Polynomial splines as examples of Chebyshevian splines - - PowerPoint PPT Presentation

Polynomial splines as examples of Chebyshevian splines

Marie-Laurence Mazure

Laboratoire Jean Kuntzmann, Université Joseph Fourier, Grenoble

SC2011 – S. Margherita di Pula, October 10-14, 2011

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Outline

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Outline

Extended Chebyshev spaces

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Outline

Extended Chebyshev spaces Chebyshevian splines

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Outline

Extended Chebyshev spaces Chebyshevian splines Geometrically continuous polynomial splines

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Outline

Extended Chebyshev spaces (Th.A) Chebyshevian splines Geometrically continuous polynomial splines

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Outline

Extended Chebyshev spaces (Th.A) Chebyshevian splines (Th.B) Geometrically continuous polynomial splines

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Outline

Extended Chebyshev spaces (Th.A) Chebyshevian splines (Th.B) Geometrically continuous polynomial splines (⇐ A + B)

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Extended Chebyshev spaces (EC-spaces)

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages :

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages :

• ffer more possibilities in Approximation or Geometric Design

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages :

• ffer more possibilities in Approximation or Geometric Design

enable us to better understand polynomial spaces

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Extended Chebyshev spaces (EC-spaces)

natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages :

• ffer more possibilities in Approximation or Geometric Design

enable us to better understand polynomial spaces

• r even to obtain new results concerning them e.g., HERE

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EC-spaces

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EC-spaces

I interval, E ⊂ C n(I) (n + 1)-dimensional

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EC-spaces

I interval, E ⊂ C n(I) (n + 1)-dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I, counting multiplicities

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EC-spaces

I interval, E ⊂ C n(I) (n + 1)-dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I, counting multiplicities Examples :

1 1, x, . . . , xn−2, cosh x, sinh x span an EC-space on I = R ; 2 1, x, . . . , xn−2, cos x, sin x span an EC-space on any

I = [a, a + 2π[ (for n ≥ 2).

3 . . . 4 / 24

EC-spaces

I interval, E ⊂ C n(I) (n + 1)-dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I, counting multiplicities Examples :

1 1, x, . . . , xn−2, cosh x, sinh x span an EC-space on I = R ; 2 1, x, . . . , xn−2, cos x, sin x span an EC-space on any

I = [a, a + 2π[ (for n ≥ 2).

3 . . .

• S. Karlin – L.L. Schumaker – T. Lyche – N. Dyn – A. Ron – G. Mühlbach –
• H. Pottmann – P.J. Barry – D. Bister – H. Prautzsch – J. Carnicer – J.-M. Peña – P. Costantini – C. Manni –. . .

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EC-spaces and weight functions

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EC-spaces and weight functions

(w0, . . . , wn) system of weight functions on I : for i = 0, . . . , n, wi ∈ C n−i(I) and is positive on I

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EC-spaces and weight functions

(w0, . . . , wn) system of weight functions on I : for i = 0, . . . , n, wi ∈ C n−i(I) and is positive on I associated generalised derivatives : D = ordinary differentiation L0F := F w0 , LiF := DLi−1F wi , i = 1, . . . , n

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EC-spaces and weight functions

(w0, . . . , wn) system of weight functions on I : for i = 0, . . . , n, wi ∈ C n−i(I) and is positive on I associated generalised derivatives : D = ordinary differentiation L0F := F w0 , LiF := DLi−1F wi , i = 1, . . . , n E := {F ∈ C n(I) | LnF constant on I} is an (n + 1)-dim. EC-space on I, denoted E = EC(w0, . . . , wn)

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EC-spaces and weight functions

(w0, . . . , wn) system of weight functions on I : for i = 0, . . . , n, wi ∈ C n−i(I) and is positive on I associated generalised derivatives : D = ordinary differentiation L0F := F w0 , LiF := DLi−1F wi , i = 1, . . . , n E := {F ∈ C n(I) | LnF constant on I} is an (n + 1)-dim. EC-space on I, denoted E = EC(w0, . . . , wn)

Ex : on I = R, for w0 = w1 = · · · = wn = 1 I, EC(w0, . . . , wn) = Pn

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E= (n + 1)-dimensional EC-space on I = [a, b]

