ATTIVITA DELL’UNIT` A DI SIENA` -...
Transcript of ATTIVITA DELL’UNIT` A DI SIENA` -...
ATTIVITA DELL’UNIT A DI SIENA
Paolo Costantini, Isabella Cravero,Paolo Costantini, Isabella Cravero,
Carla Manni, Francesca Pelosi, Maria Lucia Sampoli
Universita di Siena – Italy
COFIN 2003 – Giornate di lavoro
Roma, 16-17 Dicembre 2004
Nuovi Spazi Funzionali 1
SCHEMA DELLA PRESENTAZIONE
[1] Splines quasi-interpolanti su partizioni Powell-Sabin(C. Manni, P. Sablonniere)
[2] Studio della somma di Minkowski di due superfici parametriche(B. Juttler, M.L. Sampoli et al.)
[3] Elementi triangolari generalizzati(P. Costantini, T. Lyche, C. Manni)
[4] Superfici splines a grado variabile(P. Costantini, F. Pelosi)
[5] *****(F. Pelosi, P. Sablonniere)
Nuovi Spazi Funzionali 2
BIBLIOGRAFIA 2004
P. Costantini, T. Lyche, C. Manni: On a class of weak Tchebycheff systems,preprint, submitted.
P. Costantini, C. Manni: Geometric construction of generalized cubic splines,submitted.
P. Costantini: Properties and applications of new polynomial spaces, to appearon Inter. Jour. Wavel. Mult. Infor.
P. Lamberti, C. Manni: Tensioned quasi-interpolation via geometric continuity,Adv. Comp. Math., 20 (2004), 105-127;
C. Manni, F. Pelosi: Quasi-interpolants with tension properties from and inCAGD, Computing, 72 (2004), 143-160
P. Costantini, F. Pelosi: Shape-preserving approximation of spatial data, Adv.Comp. Math. 20 (2004), 25-51
Nuovi Spazi Funzionali 2
BIBLIOGRAFIA 2004
P. Costantini, C. Manni: Some applications of new spline spaces in computeraided geometric design, to appear on Rendiconti di Matematica,
P. Costantini, F. Pelosi: Data approximation using new spline spaces, preprint(2004), preprint, submitted,
I. Cravero, C. Manni: Detecting Shape of Spatial Data via Zero Moments,preprint, submitted.
R. T. Farouki, C. Y. Han, C. Manni, A. Sestini, Characterization and constructionof helical polynomial space curves, J. Comp. Appl. Math., 162 (2004), 365-392
F. Pelosi, R. T. Farouki, C. Manni, A. Sestini, Geometric Hermite interpolationby spatial Pythagorean–hodograph cubics, preprint, to appear on Adv. Comp.Math.
Nuovi Spazi Funzionali 2
BIBLIOGRAFIA 2004
C. Manni, P. Sablonniere: Quasi-interpolant splines on Powell-Sabin partitions,preprint.
P. Costantini, M.L. Sampoli: A general frame for the construction of constrainedcurves, to appear on Proceedings of the Conference on Applied Mathematicsand Scientific Computing.
M.L. Sampoli, B. Juttler and and M.S. Kim, Minkowski sum of free-formsurfaces, preprint.
M. Peternell, M.L. Sampoli and B. Juttler, LN-surfaces and rational convolutionsurfaces, preprint.
Nuovi Spazi Funzionali 3
QUASI-INTERPOLATION
• quasi-interpolation (q.i.) is a general approach to construct, with lowcomputational cost, efficient local approximants to a given set of data ora given function
Qf :=∑
i
µi(f)Bi
• µi local linear functionals (easy to compute) on f ;
• {Bi} suitable set of functions usually such that:- have compact support,- are non negative,- sum up to 1.
• Let S denote the linear space spanned by the functions {Bi}. In order tocompletely exploit the approximation power of S
Qp = p, ∀ p ∈ IPk ⊂ S.
Nuovi Spazi Funzionali 4
QUASI-INTERPOLATION
• one dimension: various effective q.i. schemes based on values of f, and/orits derivatives, and/or its integrals,... ([Schoenberg, 1946], [de Boor, 1973], [Lyche and
Schumaker, 1975], [Sablonniere, 1989], [Schaback and Zongmin, 1997], [Lee, Lyche, and Schumaker, 2001],
[Gori, Pitolli, and Santi, 2002] ...)
• applications to (constrained) approximation, numerical quadrature.....([Dagnino, Demichelis and Santi, 1993], [Schaback and Zongmin, 1994], [Demichelis, 1996], [Conti, Morandi
and Rabut, 1999], [Manni and Pelosi, 2004] ... )
• two dimensions: effective q.i. schemes have been proposed for
- tensor-product spaces ([Sablonniere,... ], [Lamberti and Manni, 2004],...)
- spline spaces over special partitions ∆1 and ∆2 ([Sablonniere, 1985], [Chui, Diamond
and Raphael, 1989], [Sablonniere, 1996], [Dagnino and Lamberti, 2000], [Sablonniere, 2003], ...)
• few specific results for general triangulations.
Nuovi Spazi Funzionali 5
QUASI-INTERPOLATION
•S(∆PS) := C1 quadratic splines over a PS refinement of ∆
• dim(SPS(∆)) = 3Nv (Nv:= number of vertices of ∆)
• any element of SPS(∆) is determined by its value and its gradient at thevertices of ∆;
• any element of SPS(∆) can be locally computed in each triangle of ∆providing that its values and its gradients at the three vertices are given.
Nuovi Spazi Funzionali 6
QUASI-INTERPOLATION
Let us associate three functions B(j)l , j = 1, 2, 3, l = 1, . . . , Nv to any vertex
of ∆ such that
• s(x, y) =∑Nv
l=1
∑3j=1 ci,jB
(j)l
• B(j)l (x, y) ≥ 0
•∑Nv
l=1
∑3j=1B
(j)l = 1
The functions B(j)l will be the Powell-Sabin B-splines.
The support for any B(j)l is the the union of all triangles containing
the vertex Vl.
Nuovi Spazi Funzionali 7
QUASI-INTERPOLATION
• we are interested in q.i.s not requiring derivatives of f
Qf :=Nv∑l=1
3∑j=1
µ(j)l (f)B(j)
l , µ(j)l
(f):=∑N
(j)l
k=1q(j,k)l
f(Z(j,k)l
), (D.q.i.)
