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The Bernstein Basis and Bezier Curves

The Bernstein Basis and Bezier Curves. Dr. Scott Schaefer. Problems with Interpolation. Problems with Interpolation. Bezier Curves. Polynomial curves that seek to approximate rather than to interpolate. Bernstein Polynomials. Degree 1: (1- t ), t Degree 2: (1- t ) 2 , 2(1- t ) t , t 2

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The Bernstein Basis and Bezier Curves

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  1. The Bernstein Basis and Bezier Curves Dr. Scott Schaefer

  2. Problems with Interpolation

  3. Problems with Interpolation

  4. Bezier Curves • Polynomial curves that seek to approximate rather than to interpolate

  5. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3

  6. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3 • Degree 4: (1-t)4, 4(1-t)3t, 6(1-t)2t2, 4(1-t)t3,t4

  7. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3 • Degree 4: (1-t)4, 4(1-t)3t, 6(1-t)2t2, 4(1-t)t3,t4 • Degree 5: (1-t)5, 5(1-t)4t, 10(1-t)3t2, 10(1-t)2t3 ,5(1-t)t4,t5 • … • Degree n: for

  8. Properties of Bernstein Polynomials

  9. Properties of Bernstein Polynomials

  10. Properties of Bernstein Polynomials

  11. Properties of Bernstein Polynomials Binomial Theorem:

  12. Properties of Bernstein Polynomials Binomial Theorem:

  13. Properties of Bernstein Polynomials Binomial Theorem:

  14. Properties of Bernstein Polynomials Binomial Theorem:

  15. Properties of Bernstein Polynomials

  16. Properties of Bernstein Polynomials

  17. Properties of Bernstein Polynomials

  18. Properties of Bernstein Polynomials

  19. More Properties of Bernstein Polynomials

  20. More Properties of Bernstein Polynomials

  21. More Properties of Bernstein Polynomials

  22. More Properties of Bernstein Polynomials

  23. Properties of Bernstein Polynomials

  24. Properties of Bernstein Polynomials • Base case:

  25. Properties of Bernstein Polynomials • Base case:

  26. Properties of Bernstein Polynomials • Base case: • Inductive Step: Assume

  27. Properties of Bernstein Polynomials • Base case: • Inductive Step: Assume

  28. Properties of Bernstein Polynomials • Base case: • Inductive Step: Assume

  29. Properties of Bernstein Polynomials • Base case: • Inductive Step: Assume

  30. Bezier Curves

  31. Bezier Curves

  32. Bezier Curves

  33. Bezier Curve Properties • Interpolate end-points

  34. Bezier Curve Properties • Interpolate end-points

  35. Bezier Curve Properties • Interpolate end-points

  36. Bezier Curve Properties • Interpolate end-points

  37. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  38. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  39. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  40. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  41. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  42. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge Another Bezier curve of vectors!!!

  43. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge

  44. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge • Curve lies within the convex hull of the control points

  45. Bezier Curve Properties • Interpolate end-points • Tangent at end-points in direction of first/last edge • Curve lies within the convex hull of the control points Bezier Lagrange

  46. Matrix Form of Bezier Curves

  47. Matrix Form of Bezier Curves

  48. Matrix Form of Bezier Curves

  49. Matrix Form of Bezier Curves

  50. Matrix Form of Bezier Curves Computation in monomial basis is unstable!!! Most proofs/computations are easier in Bernstein basis!!!

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