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E= (n + 1)-dimensional EC-space on I = [a, b]

Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded, then E can be written as E = EC(w0, . . . , wn)

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E= (n + 1)-dimensional EC-space on I = [a, b]

Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded, then E can be written as E = EC(w0, . . . , wn) THEOREM A ONE CAN NOW FIND ALL POSSIBLE SYSTEMS OF WEIGHT FUNCTIONS ON [a, b] SUCH THAT E = EC(w0, . . . , wn)

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E= (n + 1)-dimensional EC-space on I = [a, b]

Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded, then E can be written as E = EC(w0, . . . , wn) THEOREM A ONE CAN NOW FIND ALL POSSIBLE SYSTEMS OF WEIGHT FUNCTIONS ON [a, b] SUCH THAT E = EC(w0, . . . , wn)

MLM, Finding all systems of weight functions associated with a given Extended Chebyshev space,

• J. Approx. Theory, 2011

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Weight functions for E :=P2 restricted to I := [0, 1]

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Weight functions for E :=P2 restricted to I := [0, 1]

(B0, B1, B2) : Bernstein basis of degree 2

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Weight functions for E :=P2 restricted to I := [0, 1]

(B0, B1, B2) : Bernstein basis of degree 2 Select any α0, α1, α2 > 0 and take w0 := 2

i=0 αiBi

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Weight functions for E :=P2 restricted to I := [0, 1]

(B0, B1, B2) : Bernstein basis of degree 2 Select any α0, α1, α2 > 0 and take w0 := 2

i=0 αiBi

1 I = 2

i=0 αiBi w0

→ Bernstein basis of L0E := { F

w0 | F ∈ E}

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Weight functions for E :=P2 restricted to I := [0, 1]

(B0, B1, B2) : Bernstein basis of degree 2 Select any α0, α1, α2 > 0 and take w0 := 2

i=0 αiBi

1 I = 2

i=0 αiBi w0

→ Bernstein basis of L0E := { F

w0 | F ∈ E}

DL0E= EC-space of dimension 2 on I, Bernstein-like basis V0 := −D α0B0 w0

• ,

V1 := D α1B1 w0

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Weight functions for E :=P2 restricted to I := [0, 1]

(B0, B1, B2) : Bernstein basis of degree 2 Select any α0, α1, α2 > 0 and take w0 := 2

i=0 αiBi

1 I = 2

i=0 αiBi w0

→ Bernstein basis of L0E := { F

w0 | F ∈ E}

DL0E= EC-space of dimension 2 on I, Bernstein-like basis V0 := −D α0B0 w0

• ,

V1 := D α1B1 w0

• Select any β0, β1 > 0 and take w1 := 1

i=0 βiVi

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Weight functions for E :=P2 restricted to I := [0, 1]

1 I = 1

i=0 βiVi w1 → Bern. basis of L1E := { F w1 | F ∈ DL0E}

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Weight functions for E :=P2 restricted to I := [0, 1]

1 I = 1

i=0 βiVi w1 → Bern. basis of L1E := { F w1 | F ∈ DL0E}

DL1E := EC-space of dimension 1 on I, with Bernstein-like basis V 0 := D β1V1 w1

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Weight functions for E :=P2 restricted to I := [0, 1]

1 I = 1

i=0 βiVi w1 → Bern. basis of L1E := { F w1 | F ∈ DL0E}

DL1E := EC-space of dimension 1 on I, with Bernstein-like basis V 0 := D β1V1 w1

• Select any γ0 > 0 and take w2 := γ0V 0

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Weight functions for E :=P2 restricted to I := [0, 1]

1 I = 1

i=0 βiVi w1 → Bern. basis of L1E := { F w1 | F ∈ DL0E}

DL1E := EC-space of dimension 1 on I, with Bernstein-like basis V 0 := D β1V1 w1

• Select any γ0 > 0 and take w2 := γ0V 0

Theorem - This provides us with all ways to write E as E = EC(w0, w1, w2).

