blossoming and b-splines

1

Dr. Scott Schaefer

Blossoming and B-splines

2/105

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)tk+u wk,,…tn) = (1-u)b(t1,t2,…,tk,,…tn) + u b(t1,t2,…,wk,,…tn)

Diagonal: b(t,t,…,t) = p(t)

3/105

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)tk+u 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!!!

4/105

Examples of Blossoms

?),(1)( 21 ttbtp

5/105


1),(1)( 21 ttbtp

6/105


?),()( 21 ttbttp

1),(1)( 21 ttbtp

7/105


1),(1)( 21 ttbtp

22121),()( ttttbttp

8/105


?),()( 21

2 ttbttp

1),(1)( 21 ttbtp

22121),()( ttttbttp

9/105


1),(1)( 21 ttbtp

22121),()( ttttbttp

21212 ),()( ttttbttp

10/105


?),,(1)( 321 tttbtp

1),(1)( 21 ttbtp

22121),()( ttttbttp


11/105


1),,(1)( 321 tttbtp

1),(1)( 21 ttbtp

22121),()( ttttbttp


12/105


?),,()( 321 tttbttp

1),,(1)( 321 tttbtp

1),(1)( 21 ttbtp

22121),()( ttttbttp


13/105


1),,(1)( 321 tttbtp

1),(1)( 21 ttbtp

22121),()( ttttbttp


3321321),,()( ttttttbttp

14/105


?),,()( 321

2 tttbttp

1),,(1)( 321 tttbtp

1),(1)( 21 ttbtp

22121),()( ttttbttp



15/105


1),,(1)( 321 tttbtp


1),(1)( 21 ttbtp

22121),()( ttttbttp


33212 313221),,()( tttttttttbttp

16/105


?),,()( 321

3 tttbttp

1),,(1)( 321 tttbtp


1),(1)( 21 ttbtp

22121),()( ttttbttp



17/105


1),,(1)( 321 tttbtp



3213213 ),,()( ttttttbttp

1),(1)( 21 ttbtp

22121),()( ttttbttp


18/105

Blossoms/Polar Forms

Symmetry: b(t1,…,tn) = b(tm(1),…,tm(n)) Multi-affine: b(t1,…,(1-u)tk+u wk,,…tn) =

(1-u)b(t1,…,tk,,…tn) + u b(t1,…,wk,,…tn) Diagonal: b(t,…,t) = p(t)

)1,0,0(b )1,1,0(b

t1 t

)1,1)1(0,0( ttb

19/105

Pyramid Algorithms forBezier Curves

33

22

12

03 )1(3)1(3)1( ptpttpttpt

0p 1p 2p 3p

t1 t1 t1t tt

t1 t t1 t

t1 t

20/105


)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( tttb

t1 t

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

21/105


)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( tttb

t1 t

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

Bezier curve

Bezier control points

22/105

Subdivision Using Blossoming

)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( tttb

t1 t

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

Control points of left Bezier curve!

23/105

Subdivision Using Blossoming

)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( tttb

t1 t

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

Control points of right Bezier curve!

24/105

Change of Basis Using Blossoming

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

25/105


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

),...,()( 1 nttbtp

)1,...1,0,...,0(:tscoefficien bkn 1 k

26/105


Example: Find Bezier coefficients of p(t)=1+2t+3t2-t3

51

10003300

36301331

1321 310

35

1

Old Method

27/105


Example: Find Bezier coefficients of p(t)=1+2t+3t2-t3

21033210202110210 321),,( ttttttb ttttttttt

1)0,0,0( b 35)1,0,0( b 3

10)1,1,0( b 5)1,1,1( b

New Method

28/105

Degree Elevation

1

0

1

0

)(ˆ)(n

j

njj

n

i

nii tBptBp

n

i

niii

n

i

nii tBptpttBp

00

)())1(()(

n

i

nin

ii

n

i

nin

ini tBptBp

0

111

1

0

11

1 )()(

1

0

111

1

0

11

1 )()(n

i

nin

ii

n

i

nin

ini tBptBp

1

0

1111 )(1

n

i

nin

iin

ii tBpp

29/105

Degree Elevation Using Blossoming

1

),...,,,...,(),...,(

1

11111

111

n

ttttbttb

n

iniin

nn

30/105


Symmetry: is symmetric Multi-affine: is multi-affine

Diagonal:

),...,( 1 nn ttb

),...,( 1 nn ttb

)(1

)(

1

),...,(1

1

1

1 tpn

tp

n

ttbn

i

n

in

1

),...,,,...,(),...,(

1

11111

111

n

ttttbttb

n

iniin

nn

31/105


)1,...,1,0,...,0()1()1,...,1,0,...,0(1

1)1,...,1,0,...,0(1 nnn binbin

b

in 1 i in iin 1 1i

1

),...,,,...,(),...,(

1

11111

111

n

ttttbttb

n

iniin

nn

32/105

Homogeneous Polynomials and Blossoming Polynomial: Homogeneous Polynomial:

n

k

kktatP

0

)(

n

k

knkk wtawtP

0

),(

)()1,( tPtP

33/105

The Homogeneous Blossom

Homogenize each parameter of the blossom independently

)),(),...,,(),,(( 2211 nn wtwtwtb223)( tttp

2121

21 22

3),( ttttttb

34/105



)),(),...,,(),,(( 2211 nn wtwtwtb223)( tttp

2121

21 22

3),( ttttttb

211221

212211 22

3)),(),,(( ttwtwtwwwtwtb

22 23),( ttwwwtp

35/105



)),(),...,,(),,(( 2211 nn wtwtwtb

)),(),...,,(()),(),...,,(()),(),...,,(( 111111 vswtburwtbwtwtb vruswrut

vrustvws

homogenized combinations

36/105



)),(),...,,(),,(( 2211 nn wtwtwtb


vrustvws

))1,1(),...,,(())1,0(),...,,(()),(),...,,(( 111111 wtbwtbwtwtb vruswrut

vrustvws

37/105



)),(),...,,(),,(( 2211 nn wtwtwtb


vrustvws

))1,1(),...,,(())1,0(),...,,(()),(),...,,(( 11111111 wtbwtbwtwtb ttw

38/105

Homogeneous deCasteljau Algorithm

)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

tw tw tw t tt

tw t tw t

)~,~,~( tttb

tw t

)~,0,0( tb )1,~,0( tb )1,1,~(tb

)~,~,0( ttb )1,~,~( ttb

),(~ wtt

Really b((0,1),(0,1),(1,1))

39/105


)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

tw tw tw t tt

tw t tw t

),~,~( ttb

1 1

)~,0,0( tb )1,~,0( tb )1,1,~(tb

)~,~,0( ttb )1,~,~( ttb

),(~ wtt )0,1(

40/105


)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( ttb

1 1

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

)1,(~ tt )0,1(

41/105


)0,0,0(b )1,0,0(b )1,1,0(b )1,1,1(b

t1 t1 t1t tt

t1 t t1 t

),,( ttb

1 1

),0,0( tb )1,,0( tb )1,1,(tb

),,0( ttb )1,,( ttb

)1,(~ tt )0,1(

Homogeneous blossom evaluated at (t,1) and (1,0) yields derivatives!!!

42/105

Homogeneous Blossoms and Derivatives

),...,()( ttbtp

),,...,()(' ttnbtp

),...,,,...,()!(

!)()( ttbkn

ntp k

n

1n

kn k

43/105

Problems with Bezier Curves

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

curve

44/105



curve

45/105



curve

Solution: Use lots of Bezier curves and maintain Ck continuity!!!

46/105



curve

Solution: Use lots of Bezier curves and maintain Ck continuity!!!

Difficult to keep track of all the constraints.