• for N (j)l ≤ 3, we determine the geometric configuration of the evaluation
points Z(j,k)l , k = 1, . . . , N (j)
l in order to ensure reproduction of quadraticpolynomials;
• we give an explicit expression for all the D.q.i.s reproducing IP2 for N (j)l = 2;
• we explicitly construct a family of D.q.i.s reproducing IP2 for N (j)l = 3;
• setting Nj)l = 5 we explicitly construct a family of D.q.i.s reproducing IP2
which are projections that is
Qs = s,∀s ∈ S(∆PS);
Nuovi Spazi Funzionali 8
QUASI-INTERPOLATION
• for all the presented D.q.i.s we provide upper bounds of the infinity norm ofQ in term of the position of the evaluation points Z(j,k)
l and of the family ofused PS B-splines;
• the above mentioned upper bounds ensure that the proposed D.q.i.s providethe optimal (quadratic) approximation order in the space S(∆PS);
• we characterize the used family of PS B-splines which provide D.q.i.s withlower values of the upper bounds of the infinity norm of Q;
• we discuss conditions on the used family of PS B-splines so that theevaluation points Z(j,k)
l can be taken as vertices, or belonging to the edgesof ∆: this makes the proposed D.q.i.s particularly interesting in practicalapplications.
Nuovi Spazi Funzionali 9
QUASI-INTERPOLATIONExample : f(x, y) = exp(x(1− x)(1− y)y(x− y))
−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2
0
0.2
0.4
0.6
0.8
1
Triangulation ∆ and its Powell-Sabin refinement
Nuovi Spazi Funzionali 9
QUASI-INTERPOLATIONExample : f(x, y) = exp(x(1− x)(1− y)y(x− y))
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.985
0.99
0.995
1
1.005
1.01
1.015
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.985
0.99
0.995
1
1.005
1.01
1.015
f Qf, N(j)
l=2,
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.985
0.99
0.995
1
1.005
1.01
1.015
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0.985
0.99
0.995
1
1.005
1.01
1.015
QHf Qf : projection
Nuovi Spazi Funzionali 10
QUASI-INTERPOLATIONExample : f(x, y) = exp(x(1− x)(1− y)y(x− y))
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
f Qf, N(j)
l=2,
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
QHf Qf : projection
Nuovi Spazi Funzionali 10
QUASI-INTERPOLATIONExample : f(x, y) = exp(x(1− x)(1− y)y(x− y))
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Left to right: triangulations ∆(k), k=0,1,2, and their PS refinements
k Nv interp. q.i. N(j)
l=2 q.i. N
(j)
l=3 projection
0 5 0.016997 0.018356 0.016837 0.0135191 13 0.003767 0.003681 0.006568 0.0028562 41 0.000578 0.000573 0.001051 0.0004483 145 0.000078 0.000077 0.000143 0.000093
Error behaviour maxr,s=1,...,50 |f(xr, ys)−Qf(xr, ys)| of different q.i.s
Nuovi Spazi Funzionali 11
SOMME DI MINKOWSKI
La somma di Minkowski di due oggetti e un’operazione geometrica di basenelle teoria degli insiemi.Negli ultimi anni si e visto come questa operazione sia utile anche inmolte applicazioni, come per esempio nella descrizione delle curve e dellesuperfici ottenute con macchine a controllo numerico o nella determinazione ditraiettorie senza collisioni nell’ ambito della robotica.
In generale dati due oggetti P e Q in IR3 la loro somma di Minkowski e datadall’insieme definito come
P ⊕Q := {p+ q, con p ∈ P, q ∈ Q}
dove p e q denotano le coordinate vettoriali di punti arbitrari in P ed in Q.
Nuovi Spazi Funzionali 12
SOMME DI MINKOWSKI
Siano ora A = ∂P e B = ∂Q le superfici della frontiera di P e Qrispettivamente; il problema di determinare la superficie di frontiera dellasomma di Minkowski di P e Q, ∂(P ⊕ Q), puo essere ricondotto a quello dicalcolare la superficie di convoluzione di A e B, di solito indicata con A ? B.
Nell’ipotesi che A e B siano superfici regolari dotate ovunque di un campodi vettori normali nA ed nB rispettivamente, si puo definire la superficie diconvoluzione come
A ? B := {a+ b, con a ∈ A, b ∈ B, e nA(a)‖nB(b)}
dove nA(a) e nB(b) sono i vettori normali alle rispettive superfici in a ed in b esono tra loro paralleli.
Nuovi Spazi Funzionali 13
SOMME DI MINKOWSKI
In particolare si ha che se P e Q sono oggetti convessi, la superficie ∂(P ⊕Q)della somma di Minkowski P ⊕ Q e data esattamente dalla superficie diconvoluzione A ? B.Nel caso generale invece questo non e piu vero. Infatti per due oggettinon convessi la superficie di frontiera ∂(P ⊕ Q) della somma di Minkowskie contenuta nella superficie di convoluzione A ? B.
Studiamo in dettaglio come calcolare la superficie di convoluzione A?B di duesuperfici regolari A e B.Supponiamo che le due superfici siano parametrizzate come a(u, v) e b(s, t)rispettivamente e denotiamo con nA(u, v) e nB(s, t) i relativi vettori normali.
La superficie di convoluzione A?B e data dalla somma di quei punti (vettori) ae b le cui normali nA ed nB sono parallele (ed orientate nello stesso verso).
Nuovi Spazi Funzionali 14
SOMME DI MINKOWSKI
Pertanto il problema cruciale e quello di determinare una riparametrizzazionedi una delle due superfici a(u(s, t), v(s, t)) = a(s, t) in funzione dei parametridella seconda superficie b(s, t) per poter effettuare la somma di quelle parti diA e B per cui si ha che i vettori normali nA(s, t) e nB(s, t) siano paralleli.
In generale non e detto che esista una riparametrizzazione esplicita. Nelleapplicazioni e desiderabile poter descrivere la superficie di convoluzione(la somma di Minkowski) di due superfici parametriche con la stessarappresentazione usata per la descrizione delle superfici di partenza. Si puopero notare che in generale la convoluzione (somma di Minkowski) di duesuperfici parametriche razionali non e razionale.
Nuovi Spazi Funzionali 15
SOMME DI MINKOWSKI
L’offset di una superficie puo essere visto come un caso particolare diconvoluzione.Infatti se A e una sfera di raggio d,e B una superficie arbitraria, allora lasuperficie di convoluzione A ? B e proprio l’offset di B a distanza d.Per gli offset si sono studiate delle classi di superfici per cui si potessedescrivere esattamente gli offset (in forma razionale), (Farouki e Sederberg’95, Patrikalakis ’87, Peternell e Pottman ’96) il problema e quindi vedere sefosse possibile generalizzare queste classi.