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Chebyshevian splines : ingredients

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Chebyshevian splines : ingredients

1 a sequence of knots tk < tk+1, k ∈ Z ; 9 / 24

Chebyshevian splines : ingredients

1 a sequence of knots tk < tk+1, k ∈ Z ; 2 a sequence of section-spaces : for each k ∈ Z,

Ek ⊂ C n([tk, tk+1]) contains constants and DEk is an n-dimensional EC-space on [tk, tk+1] ;

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Chebyshevian splines : ingredients

1 a sequence of knots tk < tk+1, k ∈ Z ; 2 a sequence of section-spaces : for each k ∈ Z,

Ek ⊂ C n([tk, tk+1]) contains constants and DEk is an n-dimensional EC-space on [tk, tk+1] ;

3 a sequence Mk, k ∈ Z, of connection matrices : for each

k ∈ Z, Mk is a lower triangular matrix of order (n − 1) with positive diagonal.

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Chebyshevian splines : definition

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Chebyshevian splines : definition

S= set of all continuous functions S :] infk tk, supk tk[→ R such that, for each k ∈ Z

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Chebyshevian splines : definition

S= set of all continuous functions S :] infk tk, supk tk[→ R such that, for each k ∈ Z

1 the restriction of S to [tk, tk+1] belongs to Ek 10 / 24

Chebyshevian splines : definition

S= set of all continuous functions S :] infk tk, supk tk[→ R such that, for each k ∈ Z

1 the restriction of S to [tk, tk+1] belongs to Ek 2 at tk S satisfies the connection condition

• S′(t+

k ), . . . , S(n−1)(tk +)

T = Mk

• S′(t−

k ) . . . , S(n−1)(tk −)

T

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Chebyshevian splines : definition

S= set of all continuous functions S :] infk tk, supk tk[→ R such that, for each k ∈ Z

1 the restriction of S to [tk, tk+1] belongs to Ek 2 at tk S satisfies the connection condition

• S′(t+

k ), . . . , S(n−1)(tk +)

T = Mk

• S′(t−

k ) . . . , S(n−1)(tk −)

T Example - for each k ∈ Z, Ek :=Pn restricted to [tk, tk+1] :

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Chebyshevian splines : definition

S= set of all continuous functions S :] infk tk, supk tk[→ R such that, for each k ∈ Z

1 the restriction of S to [tk, tk+1] belongs to Ek 2 at tk S satisfies the connection condition

• S′(t+

k ), . . . , S(n−1)(tk +)

T = Mk

• S′(t−

k ) . . . , S(n−1)(tk −)

T Example - for each k ∈ Z, Ek :=Pn restricted to [tk, tk+1] : ↓ geometrically continuous polynomial splines of degree n

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Chebyshevian spline spaces “good for design"

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Chebyshevian spline spaces “good for design"

Criteria :

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Chebyshevian spline spaces “good for design"

Criteria :

1

local control : basis functions with minimal support

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Chebyshevian spline spaces “good for design"

Criteria :

1

local control : basis functions with minimal support

2

“good" control : normalised totally positive basis

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Chebyshevian spline spaces “good for design"

Criteria :

1

local control : basis functions with minimal support

2

“good" control : normalised totally positive basis

3

development of all the classical design algorithms

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Chebyshevian spline spaces “good for design"

Criteria :

1

local control : basis functions with minimal support

2

“good" control : normalised totally positive basis

3

development of all the classical design algorithms

Can be summarised as : EXISTENCE OF BLOSSOMS IN S

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ALL good spline spaces

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

1

(w k

1 , . . . , w k n ) : system of weight functions on [tk, tk+1]

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

1

(w k

1 , . . . , w k n ) : system of weight functions on [tk, tk+1]

2

Ek := EC(1 Ik, w k

1 , . . . , w k n )

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

1

(w k

1 , . . . , w k n ) : system of weight functions on [tk, tk+1]

2

Ek := EC(1 Ik, w k

1 , . . . , w k n )

3

Lk

0 = Idk, Lk 1, . . . , Lk n associated generalised derivatives

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

1

(w k

1 , . . . , w k n ) : system of weight functions on [tk, tk+1]

2

Ek := EC(1 Ik, w k

1 , . . . , w k n )

3

Lk

0 = Idk, Lk 1, . . . , Lk n associated generalised derivatives

Define S as previously but with the connection conditions

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ALL good spline spaces

Ingredients : for each k ∈ Z, take

1

(w k

1 , . . . , w k n ) : system of weight functions on [tk, tk+1]

