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Blossoming and B-splines

Blossoming and B-splines. Dr. Scott Schaefer. Blossoms/Polar Forms. A blossom b ( t 1 , t 2 ,…, t n ) of a polynomial p ( t ) is a multivariate function with the properties: Symmetry: b ( t 1 , t 2 ,…, t n ) = b ( t m (1) , t m (2) ,…, t m ( n ) ) for any permutation m of (1,2,…, n )

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Blossoming and B-splines

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  1. Blossoming and B-splines Dr. Scott Schaefer

  2. Blossoms/Polar Forms • A blossomb(t1,t2,…,tn) of a polynomial p(t) is a multivariate function with the properties: • Symmetry: b(t1,t2,…,tn) = b(tm(1),tm(2),…,tm(n)) for any permutation m of (1,2,…,n) • Multi-affine: b(t1,t2,…,(1-u)tk+uwk,,…tn) = (1-u)b(t1,t2,…,tk,,…tn) + ub(t1,t2,…,wk,,…tn) • Diagonal: b(t,t,…,t) = p(t)

  3. Blossoms/Polar Forms • A blossomb(t1,t2,…,tn) of a polynomial p(t) is a multivariate function with the properties: • Symmetry: b(t1,t2,…,tn) = b(tm(1),tm(2),…,tm(n)) for any permutation m of (1,2,…,n) • Multi-affine: b(t1,t2,…,(1-u)tk+uwk,,…tn) = (1-u)b(t1,t2,…,tk,,…tn) + ub(t1,t2,…,wk,,…tn) • Diagonal: b(t,t,…,t) = p(t) The blossom always exists and is unique!!!

  4. Examples of Blossoms

  5. Examples of Blossoms

  6. Examples of Blossoms

  7. Examples of Blossoms

  8. Examples of Blossoms

  9. Examples of Blossoms

  10. Examples of Blossoms

  11. Examples of Blossoms

  12. Examples of Blossoms

  13. Examples of Blossoms

  14. Examples of Blossoms

  15. Examples of Blossoms

  16. Examples of Blossoms

  17. Examples of Blossoms

  18. Blossoms/Polar Forms • Symmetry: b(t1,…,tn) = b(tm(1),…,tm(n)) • Multi-affine: b(t1,…,(1-u)tk+uwk,,…tn) = (1-u)b(t1,…,tk,,…tn) + ub(t1,…,wk,,…tn) • Diagonal: b(t,…,t) = p(t)

  19. Pyramid Algorithms forBezier Curves

  20. Pyramid Algorithms forBezier Curves

  21. Pyramid Algorithms forBezier Curves Bezier curve Bezier control points

  22. Subdivision Using Blossoming Control points of left Bezier curve!

  23. Subdivision Using Blossoming Control points of right Bezier curve!

  24. Change of Basis Using Blossoming • Given a polynomial p(t) of degree n, find the coefficients of the same Bezier curve

  25. Change of Basis Using Blossoming • Given a polynomial p(t) of degree n, find the coefficients of the same Bezier curve

  26. Change of Basis Using Blossoming • Example: Find Bezier coefficients of p(t)=1+2t+3t2-t3 Old Method

  27. Change of Basis Using Blossoming • Example: Find Bezier coefficients of p(t)=1+2t+3t2-t3 New Method

  28. Degree Elevation

  29. Degree Elevation Using Blossoming

  30. Degree Elevation Using Blossoming • Symmetry: is symmetric • Multi-affine: is multi-affine • Diagonal:

  31. Degree Elevation Using Blossoming

  32. Homogeneous Polynomials and Blossoming • Polynomial: • Homogeneous Polynomial:

  33. The Homogeneous Blossom • Homogenize each parameter of the blossom independently

  34. The Homogeneous Blossom • Homogenize each parameter of the blossom independently

  35. The Homogeneous Blossom • Homogenize each parameter of the blossom independently homogenized combinations

  36. The Homogeneous Blossom • Homogenize each parameter of the blossom independently

  37. The Homogeneous Blossom • Homogenize each parameter of the blossom independently

  38. Homogeneous deCasteljau Algorithm Really b((0,1),(0,1),(1,1))

  39. Homogeneous deCasteljau Algorithm

  40. Homogeneous deCasteljau Algorithm

  41. Homogeneous deCasteljau Algorithm Homogeneous blossom evaluated at (t,1) and (1,0) yields derivatives!!!

  42. Homogeneous Blossoms and Derivatives

  43. Problems with Bezier Curves • More control points means higher degree • Moving one control point affects the entire curve

  44. Problems with Bezier Curves • More control points means higher degree • Moving one control point affects the entire curve

  45. Problems with Bezier Curves • More control points means higher degree • Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain Ck continuity!!!

  46. Problems with Bezier Curves • More control points means higher degree • Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain Ck continuity!!! Difficult to keep track of all the constraints. 

  47. B-spline Curves • Not a single polynomial, but lots of polynomials that meet together smoothly • Local control

  48. B-spline Curves • Not a single polynomial, but lots of polynomials that meet together smoothly • Local control

  49. History of B-splines • Designed to create smooth curves • Similar to physical process of bending wood • Early Work • de Casteljau at Citroen • Bezier at Renault • de Boor at General Motors

  50. B-spline Curves • Curve defined over a set of parameters t0,…,tk (titi+1) with a polynomial of degree n in each interval [ti, ti+1] that meet with Cn-1 continuity • ti do not have to be evenly spaced • Commonly called NURBS • Non-Uniform Rational B-Splines

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