Nuovi Spazi Funzionali 16
SOMME DI MINKOWSKI
Si e pertanto studiata una nuova classe di superfici parametriche (le superficiLN) per cui sia possibile dare una forma chiusa per la loro convoluzione(somma di Minkowski) che in particolare risulta essere razionale.
Le superfici LN sono particolari superfici (Juttler ’98, Juttler e S. ’00).Esse hanno la caratteristica di possedere un campo lineare di vettori normali.Tale proprieta, fa si che partendo da superfici polinomiali e possibiledeterminare una riparametrizzazione razionale che permette di esprimere lesuperfici offset in forma razionale.
L’idea e stata quindi quella di estendere tale proprieta al caso generale dellesuperfici di convoluzione.
In un primo lavoro si e studiato il problema della convoluzione (e quindi dellasomma di Minkowski) di due superfici LN.Attualmente si sta studiando il caso ”misto” in cui una superficie siaparametrizzabile in forma razionale mentre la seconda sia una superficie LN.
Nuovi Spazi Funzionali 17
ELEMENTI TRIANGOLARI GENERALIZZATI
[1] Motivations
[2] Recent univariate results
[3] First results on bivariate triangular extensions
[4] Tensor product extensions and their practical applications
Nuovi Spazi Funzionali 17
SKETCH OF THE TALK
[1] Motivations
[2] Recent univariate results
[3] First results on bivariate triangular extensions
[4] Tensor product extensions and their practical applications
Nuovi Spazi Funzionali 17
SKETCH OF THE TALK
[1] Motivations
[2] Recent univariate results (generalization of polynomial spaces)
[3] First results on bivariate triangular extensions
[4] Tensor product extensions and their practical applications
Nuovi Spazi Funzionali 17
SKETCH OF THE TALK
[1] Motivations
[2] Recent univariate results (generalization of polynomial spaces)
[3] First results on bivariate triangular extensions
[4] Tensor product extensions and their practical applications
Nuovi Spazi Funzionali 17
SKETCH OF THE TALK
[1] Motivations
[2] Recent univariate results (generalization of polynomial spaces)
[3] First results on bivariate triangular extensions
[4] Tensor product extensions and their practical applications
Nuovi Spazi Funzionali 18
[1] – Motivations
Why polynomials and polynomial splines are so popular in CAGD?
� Easy to compute and manipulate
� Positive, normalized bases
� Control polygon
Clssical extensions
� Rational (NURBS), exponential
Recent extensions
� Extension of polynomial spaces
�� – More flexibility on the shape
�� – “B ezier” basis and control polygon
Nuovi Spazi Funzionali 18
[1] – Motivations
Why polynomials and polynomial splines are so popular in CAGD?
� Easy to compute and manipulate
� Positive, normalized bases
� Control polygon
Clssical extensions
� Rational (NURBS), exponential
Recent extensions
� Extension of polynomial spaces
�� – More flexibility on the shape
�� – “B ezier” basis and control polygon
Nuovi Spazi Funzionali 18
[1] – Motivations
Why polynomials and polynomial splines are so popular in CAGD?
� Easy to compute and manipulate
� Positive, normalized bases
� Control polygon
Classical extensions
� Rational (NURBS), exponential
Recent extensions
� Extension of polynomial spaces
�� – More flexibility on the shape
�� – “B ezier” basis and control polygon
Nuovi Spazi Funzionali 18
[1] – Motivations
Why polynomials and polynomial splines are so popular in CAGD?
� Easy to compute and manipulate
� Positive, normalized bases
� Control polygon
Classical extensions
� Rational (NURBS), exponential
Recent extensions
� Extension of polynomial spaces
�� – More flexibility on the shape
�� – “Bezier” basis and control polygon
Nuovi Spazi Funzionali 19
[1] – Motivations
Few examples:
span {1, x, cos(x), sin(x)}, x ∈ [0, α], α < π ;
span{1, x, x2, cos(x), sin(x)
}, x ∈ [0, α], α < π ;
span {1, x, (1− x)m0, xm1}, x ∈ [0, 1]
(Carnicer, Mainar, Pena, Chen, Wang, Mazure, Goodman, Kvasov,Sablonniere, Lyche, Manni, Pelosi, Costantini ...)
Nuovi Spazi Funzionali 20
[2] – Recent results
[P. C., T. Lyche, C. Manni: On a Class of Weak Tchebycheff Systems, (2003) submitted.]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2,v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2, v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2, v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2, v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2, v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 21
[2] – Basic ideas
Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2
IPn = span{(
b− t
h
)n
, IPn−2,
(t− a
h
)n}⇓ ⇓ ⇓
IPu,vn := span {u(t), IPn−2, v(t)}
where:
(1) u, v ∈ Cn+1([a, b])
(2) dim (IPu,vn ) = n+ 1
(3) any non-zero element of span{u(n−1), v(n−1)
}. cannot have two distinct zeros in [a, b]
Nuovi Spazi Funzionali 22
[2] – Basic results
“Bernstein” basis for the space IPu,vn
• For ` = 0, . . . , n− 1 there exists unique bu,v` ∈ IPu,v
n such thatD(i)bu,v
` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,
D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1
• There exists unique bu,vn ∈ IPu,v
n such thatD(i)bu,v
n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = 1.
• The elements bu,v` ∈ IPu,v
n , ` = 0, · · · , n are linearly independent .
• It possible to select positive values β` so that• - 1 ≡
∑n`=0 b
u,v`
• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.
(positive, normalized basis)
Nuovi Spazi Funzionali 22
[2] – Basic results
“Bernstein” basis for the space IPu,vn
• For ` = 0, . . . , n− 1 there exists unique bu,v` ∈ IPu,v
n such thatD(i)bu,v
` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,
D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1
• There exists unique bu,vn ∈ IPu,v
n such thatD(i)bu,v
n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.
• The elements bu,v` ∈ IPu,v
n , ` = 0, · · · , n are linearly independent .
• It possible to select positive values β` so that• - 1 ≡
∑n`=0 b
u,v`
• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.
(positive, normalized basis)
Nuovi Spazi Funzionali 22
[2] – Basic results
“Bernstein” basis for the space IPu,vn
• For ` = 0, . . . , n− 1 there exists unique bu,v` ∈ IPu,v
n such thatD(i)bu,v
` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,
D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1
• There exists unique bu,vn ∈ IPu,v
n such thatD(i)bu,v
n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.