47/105

B-spline Curves

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

Local control

48/105

B-spline Curves

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

Local control

49/105

History of B-splines

Designed to create smooth curves Similar to physical process of bending wood Early Work

de Casteljau at CitroenBezier at Renaultde Boor at General Motors

50/105

B-spline Curves

Curve defined over a set of parameters t0,…,tk (ti ti+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

51/105

B-Spline Basis Functions

..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(03 tN

52/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(04 tN

53/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(13 tN

54/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(14 tN

55/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(23 tN

56/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(24 tN

57/105


..01)( 10

wottttN ii

i

)()()( 11

1

11

1 tNtttttN

tttttN n

iini

inni

ini

ini

)(33 tN

58/105

B-Spline Curves

m

i

nii tNptp

1

)()(

59/105

B-Splines Via Blossoming

),,( 321 tttb

14

4

tttt

14

1

tttt

),,( 32 tttb

),,( 432 tttb

60/105


),,( 321 tttb

tt 4 1tt

),,( 32 tttb

),,( 432 tttb

61/105


),,( 321 tttb

tt 4 tt 5tt 61tt 3tt 2tt

tt 4 2tt tt 5 3tt

),,( tttbtt 4 3tt

),,( 32 tttb ),,( 43 tttb ),,( 54 tttb

),,( 3 tttb ),,( 4 tttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

Single polynomial

62/105


),,( 321 tttb


tt 4 2tt tt 5 3tt

),,( tttbtt 4 3tt

),,( 32 tttb ),,( 43 tttb ),,( 54 tttb

),,( 3 tttb ),,( 4 tttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

43 ttt

63/105


),,( 321 tttb

tt 4 tt 51tt 2tt

tt 4 2tt

),,( 32 tttb ),,( 43 tttb

),,( 3 tttb

),,( 432 tttb ),,( 543 tttb),,( 210 tttb

tt 3 0tt

tt 3 1tt

),,( tttbtt 3 2tt

),,( 21 tttb

),,( 2 tttb

32 ttt

64/105


),,( 321 tttb


tt 4 2tt tt 5 3tt

),,( tttbtt 4 3tt

),,( 32 tttb ),,( 43 tttb ),,( 54 tttb

),,( 3 tttb ),,( 4 tttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb),,( 210 tttb

tt 3 0tt

tt 3 1tt

),,( tttbtt 3 2tt

),,( 21 tttb

),,( 2 tttb

65/105


),,( 321 tttb

1 1 11 11

tt 4 2tt tt 5 3tt

),,( ttbtt 4 3tt

),,( 32 ttb ),,( 43 ttb ),,( 54 ttb

),,( 3 ttb ),,( 4 ttb


1 1

tt 3 1tt

),,( ttbtt 3 2tt

),,( 21 ttb

),,( 2 ttb

66/105


),,( 321 tttb

1 1 11 11

1 1 1 1

),,( tbtt 4 3tt

),,( 32 ttb ),,( 43 ttb ),,( 54 ttb

),,( 3 tb ),,( 4 tb


1 1

1 1

),,( tbtt 3 2tt

),,( 21 ttb

),,( 2 tb

67/105


),,( 321 tttb

1 11 1

1 1

),,( 32 ttb ),,( 43 ttb

),,( 3 tb

),,( 432 tttb ),,( 543 tttb

n-1 derivatives are equal yielding Cn-1 continuity!!!

68/105


),,( 221 tttb


tt 3 2tt tt 4 2tt

),,( tttbtt 4 3tt

),,( 22 tttb ),,( 32 tttb ),,( 43 tttb

),,( 2 tttb ),,( 3 tttb


tt 2 0tt

tt 2 1tt

),,( 21 tttb

),,( 2 tttb

),,( 101 tttb

tt 2

tt 2

),,( tttbtt 2 1tt

),,( 10 tttb

),,( 1 tttb0tt

1 tt

69/105


),,( 221 tttb

1 1),,( 22 ttb

),,( 322 tttb

n-2 derivatives are equal yielding

Cn-2 continuity at doubled knot!!!

