] + (1 / N * (Element[list5, i + N + 1] - Element[list5, i + 1])), Element[list5, i] + (1 / N * (Element[list5, i + N] - Element[list5, i]))], i, 1, mx - 1 - 1) → list7
  8. PolyLine(First[list5, mx - 1]) → g
  9. Intersect(LineBisector[Element[list5, 1] + (1 / N * (Element[list5, N + 1] - Element[list5, 1])), Element[list5, 2]], LineBisector[Element[list5, 2] + (1 / N * (Element[list5, N + 2] - Element[list5, 2])), Element[list5, 3]]) → B
  10. Sequence(Rotate[list7, ((i * 360) * ° / N), B], i, 1, N) → list8
  11. Textfield(mx) → InputBox1
  12. Element(list8, 1) → R_{1}
  13. Element(list8, 2) → R_{2}
  14. Element(list8, 3) → R_{3}
  15. Element(list8, 4) → R_{4}
  16. Element(list8, 5) → R_{5}
  17. Element(list8, 6) → R_{6}