• The elements bu,v` ∈ IPu,v
n , ` = 0, · · · , n are linearly independent.
• It possible to select positive values β` so that• - 1 ≡
∑n`=0 b
u,v`
• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.
(positive, normalized basis)
Nuovi Spazi Funzionali 22
[2] – Basic results
“Bernstein” basis for the space IPu,vn
• For ` = 0, . . . , n− 1 there exists unique bu,v` ∈ IPu,v
n such thatD(i)bu,v
` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,
D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1
• There exists unique bu,vn ∈ IPu,v
n such thatD(i)bu,v
n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.
• The elements bu,v` ∈ IPu,v
n , ` = 0, · · · , n are linearly independent.
• It possible to select positive values β` so that• - 1 ≡
∑n`=0 b
u,v`
• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.
(positive, normalized basis)
Nuovi Spazi Funzionali 23
[2] – Basic resultsExample: “Bernstein” basis for IPu,v
3 = span {u(t), IP1, v(t)}, [a, b] = [0, 1]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
u(t) = (1− t)3, v(t) = t3, u(t) =√
(1− t)9, v(t) = t3π
Nuovi Spazi Funzionali 24
[2] – Basic results
B-basis for the space IPu,vn
Theorem.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is strictly totally positive in (a, b) and
totally positive in [a, b].
Corollary.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IP u,v
n .
Remark:all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span
{u(n−1), v(n−1)
}.
Nuovi Spazi Funzionali 24
[2] – Basic results
B-basis for the space IPu,vn
Theorem.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is strictly totally positive in (a, b) and
totally positive in [a, b].
Corollary.
The basis {bu,v` ∈ IPu,v
n , ` = 0, · · · , n} is the normalized B-basis for IP u,vn .
Remark:
all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span
{u(n−1), v(n−1)
}.
Nuovi Spazi Funzionali 24
[2] – Basic results
B-basis for the space IPu,vn
Theorem.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is strictly totally positive in (a, b) and
totally positive in [a, b].
Corollary.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IPu,v
n .
Remark:
all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span
{u(n−1), v(n−1)
}.
Nuovi Spazi Funzionali 24
[2] – Basic results
B-basis for the space IPu,vn
Theorem.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is strictly totally positive in (a, b) and
totally positive in [a, b].
Corollary.The basis {bu,v
` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IPu,v
n .
Remark:all the previous results can be proved in an elementary way by using propertiesof the zeros of elements belonging to span
{u(n−1), v(n−1)
}.
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additionalresults:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(
i− 1n− 2
)α− 2α
, i = 1, · · · ,n− 1 .
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additionalresults:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(
i− 1n− 2
)α− 2α
, i = 1, · · · ,n− 1 .
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(
i− 1n− 2
)α− 2α
, i = 1, · · · ,n− 1 .
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(i− 1n− 2
)α− 2α
, i = 1, · · · , n− 1 .
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(i− 1n− 2
)α− 2α
, i = 1, · · · , n− 1 .
0 1 1/α 1−(1/α)
ξ0 ξ
1
ξ2
ξ3
P3α , α=3π
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(i− 1n− 2
)α− 2α
, i = 1, · · · , n− 1 .
P4α , α=3π
0 1/α 1−1/α 1
ξ0
ξ1 ξ
2 ξ
3ξ
4
Nuovi Spazi Funzionali 25
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
Marsden identity : x =n∑
i=0
ξn,αi bn,α
i ;
ξn,α0 = 0, ξn,α
n = 1; ξn,αi =
1α
+(i− 1n− 2
)α− 2α
, i = 1, · · · , n− 1 .
P6α , α=3π
ξ0 ξ
1 ξ
2 ξ3 ξ
4 ξ
5 ξ
6
0 1/α 1−1/α 1
Nuovi Spazi Funzionali 26
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
IPαn ⊂ IPα
n+1 ⇒ degree elevation
bn,αi = θn
i bn+1,αi +
(1− θn
i+1
)bn+1,α
i+1 , i = 0,1, . . . ,n ,
whereθn0 = 1 ; θn
i =n− in− 1
, i = 1, . . . ,n ; θnn+1 = 0
Nuovi Spazi Funzionali 26
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
IPαn ⊂ IPα
n+1 ⇒ degree elevation
bn,αi = θn
i bn+1,αi +
(1− θn
i+1
)bn+1,α
i+1 , i = 0,1, . . . ,n ,
whereθn0 = 1 ; θn
i =n− in− 1
, i = 1, . . . ,n ; θnn+1 = 0
Nuovi Spazi Funzionali 26
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
IPαn ⊂ IPα
n+1 ⇒ degree elevation
bn,αi = θn
i bn+1,αi +
(1− θn
i+1
)bn+1,αi+1 , i = 0, 1, . . . , n ,
whereθn0 = 1 ; θn
i =n− in− 1
, i = 1, . . . ,n ; θnn+1 = 0
Nuovi Spazi Funzionali 26
[2] – Special caseSpecial case: IPu,v
n = IPαn
[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.
The particular and symmetric structure of the space ⇒ additional results:
IPαn ⊂ IPα
n+1 ⇒ degree elevation
bn,αi = θn
i bn+1,αi +
(1− θn
i+1
)bn+1,αi+1 , i = 0, 1, . . . , n ,
whereθn0 = 1 ; θn
i =n− i
n− 1, i = 1, . . . , n ; θn
n+1 = 0
Nuovi Spazi Funzionali 27
[3] – Triangular extensions
Work in progress
• no general result
• only “cubic” and “quartic” cases
[Joint work with Tom Lyche and Carla Manni]
Nuovi Spazi Funzionali 27
[3] – Triangular extensions
Work in progress
• no general result
• only “cubic” and “quartic” cases
[Joint work with Tom Lyche and Carla Manni]
Nuovi Spazi Funzionali 27
[3] – Triangular extensions
Work in progress
• no general result
• only “cubic” and “quartic” cases
[Joint work with Tom Lyche and Carla Manni]
Nuovi Spazi Funzionali 27
[3] – Triangular extensions
Work in progress
• no general result
• only “cubic” and “quartic” cases
[Joint work with Tom Lyche and Carla Manni]
Nuovi Spazi Funzionali 28
[3] – NotationsStandard triangle T :
P1=(0,0,1) P
2=(1,0,0)
P3=(0,1,0)
e1
e2
e3
Barycentric coordinates r, s, t ; r + s+ t = 1
Nuovi Spazi Funzionali 29
[3] – Notations
space of univariate ν-degree polynomials in ξ, η:
IPν(ξ, η) := span {bνi (ξ, η), i = 0, . . . , ν} , ξ ∈ [0, 1], η = 1− ξ
Bernstein polynomials:
bνi (ξ, η) :=(ν
i
)ξiην−i
space of bivariate polynomials of total degree ν:
IΠν := span{Bν
i,j,k(r, s, t), i+ j + k = ν}
Bernstein polynomials on the standard triangle:
Bνi,j,k(r, s, t) =
ν!i!j!k!