70/105


),,( 221 tttb

1 1),,( 22 ttb

),,( 322 tttb

In general, curves have Cn-u

continuity at knot of multiplicity u

71/105

Conversion to Bezier Form

),,( 321 tttb


tt 4 2tt tt 5 3tt

),,( tttbtt 4 3tt

),,( 32 tttb ),,( 43 tttb ),,( 54 tttb

),,( 3 tttb ),,( 4 tttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

72/105


),,( 321 tttb

34 tt 35 tt 13 tt 23 tt

tt 4 23 tt

),,( 332 tttb ),,( 343 tttb

),,( 333 tttb

),,( 432 tttb ),,( 543 tttb

73/105


45 tt 46 tt 34 tt 24 tt

45 tt 34 tt

),,( 443 tttb ),,( 454 tttb

),,( 444 tttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

74/105

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)

75/105



32 4321)( ttttp

76/105



32 4321)( ttttp

32133321 4321),,( 313221321 uuuuuub uuuuuuuuu

77/105



32 4321)( ttttp

32133321 4321),,( 313221321 uuuuuub uuuuuuuuu

),...,,(),,,(),,,(),,,( 654543432321 tttbtttbtttbtttb

Control points!!!

78/105

Knot Insertion

Given a B-spline curve with knot sequence …, tk-2, tk-1, tk, tk+1, tk+2, tk+3, … generate the control points for an identical B-spline curve over the knot sequence …, tk-2, tk-1, tk, u, tk+1, tk+2, tk+3, …

79/105

Boehm’s Knot Insertion Algorithm

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

80/105


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

),,( 321 tttb

ut 4 ut 5ut 61tu 3tu 2tu

),,( 32 uttb ),,( 43 uttb ),,( 54 uttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

81/105


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

),,( 321 tttb

ut 4 ut 5ut 61tu 3tu 2tu

),,( 32 uttb ),,( 43 uttb ),,( 54 uttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

ut 4 2tu ut 5 3tu ),,( 3 uutb ),,( 4 uutb

82/105

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

83/105

The Oslo Algorithm


),,( 321 tttb

14 ut 15 ut 16 ut 11 tu 31 tu 21 tu

),,( 143 uttb ),,( 154 uttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

24 ut 22 tu 25 ut 32 tu ),,( 213 uutb ),,( 214 uutb

),,( 132 uttb

),,( 321 uuub34 ut 33 tu

84/105

The Oslo Algorithm


),,( 321 tttb

44 ut 45 ut 46 ut 14 tu 34 tu 24 tu

),,( 443 uttb ),,( 454 uttb

),,( 432 tttb ),,( 543 tttb ),,( 654 tttb

34 ut 23 tu 35 ut 33 tu ),,( 433 uutb ),,( 434 uutb

),,( 432 uttb

),,( 432 uuub

24 ut 32 tu

85/105

Degree Elevation for B-splines

Degree n B-splines meet with Cn-1 continuity Degree n+1 B-splines meet with Cn continuity Must double knots to maintain same degree

of continuity!!!

1

),...,,,...,(),...,(

1

11111

111

n

ttttbttb

n

iniin

nn

86/105

Subdivision for Non-Uniform B-splines Given a knot sequence t1, t2, t3, t4, …, insert

knots u1, u2, u3, u4, … such that ti ui ti+1

87/105

Subdivision for Non-Uniform B-splines Given a knot sequence t1, t2, t3, t4, …, insert