2

Ek := EC(1 Ik, w k

1 , . . . , w k n )

3

Lk

0 = Idk, Lk 1, . . . , Lk n associated generalised derivatives

Define S as previously but with the connection conditions

• Lk

1S(t+ k ), . . . , Lk n−1S(t+ k )

T=

• Lk−1

1

S(t−

k ), . . . , Lk−1 n−1S(tk −)

T

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ALL good spline spaces

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ALL good spline spaces

THEOREM B

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ALL good spline spaces

THEOREM B THIS PROCEDURE YIELDS ALL CHEBYSHEVIAN SPLINE SPACES GOOD FOR DESIGN

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ALL good spline spaces

THEOREM B THIS PROCEDURE YIELDS ALL CHEBYSHEVIAN SPLINE SPACES GOOD FOR DESIGN

MLM, How to build all Chebyshevian spline spaces good for Geometric Design, to appear in Num. Math. 13 / 24

ALL good spline spaces

THEOREM B THIS PROCEDURE YIELDS ALL CHEBYSHEVIAN SPLINE SPACES GOOD FOR DESIGN

MLM, How to build all Chebyshevian spline spaces good for Geometric Design, to appear in Num. Math.

Remarks : P.J. Barry, Constr. Approx., 96, connection conditions

• Lk

1S(t+ k ), . . . , Lk n−1S(t+ k )

T= Mk

• Lk−1

1

S(t−

k ), . . . , Lk−1 n−1S(tk−)

T ,

• Mk totally positive (TP, i.e., all minors are ≥ 0) for all k + . . . ⇒ S

good for design

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ALL good spline spaces

THEOREM B THIS PROCEDURE YIELDS ALL CHEBYSHEVIAN SPLINE SPACES GOOD FOR DESIGN

MLM, How to build all Chebyshevian spline spaces good for Geometric Design, to appear in Num. Math.

Remarks : P.J. Barry, Constr. Approx., 96, connection conditions

• Lk

1S(t+ k ), . . . , Lk n−1S(t+ k )

T= Mk

• Lk−1

1

S(t−

k ), . . . , Lk−1 n−1S(tk−)

T ,

• Mk totally positive (TP, i.e., all minors are ≥ 0) for all k + . . . ⇒ S

good for design Polynomial case (Mk TP for all k) : TNT Goodman (85) – N. Dyn+C. Micchelli (88)

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S = space of geometrically continuous polynomial splines

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S = space of geometrically continuous polynomial splines

Question : is S good for design ?

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S = space of geometrically continuous polynomial splines

Question : is S good for design ? THEOREM B ⇓ Equivalent question : can we find systems of weight functions (wk

1 , . . . , wk n ) on [tk, tk+1], k ∈ Z, such that :

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S = space of geometrically continuous polynomial splines

Question : is S good for design ? THEOREM B ⇓ Equivalent question : can we find systems of weight functions (wk

1 , . . . , wk n ) on [tk, tk+1], k ∈ Z, such that :

1 for each k, EC(1

Ik, wk

1 , . . . , wk n ) = Pn restricted to [tk, tk+1]

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S = space of geometrically continuous polynomial splines

Question : is S good for design ? THEOREM B ⇓ Equivalent question : can we find systems of weight functions (wk

1 , . . . , wk n ) on [tk, tk+1], k ∈ Z, such that :

1 for each k, EC(1

Ik, wk

1 , . . . , wk n ) = Pn restricted to [tk, tk+1]

2 for each k, the connection conditions

• Lk

1S(t+ k ), . . . , Lk n−1S(t+ k )

T=

• Lk−1

1

S(t−

k ), . . . , Lk−1 n−1S(tk −)

T and

• S′(t+

k ), . . . , S(n−1)(tk +)

T = Mk

• S′(t−

k ) . . . , S(n−1)(tk −)

T will be equivalent ?

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Geometrically continuous polynomial splines

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Geometrically continuous polynomial splines

ANSWER ACHIEVABLE THANKS TO THEOREM A

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Geometrically continuous polynomial splines of degree 3

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Geometrically continuous polynomial splines of degree 3

connection matrices Mk = ak bk ck

• , ak, ck > 0

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Geometrically continuous polynomial splines of degree 3

connection matrices Mk = ak bk ck

• , ak, ck > 0

Theorem S good for design ⇔ bk + 2(ak + ck) > 0 for all k.