risjtk ; r + s+ t = 1
Nuovi Spazi Funzionali 29
[3] – Notations
space of univariate ν-degree polynomials in ξ, η:
IPν(ξ, η) := span {bνi (ξ, η), i = 0, . . . , ν} , ξ ∈ [0, 1], η = 1− ξ
Bernstein polynomials:
bνi (ξ, η) :=(ν
i
)ξiην−i
space of bivariate polynomials of total degree ν:
IΠν := span{Bν
i,j,k(r, s, t), i+ j + k = ν}
Bernstein polynomials on the standard triangle:
Bνi,j,k(r, s, t) =
ν!i!j!k!
risjtk ; r + s+ t = 1
Nuovi Spazi Funzionali 30
[3] – Basic ideas, “quartic” case
Replace IΠ4 with IΠα4
� Extend IΠ4
- IΠ44 ≡ IΠ4
- dim (IΠα4 ) = dim (IΠ4) = 15
- IΠ4 and IΠα4 should have similar “boundary properties”
� Extend the 1D ideas of IP α4
- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4
}→ IPα
4 (ξ, η) = span {ηα, IP2(ξ, η), ξα}
- As α→∞ , IP2(ξ, η) “covers” [0, 1]
Nuovi Spazi Funzionali 30
[3] – Basic ideas, “quartic” case
Replace IΠ4 with IΠα4
� Extend IΠ4
- IΠ44 ≡ IΠ4
- dim (IΠα4 ) = dim (IΠ4) = 15
- IΠ4 and IΠα4 should have similar “boundary properties”
� Extend the 1D ideas of IP α4
- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4
}→ IPα
4 (ξ, η) = span {ηα, IP2(ξ, η), ξα}
- As α→∞ , IP2(ξ, η) “covers” [0, 1]
Nuovi Spazi Funzionali 30
[3] – Basic ideas, “quartic” case
Replace IΠ4 with IΠα4
� Extend IΠ4
- IΠ44 ≡ IΠ4
- dim (IΠα4 ) = dim (IΠ4) = 15
- IΠ4 and IΠα4 should have similar “boundary properties”
� Extend the 1D ideas of IPα4
- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4
}→ IPα
4 (ξ, η) = span {ηα, IP2(ξ, η), ξα}
- As α→∞ , IP2(ξ, η) “covers” [0, 1]
Nuovi Spazi Funzionali 31
[3] – Basic ideas, “quartic” caseB
0404 =s4
B0044 =t4 B
4004 =r4 B
1034 =4rt3 B
2024 =6r2t2 B
3014 =4r3t
B0134 =4t3s
B0224 =6t2s2
B0314 =4ts3
B3104 =4sr3
B2204 =6s2r2
B1304 =4s3r
B2114 =12r2st
B1214 =12rs2t
B1124 =12rst2
IΠ4 = span{B4
0,0,4, B41,0,3, B
42,0,2, B
43,0,1, B
44,0,0, B
43,1,0, B
42,2,0, B
41,3,0,
B40,4,0, B
40,3,1, B
40,2,2, B
40,1,3, B
41,1,2, B
42,1,1, B
41,2,1
}
Nuovi Spazi Funzionali 32
[3] – Basic ideas, “quartic” case
B0404 =s4
B0044 =t4 B
4004 =r4
First step: “vertex” functions
Nuovi Spazi Funzionali 33
[3] – Basic ideas, “quartic” caseB
0404 =s4
B0044 =t4 B
4004 =r4 B
1034 =4rt3 B
2024 =6r2t2 B301
4 =4r3t
B0134 =4t3s
B0224 =6t2s2
B0314 =4ts3
B3104 =4sr3
B2204 =6s2r2
B1304 =4s3r
Second step: “boundary” functions
Nuovi Spazi Funzionali 34
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
(1) B4,αi,j,0(r, s,0) ∈ IP4(s, r) ≡ span
{r4, IP2(s, r), s4
}(2)
∑i+j=4 B4,α
i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)
(2) B4,αi,j,0(r, s, t) ∈ span
{r4, (r + s)2IP2, s4
}
Nuovi Spazi Funzionali 34
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span
{r4, IP2(s, r), s4
}(2)
∑i+j=4 B4
i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)
(2) B4i,j,0(r, s, t) ∈ span
{r4, (r + s)2IP2, s4
}
Nuovi Spazi Funzionali 34
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span
{r4, IP2(s, r), s4
}(2)
∑i+j=4B
4i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)
(2) W4i,j,0(r, s, t) ∈ span
{r4, (r + s)2IP2, s4
}
Nuovi Spazi Funzionali 34
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span
{r4, IP2(s, r), s4
}(2)
∑i+j=4B
4i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)
(3) B4i,j,0(r, s, t) ∈ span
{r4, (r + s)2IP2(s, r), s4
}
Nuovi Spazi Funzionali 35
[3] – Basic ideas, “quartic” caseB
0404 =s4
B0044 =t4 B
4004 =r4
(s+r)2P2(s,r) (t+s)2P
2(t,s)
(r+t)2P2(r,t)
Second step: Reformulation of the “boundary” functions
Nuovi Spazi Funzionali 36
[3] – Basic ideas, “quartic” caseB
0404 =s4
B0044 =t4 B
4004 =r4 B
1034 =4rt3 B
2024 =6r2t2 B
3014 =4r3t
B0134 =4t3s
B0224 =6t2s2
B0314 =4ts3
B3104 =4sr3
B2204 =6s2r2
B1304 =4s3r
Third step: “internal” functions
Nuovi Spazi Funzionali 36
[3] – Basic ideas, “quartic” caseB
0404 =s4
B0044 =t4 B
4004 =r4 B
1034 =4rt3 B
2024 =6r2t2 B
3014 =4r3t
B0134 =4t3s
B0224 =6t2s2
B0314 =4ts3
B3104 =4sr3
B2204 =6s2r2
B1304 =4s3r
Π1=span{t,r,s}
Third step: “internal” functions
Nuovi Spazi Funzionali 37
[3] – Basic ideas, “quartic” case
Requirement for the first step:- More flexibility in the vertex functions
Nuovi Spazi Funzionali 37
[3] – Basic ideas, “quartic” case
B0404,α =sα
B0044,α =tα B
4004,α =rα
Requirement for the first step:- More flexibility in the vertex functions
Nuovi Spazi Funzionali 38
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0.
