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Bezier Curve: Dynamic Construction and Geometric Visualization

Bezier Curve: Dynamic Construction and Geometric Visualization
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2017-07-25 08:01
FunctionParametric EquationGeometric ConstructionSlider AnimationLocus
This GeoGebra example dynamically demonstrates the construction of a Bezier curve, showing how a parametric curve defined by control points is generated through recursive linear interpolation, gradually forming a smooth path. The core mathematical concepts include parametric equations, control points, Bernstein polynomials, and the notion of locus, with related topics such as geometric mean, right triangles, and fractal geometry. The construction uses the Segment command to connect control points, Sequence to generate point series, and Locus to trace the moving point's path, ultimately producing the Bezier curve; objects include control points A, B, C, etc., and auxiliary segments and loci like f, g. This example is suitable for high school to undergraduate students learning analytic geometry, parametric curves, or introductory computer graphics, and also serves as a teaching tool for instructors to illustrate curve construction principles and interactive parametric modeling.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, D) → h
  4. Segment(D, E) → i
  5. Segment(E, F) → j
  6. Segment(F, G) → k
  7. Segment(G, H) → l
  8. Segment(H, I) → m
  9. Segment(I, J) → n
  10. Segment(B_1, C_1) → p
  11. Segment(C_1, D_1) → q
  12. Segment(D_1, E_1) → r
  13. Segment(E_1, F_1) → s
  14. Segment(F_1, G_1) → a
  15. Segment(G_1, H_1) → b
  16. Segment(H_1, I_1) → c
  17. Segment(A_1, B_1) → d
  18. Segment(A_2, B_2) → e
  19. Segment(B_2, C_2) → f_1
  20. Segment(C_2, D_2) → g_1
  21. Segment(D_2, E_2) → h_1
  22. Segment(E_2, F_2) → i_1
  23. Segment(F_2, G_2) → j_1
  24. Segment(G_2, H_2) → k_1
  25. Segment(A_3, B_3) → l_1
  26. Segment(B_3, C_3) → m_1
  27. Segment(C_3, D_3) → n_1
  28. Segment(D_3, E_3) → p_1
  29. Segment(E_3, F_3) → q_1
  30. Segment(F_3, G_3) → r_1
  31. Segment(A_4, B_4) → s_1
  32. Segment(B_4, C_4) → t_1
  33. Segment(C_4, D_4) → a_1
  34. Segment(D_4, E_4) → b_1
  35. Segment(E_4, F_4) → c_1
  36. Segment(A_5, B_5) → d_1
  37. Segment(B_5, C_5) → e_1
  38. Segment(C_5, D_5) → f_2
  39. Segment(D_5, E_5) → g_2
  40. Segment(A_6, B_6) → h_2
  41. Segment(B_6, C_6) → i_2
  42. Segment(C_6, D_6) → j_2
  43. Segment(A_7, B_7) → k_2
  44. Segment(B_7, C_7) → l_2
  45. Segment(A_8, B_8) → m_2
  46. Locus(A_9, t) → loc1
  47. Sequence((x(A_9), y(A_9)), t, x(A), x(J)) → list1