knots u1, u2, u3, u4, … such that ti ui ti+1

Double control points At level 0<k<n+1, insert knots uk-1, uk, uk,

uk+1, uk+1, … into pyramid

88/105

Schaefer’s Algorithm

)( 1tb )( 1tb )( 2tb )( 2tb )( 3tb )( 3tb )( 4tb )( 4tb

)( 1ub )( 2ub )( 3ub

3u

)( 4tb)( 3tb)( 2tb)( 1tb

0u 1u 1u 2u 2u 3u

89/105


),( 2ub

),( 1tb 2t ),( 2tb 3t ),( 3tb 4t ),( 4tb 5t

),( 1ub 2t 3t ),( 3ub 4t

0u 1u 1u 2u 2u 3u 3u),( 1tb 2t

),( 1tb 2t

),( 2tb 3t

),( 2tb 3t

),( 3tb 4t

),( 3tb 4t

),( 4tb 5t

),( 4tb 5t

90/105


),( 2ub

),( 1tb 2t ),( 2tb 3t ),( 3tb 4t ),( 4tb 5t

),( 1ub 2t 3t ),( 3ub 4t

),( 21 tub ),( 22 utb ),( 32 tub ),( 33 utb ),( 43 tub ),( 44 utb

0u 1u 1u 2u 2u 3u 3u

1u 2u 2u 3u 3u 4u

),( 1tb 2t

),( 1tb 2t

),( 2tb 3t

),( 2tb 3t

),( 3tb 4t

),( 3tb 4t

),( 4tb 5t

),( 4tb 5t

91/105


),,( 21 tub ),,( 22 utb ),,( 32 tub ),,( 33 utb ),,( 43 tub ),,( 44 utb

),,( 1tb 2t 3t ),,( 2tb 3t 4t ),,( 3tb 4t 5t ),,( 4tb 5t 6t

),,( 1ub 2t 3t ),,( 2ub 3t 4t ),,( 3ub 4t 5t

3t 3t 4t 4t 5t 5t

0u 1u 1u 2u 2u 3u 3u

1u 2u 2u 3u 3u 4u

),,( 4tb 5t 6t

),,( 4tb 5t 6t

),,( 1tb 2t 3t

),,( 1tb 2t 3t

),,( 2tb 3t 4t

),,( 2tb 3t 4t

),,( 3tb 4t 5t

),,( 3tb 4t 5t

92/105




),,( 1ub 2t 3t ),,( 2ub 3t 4t ),,( 3ub 4t 5t

3t 3t 4t 4t 5t 5t

),,( 322 tutb ),,( 332 utub ),,( 433 tutb ),,( 443 utub ),,( 544 tutb

0u 1u 1u 2u 2u 3u 3u

1u 2u 2u 3u 3u 4u

2u 3u 3u 4u 4u

),,( 4tb 5t 6t

),,( 4tb 5t 6t

),,( 1tb 2t 3t

),,( 1tb 2t 3t

),,( 2tb 3t 4t

),,( 2tb 3t 4t

),,( 3tb 4t 5t

),,( 3tb 4t 5t

93/105




),,( 1ub 2t 3t ),,( 2ub 3t 4t ),,( 3ub 4t 5t

3t 3t 4t 4t 5t 5t

),,( 322 tutb ),,( 332 utub ),,( 433 tutb ),,( 443 utub ),,( 544 tutb

0u 1u 1u 2u 2u 3u 3u

1u 2u 2u 3u 3u 4u

2u 3u 3u 4u 4u

),,( 4tb 5t 6t

),,( 4tb 5t 6t

),,( 1tb 2t 3t

),,( 1tb 2t 3t

),,( 2tb 3t 4t

),,( 2tb 3t 4t

),,( 3tb 4t 5t

),,( 3tb 4t 5t

94/105

B-spline Properties

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

property Variation Diminishing

95/105

B-spline Curve Example

96/105


97/105


98/105

Choosing Knot Values

B-splines dependent on choice of knots ti

Can we choose ti automatically?

99/105




kikii ppt 1

100/105




kikii ppt 1

0Uniform parameterization

101/105




kikii ppt 1

21

Centripetal parameterization

102/105




kikii ppt 1

1

Chord length parameterization

103/105




kikii ppt 1

0Uniform parameterization

104/105




kikii ppt 1

21

Centripetal parameterization

105/105




kikii ppt 1

1

Chord length parameterization

blossoming and b-splines

Documents

examples of blossoms

homogeneous blossoms

tn u bt1

homogeneous blossomhomogenize

bezier coefficients

homogeneous polynomials

polynomial pt of degree

degree elevation