Requirements for the boundary functions
(1) B4,αi,j,0(r, s,0) ∈ IPα
4 (s, r) ≡ span {rα, IP2(s, r), sα}
(2)∑
i+j=4 B4,αi,j,0(r, s, t) ≡ (r + s)α (“α− decay′′)
(2) B4,αi,j,0(r, s, t) ∈ span
{r4, (r + s)(α− 2)IP2, s4
}
Nuovi Spazi Funzionali 38
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
Requirements for the boundary functions
(1) B4,αi,j,0(r, s, 0) ∈ IPα
4 (s, r) ≡ span {rα, IP2(s, r), sα}
(2)∑
i+j=4 B4,αi,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)
(2) B4,αi,j,0(r, s, t) ∈ span
{r4, (r + s)2IP2, s4
}
Nuovi Spazi Funzionali 38
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
Requirements for the boundary functions
(1) B4,αi,j,0(r, s, 0) ∈ IPα
4 (s, r) ≡ span {rα, IP2(s, r), sα}
(2)∑
i+j=4B4,αi,j,0(r, s, t) ≡ (s+ r)α (“α− decay′′)
(2) W4,αi,j,0(r, s, t) ∈ span
{r4, (s + r)2IP2, s4
}
Nuovi Spazi Funzionali 38
[3] – Basic ideas, “quartic” case
Take, e.g., the edge e2 : r + s = 1 ↔ t = 0
Requirements for the boundary functions
(1) B4,αi,j,0(r, s, 0) ∈ IPα
4 (s, r) ≡ span {rα, IP2(s, r), sα}
(2)∑
i+j=4B4,αi,j,0(r, s, t) ≡ (s+ r)α (“α− decay′′)
(3) B4,αi,j,0(r, s, t) ∈ span
{rα, (s+ r)α−2IP2(s, r), sα
}
Nuovi Spazi Funzionali 39
[3] – Basic ideas, “quartic” case
B0404,α =sα
(s+r)α−2P2(s,r) (t+s)α−2P
2(t,s)
(r+t)α−2P2(r,t) B
4004,α =rα B
0044,α =tα
Second step: “boundary” functions
Nuovi Spazi Funzionali 40
[3] – Basic ideas, “quartic” case
B0404,α =sα
(s+r)α−2P2(s,r) (t+s)α−2P
2(t,s)
(r+t)α−2P2(r,t) B
4004,α =rα B
0044,α =tα
Π1=span{t,r,s}
Third step: “internal” functions
Nuovi Spazi Funzionali 41
[3] – Triangular extension, “quartic” case
Definition
IΠα4 := span
{tα, (r + t)α−2IP2(r, t), rα, (s+ r)α−2IP2(s, r),
sα, (t+ s)α−2IP2(t, s), IΠ1
}that is
IΠα4 := span
{tα, (t+ r)α−2t2, (t+ r)α−2tr, (t+ r)α−2r2,rα, (r + s)α−2r2, (r + s)α−2rs, (r + s)α−2s2,sα, (s+ t)α−2s2, (s+ t)α−2st, (s+ t)α−2t2,IΠ1}
Theorem
The above functions are linearly independent
Nuovi Spazi Funzionali 42
[3] – Some Hermite problems
Problem H(P1): find ψ ∈ IΠα4 s.t.
ψ(P1) = f1 ; D`(−e1)`ψ(P2) = 0 , ` ≤ 3
D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .
Nuovi Spazi Funzionali 43
[3] – Some Hermite problems
Problem He1(P1): find ψ ∈ IΠα4 s.t.
ψ(P1) = f1 ; De1ψ(P1) = fe11 ; D`
(−e1)`ψ(P2) = 0 , ` ≤ 2
D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .
Nuovi Spazi Funzionali 44
[3] – Some Hermite problems
Problem H(e1)2(P1): find ψ ∈ IΠα
4 s.t.
ψ(P1) = f1 ; De1ψ(P1) = fe11 ; D2
(e1)2ψ(P1) = f
(e1)2
1 ; D`(−e1)`ψ(P2) = 0 , ` ≤ 1 ;
D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .
Nuovi Spazi Funzionali 45
[3] – Some Hermite problems
Problem H(e1)3(P1): find ψ ∈ IΠα
4 s.t.
ψ(P1) = fν ; De1ψ(P1) = fe11 ; D2
(e1)2ψ(P1) = f
e21
1 ; D3(e1)3
ψ(P1) = fe31
1 ;
ψ(P2) = 0 ; D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .
Nuovi Spazi Funzionali 46
[3] – Some Hermite problems
Problem He1(−e2)(P1): find ψ ∈ IΠα4 s.t.
ψ(P1) = f1 ; D`+r(e1)`(−e2)rψ(P1) = f
(e1)`(−e2)
r
1 ; `+ r = 1, 2 ;D`
(e2)`ψ(P2) = D`(−e2)`ψ(P3) = 0, ` = 0, 1, 2;D2
(−e1)e2ψ(P2) = D2
e3(−e2)ψ(P3) = 0 .
Nuovi Spazi Funzionali 47
[3] – Some Hermite problems
Similar Hermite problems can be “centered” in P2 and P3.
TheoremThere exists an unique solution to each of the above Hermite problems.
Nuovi Spazi Funzionali 47
[3] – Some Hermite problems
Similar Hermite problems can be “centered” in P2 and P3.
TheoremThere exists an unique solution to each of the above Hermite problems.