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Geometrically continuous polynomial splines of degree 3

connection matrices Mk = ak bk ck

• , ak, ck > 0

Theorem S good for design ⇔ bk + 2(ak + ck) > 0 for all k. Comparison with TP of the Mk’s : bk ≥ 0

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Geometrically continuous polynomial splines of degree 3

connection matrices Mk = ak bk ck

• , ak, ck > 0

Theorem S good for design ⇔ bk + 2(ak + ck) > 0 for all k. Comparison with TP of the Mk’s : bk ≥ 0 Ex : ak = 1, ck = 9, bk > −20 instead of bk ≥ 0

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Geometrically continuous polynomial splines of degree 4

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0 Théorème S good for design ⇔ for all k ∈ Z :    Bk := bk + 3(ak + ck) > 0 Ek := ek + 4(ck + fk) > 0 −Ak < dk < BkEk

ck

− Ak , Ak := 4bk+3ek+6(ak+2ck+fk).

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0 Théorème S good for design ⇔ for all k ∈ Z :    Bk := bk + 3(ak + ck) > 0 Ek := ek + 4(ck + fk) > 0 −Ak < dk < BkEk

ck

− Ak , Ak := 4bk+3ek+6(ak+2ck+fk). Comparison with TP of the Mk’s :    bk ≥ 0 ek ≥ 0 0 ≤ dk ≤ bkek

ck

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0 Théorème S good for design ⇔ for all k ∈ Z :    Bk := bk + 3(ak + ck) > 0 Ek := ek + 4(ck + fk) > 0 −Ak < dk < BkEk

ck

− Ak , Ak := 4bk+3ek+6(ak+2ck+fk). Comparison with TP of the Mk’s :    bk ≥ 0 ek ≥ 0 0 ≤ dk ≤ bkek

ck

Ex : bk = ek = 0, ak = ck = fk = 1

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0 Théorème S good for design ⇔ for all k ∈ Z :    Bk := bk + 3(ak + ck) > 0 Ek := ek + 4(ck + fk) > 0 −Ak < dk < BkEk

ck

− Ak , Ak := 4bk+3ek+6(ak+2ck+fk). Comparison with TP of the Mk’s :    bk ≥ 0 ek ≥ 0 0 ≤ dk ≤ bkek

ck

Ex : bk = ek = 0, ak = ck = fk = 1

1

TP : dk = 0

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Geometrically continuous polynomial splines of degree 4

connection matrices Mk =   ak bk ck dk ek fk  , ak, ck, fk > 0 Théorème S good for design ⇔ for all k ∈ Z :    Bk := bk + 3(ak + ck) > 0 Ek := ek + 4(ck + fk) > 0 −Ak < dk < BkEk

ck

− Ak , Ak := 4bk+3ek+6(ak+2ck+fk). Comparison with TP of the Mk’s :    bk ≥ 0 ek ≥ 0 0 ≤ dk ≤ bkek

ck

Ex : bk = ek = 0, ak = ck = fk = 1

1

TP : dk = 0

2

NSC of Th B : −24 < dk < 24 ! !

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Geometrically continuous polynomial splines of degree 3

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Geometrically continuous polynomial splines of degree 3

At each knot, connection matrix M = 1 0 b 1

• , with, from left to

right b = 0 ; 4 ; 16.

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Geometrically continuous polynomial splines of degree 3

At each knot, connection matrix M = 1 0 b 1

• , with, from left to

right b = −1 ; −2 ; −3 ; −3, 9.

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Geometrically continuous polynomial splines of degree 3

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Geometrically continuous polynomial splines of degree 3

M = 1 0 b 9

• , b = 0 ; 30.

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Geometrically continuous polynomial splines of degree 3

M = 1 0 b 9

• , b = −5 ; −10 ; −15 ; −19, 9.

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Geometrically continuous polynomial splines of degree 3

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Geometrically continuous polynomial splines of degree 3

M = 9 0 b 1

• , b = 0 ; 30.

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Geometrically continuous polynomial splines of degree 3

M = 9 0 b 1

• , b = −5 ; −10 ; −15 ; −19, 9.

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THE END !

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