Dr' Scott Schaefer

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Blossoming and B-splines

• Dr. Scott Schaefer

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Blossoms/Polar Forms

• A blossom b(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)tku wk,,tn)
• (1-u)b(t1,t2,,tk,,tn) u b(t1,t2,,wk,,tn)
• Diagonal
• b(t,t,,t) p(t)

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Blossoms/Polar Forms

• A blossom b(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)tku wk,,tn)
• (1-u)b(t1,t2,,tk,,tn) u b(t1,t2,,wk,,tn)
• Diagonal
• b(t,t,,t) p(t)

The blossom always exists and is unique!!!
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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Examples of Blossoms

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Blossoms/Polar Forms

• Symmetry b(t1,,tn) b(tm(1),,tm(n))
• Multi-affine b(t1,,(1-u)tku wk,,tn)
• (1-u)b(t1,,tk,,tn) u b(t1,,wk,,tn)
• Diagonal b(t,,t) p(t)

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Pyramid Algorithms forBezier Curves
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Pyramid Algorithms forBezier Curves
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Pyramid Algorithms forBezier Curves
Bezier curve
Bezier control points
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Subdivision Using Blossoming
Control points of left Bezier curve!
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Subdivision Using Blossoming
Control points of right Bezier curve!
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Change of Basis Using Blossoming

• Given a polynomial p(t) of degree n, find the
  coefficients of the same Bezier curve

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Change of Basis Using Blossoming

• Given a polynomial p(t) of degree n, find the
  coefficients of the same Bezier curve

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Change of Basis Using Blossoming

• Example Find Bezier coefficients of
  p(t)12t3t2-t3

Old Method
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Change of Basis Using Blossoming

• Example Find Bezier coefficients of
  p(t)12t3t2-t3

New Method
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Degree Elevation
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Degree Elevation Using Blossoming
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Degree Elevation Using Blossoming

• Symmetry is symmetric
• Multi-affine is multi-affine
• Diagonal

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Degree Elevation Using Blossoming
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Homogeneous Polynomials and Blossoming

• Polynomial
• Homogeneous Polynomial

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The Homogeneous Blossom

• Homogenize each parameter of the blossom
  independently

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The Homogeneous Blossom

• Homogenize each parameter of the blossom
  independently

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The Homogeneous Blossom

• Homogenize each parameter of the blossom
  independently

homogenized combinations
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The Homogeneous Blossom

• Homogenize each parameter of the blossom
  independently

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The Homogeneous Blossom

• Homogenize each parameter of the blossom
  independently

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Homogeneous deCasteljau Algorithm
Really b((0,1),(0,1),(1,1))
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Homogeneous deCasteljau Algorithm
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Homogeneous deCasteljau Algorithm
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Homogeneous deCasteljau Algorithm
Homogeneous blossom evaluated at (t,1) and (1,0)
yields derivatives!!!
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Homogeneous Blossoms and Derivatives
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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

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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!!!
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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. ?
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B-spline Curves

• Not a single polynomial, but lots of polynomials
  that meet together smoothly
• Local control

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B-spline Curves

• Not a single polynomial, but lots of polynomials
  that meet together smoothly
• Local control

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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

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B-spline Curves

• Curve defined over a set of parameters t0,,tk
  (ti ti1) with a polynomial of degree n in
  each interval ti, ti1 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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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Basis Functions
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B-Spline Curves
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B-Splines Via Blossoming
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B-Splines Via Blossoming
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B-Splines Via Blossoming
Single polynomial
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B-Splines Via Blossoming
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B-Splines Via Blossoming
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B-Splines Via Blossoming
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B-Splines Via Blossoming
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B-Splines Via Blossoming
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B-Splines Via Blossoming
n-1 derivatives are equal yielding Cn-1
continuity!!!
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B-Splines Via Blossoming
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B-Splines Via Blossoming
n-2 derivatives are equal yielding Cn-2
continuity at doubled knot!!!
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B-Splines Via Blossoming
In general, curves have Cn-u continuity at knot
of multiplicity u
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Conversion to Bezier Form
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Conversion to Bezier Form
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Conversion to Bezier Form
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Polynomial Reproduction

• Given a polynomial p(t) and a set of knots t1,
  t2, t3, , find control points for the b-spline
  curve that produces p(t)

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Polynomial Reproduction

• Given a polynomial p(t) and a set of knots t1,
  t2, t3, , find control points for the b-spline
  curve that produces p(t)

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Polynomial Reproduction

• Given a polynomial p(t) and a set of knots t1,
  t2, t3, , find control points for the b-spline
  curve that produces p(t)

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Polynomial Reproduction

• Given a polynomial p(t) and a set of knots t1,
  t2, t3, , find control points for the b-spline
  curve that produces p(t)

Control points!!!
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Knot Insertion

• Given a B-spline curve with knot sequence ,
  tk-2, tk-1, tk, tk1, tk2, tk3, generate the
  control points for an identical B-spline curve
  over the knot sequence , tk-2, tk-1, tk, u,
  tk1, tk2, tk3,

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Boehms Knot Insertion Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u, t4, t5, t6

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Boehms Knot Insertion Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u, t4, t5, t6

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Boehms Knot Insertion Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u, u, t4, t5, t6

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The Oslo Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u1, u2, u3, u4,
  t4, t5, t6

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The Oslo Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u1, u2, u3, u4,
  t4, t5, t6

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The Oslo Algorithm

• Given curve with knots t1, t2, t3, t4, t5, t6,
  find curve with knots t1, t2, t3, u1, u2, u3, u4,
  t4, t5, t6

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Degree Elevation for B-splines

• Degree n B-splines meet with Cn-1 continuity
• Degree n1 B-splines meet with Cn continuity
• Must double knots to maintain same degree of
  continuity!!!

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Subdivision for Non-Uniform B-splines

• Given a knot sequence t1, t2, t3, t4, , insert
  knots u1, u2, u3, u4, such that ti ui
  ti1

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Subdivision for Non-Uniform B-splines

• Given a knot sequence t1, t2, t3, t4, , insert
  knots u1, u2, u3, u4, such that ti ui
  ti1
• Double control points
• At level 0ltkltn1, insert knots uk-1, uk, uk,
  uk1, uk1, into pyramid

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Schaefers Algorithm
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Schaefers Algorithm
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Schaefers Algorithm
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Schaefers Algorithm
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Schaefers Algorithm
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Schaefers Algorithm
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B-spline Properties

• Piecewise polynomial
• Cn-u continuity at knots of multiplicity u
• Compact support
• Non-negativity implies local convex hull property
• Variation Diminishing

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B-spline Curve Example
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B-spline Curve Example
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B-spline Curve Example
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Uniform parameterization
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Centripetal parameterization
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Chord length parameterization
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Uniform parameterization
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Centripetal parameterization
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Choosing Knot Values

• B-splines dependent on choice of knots ti
• Can we choose ti automatically?

Chord length parameterization