Nuovi Spazi Funzionali 48
[3] – B ezier basis
Theorem
It is possible to find values f∗∗ such that the solutions of the properinterpolation problems produce linearly independent functions
B4,αi,j,k ; i+ j + k = 4
such that:(1) B4,4
i,j,k = B4i,j,k ;
(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4
i,j,k(r, s, t) ≥ 0 ;
(3)∑
i+j+k=4B4,αi,j,k ≡ 1
Nuovi Spazi Funzionali 48
[3] – B ezier basis
Theorem
It is possible to find values f∗∗ such that the solutions of the properinterpolation problems produce linearly independent functions
B4,αi,j,k ; i+ j + k = 4
such that:(1) B4,4
i,j,k = B4i,j,k ;
(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4
i,j,k(r, s, t) ≥ 0 ;
(3)∑
i+j+k=4 B4,αi,j,k ≡ 1
Nuovi Spazi Funzionali 48
[3] – B ezier basis
Theorem
It is possible to find values f∗∗ such that the solutions of the properinterpolation problems produce linearly independent functions
B4,αi,j,k ; i+ j + k = 4
such that:(1) B4,4
i,j,k = B4i,j,k ;
(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4
i,j,k(r, s, t) ≥ 0 ;
(3) sumi+j+k=4B4,αi,j,k ≡ 1
Nuovi Spazi Funzionali 48
[3] – B ezier basis
Theorem
It is possible to find values f∗∗ such that the solutions of the properinterpolation problems produce linearly independent functions
B4,αi,j,k ; i+ j + k = 4
such that:(1) B4,4
i,j,k = B4i,j,k ;
(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4
i,j,k(r, s, t) ≥ 0 ;
(3)∑
i+j+k=4B4,αi,j,k ≡ 1
Nuovi Spazi Funzionali 49
[3] – B ezier basis
Theorem
B4,αi,j,0(r, s, 0) = b4,α
i (s, r)
(On the edge e2 the basis functions of IΠα4 and IPα
4 (r, s) coincide)
Nuovi Spazi Funzionali 49
[3] – B ezier basis
Theorem
B4,αi,j,0(r, s, 0) = b4,α
i (s, r)
(On the edge e2 the basis functions of IΠα4 and IPα
4 (r, s) coincide)
Nuovi Spazi Funzionali 50
[3] – B ezier basis
B4,α0,0,4 : H(P1) with f1 = 1.
Nuovi Spazi Funzionali 51
[3] – B ezier basis
B4,α1,0,3: He1(P1) with f1 = 0, fe1
1 = −De1B4,α0,0,4(P1) .
Nuovi Spazi Funzionali 51
[3] – B ezier basis
B4,α1,0,3 example: α = 4
Nuovi Spazi Funzionali 51
[3] – B ezier basis
B4,α1,0,3 example: α = 4 + π
Nuovi Spazi Funzionali 51
[3] – B ezier basis
B4,α1,0,3 example: α = 8 + e
Nuovi Spazi Funzionali 52
[3] – B ezier basis
B4,α2,0,2 : H(e1)2
(P1) with f1 = 0 ;
f(e1)
`
1 = −D`(e1)`(B
4,α0,0,4(P1) +B4,α
1,0,3(P1)) ; ` = 1, 2.
Nuovi Spazi Funzionali 53
[3] – B ezier basis
B4,α1,1,2 : He1(−e3)(P1) with f1 = 0 ;
fe`1(−e3)
m
1 = −D`+m
e`1(−e3)m
∑2k=4B
4,αi,j,k(P1) ; 1 ≤ `+m ≤ 2
Nuovi Spazi Funzionali 53
[3] – B ezier basis
B4,α1,1,2 example: α = 4
Nuovi Spazi Funzionali 53
[3] – B ezier basis
B4,α1,1,2 example: α = 4 + π
Nuovi Spazi Funzionali 53
[3] – B ezier basis
B4,α1,1,2 example: α = 8 + e
Nuovi Spazi Funzionali 54
[3] – Control polygon
t ∈ IΠα4 ⇔ t =
∑i+j+k=4
ξ(t)i,j,kB
4,αi,j,k(r, s, t) .
Theoremξ(t)0,0,4 = 1 ; ξ(t)1,0,3 = ξ
(t)0,1,3 = 1− 1
α; ξ(t)1,1,2 = 1− 2
α
ξ(t)2,0,2 = ξ
(t)0,2,2 =
12
; ξ(t)3,0,1 = ξ(t)2,1,1 = ξ
(t)1,2,1 = ξ
(t)0,3,1 =
1α
;
ξ(t)4,0,0 = ξ
(t)3,1,0 = ξ
(t)2,2,0 = ξ
(t)1,3,0 = ξ
(t)0,4,0 = 0 .
Nuovi Spazi Funzionali 54
[3] – Control polygon
t ∈ IΠα4 ⇔ t =
∑i+j+k=4
ξ(t)i,j,kB
4,αi,j,k(r, s, t) .
Theoremξ(t)0,0,4 = 1 ; ξ(t)1,0,3 = ξ
(t)0,1,3 = 1− 1
α; ξ(t)1,1,2 = 1− 2
α
ξ(t)2,0,2 = ξ
(t)0,2,2 =
12
; ξ(t)3,0,1 = ξ(t)2,1,1 = ξ
(t)1,2,1 = ξ
(t)0,3,1 =
1α
;
ξ(t)4,0,0 = ξ
(t)3,1,0 = ξ
(t)2,2,0 = ξ
(t)1,3,0 = ξ
(t)0,4,0 = 0 .
Nuovi Spazi Funzionali 55
[3] – Control polygonSimilar representations hold for r ∈ IΠα
4 and s ∈ IΠα4
ψ =∑
i+j+k=4
βi,j,kB4,αi,j,k
the piecewise linear function L3,α connecting the coefficients βi,j,k at thepoints Qi,j,k given by
Q0,0,4 = (0, 0, 1), Q1,0,3 =
(1
α, 0, 1−
1
α
), Q2,0,2 =
(1
2, 0,
1
2
), Q3,0,1 =
(1−
1
α, 0,
1
α
),
Q4,0,0 = (1, 0, 0), Q3,1,0 =
(1−
1
α,1
α, 0
), Q2,2,0 =
(1
2,1
2, 0
), Q1,3,0 =
(1
α, 1−
1
α, 0
),
Q0,4,0 = (0, 1, 0), Q0,3,1 =
(0, 1−
1
α,1
α
), Q0,2,2 =
(0,
1
2,1
2
), Q0,1,3 =
(0,
1
α, 1−
1
α
),
Q1,1,2 =
(1
α,1
α, 1−
2
α
), Q2,1,1 =
(1−
2
α,1
α,1
α
), Q1,2,1 =
(1
α, 1−
2
α,1
α
)is the control polygon with the usual end tangent properties.
