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Bernstein Polynomials

Bernstein Polynomials
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2018-04-09 11:04
FunctionSequenceCalculusGeometric ConstructionGraphing
In collections Mathematics of Computer Graphics
This example demonstrates the graph and properties of Bernstein polynomials, which are important tools in approximation theory for polynomial interpolation. By adjusting parameters, observe the shape changes of the polynomial. The construction uses commands like Sequence, Text, Integral, suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Text(Simplify[((Binomial[n - 1, i - 1] * x^(i - 1)) * (1 - x)^(n - i))]) → Tform0
  2. Length(Tform0) → lä0
  3. Take(Tform0, (läg0 * zl_0) + 1, lä0) → Liste0R
  4. Sequence(Take[Tform0, ((i - 1) * zl_0) + 1, (i * zl_0)], i, 1, läg0) → Liste0
  5. Text(Simplify[((Binomial[n - 1, i] * x^(i)) * (1 - x)^(n - i - 1))]) → TTerm1
  6. Length(TTerm1) → lä1
  7. Take(TTerm1, (läg1 * zl_1) + 1, lä1) → Liste1R
  8. Sequence(Take[TTerm1, ((i - 1) * zl_1) + 1, (i * zl_1)], i, 1, läg1) → Liste1
  9. Text(Simplify[((Binomial[n, i] * x^(i)) * (1 - x)^(n - i))]) → Tform2
  10. Length(Tform2) → lä2
  11. Take(Tform2, (läg2 * zl_2) + 1, lä2) → Liste2R
  12. Sequence(Take[Tform2, ((i - 1) * zl_2) + 1, (i * zl_2)], i, 1, läg2) → Liste2
  13. Sequence(((Binomial[n, i] * t_1^(i)) * (1 - t_1)^(n - i)), i, 0, n) → BernsteinP
  14. Join({{0}, BernsteinP}) → H_2
  15. Sequence(Sum[Take[H_2, 1, j]], j, 1, n + 2) → H_4
  16. Point(Segment[T_0, T_{10}]) → V_1
  17. Point(Segment[T_0, T_{10}]) → V_2
  18. Sequence(Polygon[T_0 + ((scl * Element[H_4, i + 1]), 0), T_0 + ((scl * Element[H_4, i + 2]), 0), T_0 + ((scl * Element[H_4, i + 2]), scl_2), T_0 + ((scl * Element[H_4, i + 1]), scl_2)], i, 0, n, 2) → H_5
  19. Sequence(Polygon[T_0 + ((scl * Element[H_4, i + 1]), 0), T_0 + ((scl * Element[H_4, i + 2]), 0), T_0 + ((scl * Element[H_4, i + 2]), scl_2), T_0 + ((scl * Element[H_4, i + 1]), scl_2)], i, 1, n + 1, 2) → H_6
  20. Sequence((t_1, Element[BernsteinP, i + 1]), i, 0, n) → Q
  21. Sequence(If[0 ≤ x ≤ 1, ((Binomial[n, i] * x^(i)) * (1 - x)^(n - i))], i, 0, n, 2) → Bernstein1
  22. Sequence(If[0 ≤ x ≤ 1, ((Binomial[n, i] * x^(i)) * (1 - x)^(n - i))], i, 1, n, 2) → Bernstein2
  23. TextToUnicode(Tform2) → Tfrmuni
  24. Degree(p) → m
  25. Sequence(i / m, i, 0, m) → Pktp
  26. Sequence(((Binomial[m, i] * x^(i)) * (1 - x)^(m - i)), i, 0, m) → Basis
  27. Textfield(p) → EF1
  28. Sequence(p(j / m), j, 0, m) → Lexp
  29. Sequence(Text[Element[Lexp, j + 1]] + "\cdot " + (Text[((Binomial[m, j] * x^(j)) * (1 - x))]) + "^{" + (Text[m - j]) + "}", j, 0, m) → LBZ
  30. Integral(f, 0, 1) → if
  31. Corner(1, 1) → Ep_1
  32. Corner(1, 2) → Ep_2
  33. Corner(1, 3) → Ep_3
  34. Corner(1, 4) → Ep_4

AI Drawing Prompt Reference

Draw the graph of Bernstein polynomials to demonstrate their function curves and properties, observing shape changes by adjusting parameters. First, plot the function curve of the polynomial, then construct a stepped shape using multiple polygons to represent the polynomial's behavior over the interval [0,1]. Finally, label key points and parameters for clarity.
The prompt above was compiled from the original drawing commands and is for reference only.