Nuovi Spazi Funzionali 55
[3] – Control polygonSimilar representations hold for r ∈ IΠα
4 and s ∈ IΠα4
ψ =∑
i+j+k=4
βi,j,kB4,αi,j,k
the piecewise linear function L3,α connecting the coefficients βi,j,k at thepoints Qi,j,k given by
Q0,0,4 = (0, 0, 1), Q1,0,3 =
(1
α, 0, 1−
1
α
), Q2,0,2 =
(1
2, 0,
1
2
), Q3,0,1 =
(1−
1
α, 0,
1
α
),
Q4,0,0 = (1, 0, 0), Q3,1,0 =
(1−
1
α,1
α, 0
), Q2,2,0 =
(1
2,1
2, 0
), Q1,3,0 =
(1
α, 1−
1
α, 0
),
Q0,4,0 = (0, 1, 0), Q0,3,1 =
(0, 1−
1
α,1
α
), Q0,2,2 =
(0,
1
2,1
2
), Q0,1,3 =
(0,
1
α, 1−
1
α
),
Q1,1,2 =
(1
α,1
α, 1−
2
α
), Q2,1,1 =
(1−
2
α,1
α,1
α
), Q1,2,1 =
(1
α, 1−
2
α,1
α
)is the control polygon with the usual end tangent properties.
Nuovi Spazi Funzionali 55
[3] – Control polygonSimilar representations hold for r ∈ IΠα
4 and s ∈ IΠα4
ψ =∑
i+j+k=4
βi,j,kB4,αi,j,k
the piecewise linear function L4,α connecting the coefficients βi,j,k at thepoints Qi,j,k given by
Q0,0,4 = (0, 0, 1), Q1,0,3 =
(1
α, 0, 1−
1
α
), Q2,0,2 =
(1
2, 0,
1
2
), Q3,0,1 =
(1−
1
α, 0,
1
α
),
Q4,0,0 = (1, 0, 0), Q3,1,0 =
(1−
1
α,1
α, 0
), Q2,2,0 =
(1
2,1
2, 0
), Q1,3,0 =
(1
α, 1−
1
α, 0
),
Q0,4,0 = (0, 1, 0), Q0,3,1 =
(0, 1−
1
α,1
α
), Q0,2,2 =
(0,
1
2,1
2
), Q0,1,3 =
(0,
1
α, 1−
1
α
),
Q1,1,2 =
(1
α,1
α, 1−
2
α
), Q2,1,1 =
(1−
2
α,1
α,1
α
), Q1,2,1 =
(1
α, 1−
2
α,1
α
)is the control polygon with the usual end tangent properties.
Nuovi Spazi Funzionali 56
[3] – Control polygon: example
α = 4
Nuovi Spazi Funzionali 56
[3] – Control polygon: example
α = 5
Nuovi Spazi Funzionali 56
[3] – Control polygon: example
α = 7.1416
Nuovi Spazi Funzionali 56
[3] – Control polygon: example
α = 10.7183
Nuovi Spazi Funzionali 57
[3] – Triangular extension, examples
α = 4
Nuovi Spazi Funzionali 57
[3] – Triangular extension, examples
α = 4 + π
Nuovi Spazi Funzionali 57
[3] – Triangular extension, examples
α = 8 + e
Nuovi Spazi Funzionali 58
[3] – Triangular extension, “cubic” case
Similar construction
tα rα
sα
(r+t)α−1P1(r,t)
(s+r)α−1P1(s,r)
(t+s)α−1P1(t+s)
Π0=span{1}
IΠ3 → IΠα3
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
Nuovi Spazi Funzionali 59
[3] – Triangular extension, comments
� Extension of cubic and quartic bivariate polynomials� IΠα
3 → IΠ0 , IΠα4 → IΠ1
� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα
4
� • in practical tension schemes)� • IΠα
n
tα rα
sα
(r+t)βPn−2
(r,t)
(s+r)βPn−2
(s,r) (t+s)βPn−2
(t+s)
Πn−3
β=α−(n−2)
Nuovi Spazi Funzionali 60
[4] – Tensor-product extensions
[Based on joint works with Francesca Pelosi (2000-2004)]
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 2IPk0,k1
2 := span{(1− x)k0, IP0, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP0 ; ψ ∈ IPk0,k12 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k12 → IP0
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 2IPk0,k1
2 := span{(1− x)k0, IP0, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP0 ; ψ ∈ IPk0,k12 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k12 → IP0
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 2IPk0,k1
2 := span{(1− x)k0, IP0, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP0 ; ψ ∈ IPk0,k12 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k12 → IP0
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 2IPk0,k1
2 := span{(1− x)k0, IP0, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP0 ; ψ ∈ IPk0,k12 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k12 → IP0
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 3IPk0,k1
3 := span{(1− x)k0, IP1, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP1 ; ψ ∈ IPk0,k13 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k13 → IP0
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 3IPk0,k1
3 := span{(1− x)k0, IP1, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP1 ; ψ ∈ IPk0,k13 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k13 → IP1
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 4IPk0,k1
4 := span{(1− x)k0, IP2, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP2 ; ψ ∈ IPk0,k14 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k14 → IP2
)
Nuovi Spazi Funzionali 61
[4] – Basic univariate results
Particular case: IP u,vn = IPk0,k1
n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.
u(x) := (1− x)k0 , v(x) := xk1
Case n = 4IPk0,k1
4 := span{(1− x)k0, IP2, x
k1}
x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP2 ; ψ ∈ IPk0,k14 , ψ(x) = a0(1− x)k0 + p(x) + a1x
k1
limk0,k1→∞
ψ(x) = p(x)(
IPk0,k14 → IP2
)
Nuovi Spazi Funzionali 62
[4] – Basic univariate resultsSPLINE SPACES
[possibly extended] knot sequence
[· · · < r−2 < r−1 <]r0 < r1 < · · · < rN [< rN+1 < rN+2 < · · ·]
[possibly extended] degree sequence
[· · · < k−2 < k−1 <]k0 < k1 < · · · < kN [< kN+1 < kN+2 < · · ·]
IPki,ki+1n,i := span
{(1− ρ)ki, IPn−2, ρ
ki+1}, ρ =
r − riri+1 − ri
GENERAL SPLINE SPACES
VSkn,r =
{s ∈ Cm[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
n,i
}
Nuovi Spazi Funzionali 63
[4] – “Quartic” splines
n = 4 ,m = 2 (the case m = 3 is also possible)
VSk4,r =
{s ∈ C2[r0, rN] : s|[ri,ri+1] ∈ IPki,ki+1
4,i
}
• For limit values of the degrees: k0, . . . ,kN →∞
VSk4,r → S0
2
• For technical reasons in data approximation
IPk0,k14,0 = span
{IP2, rk1
}, IPkN−1,kN
4,N−1 = span{(1− r)kN−1, IP2
}so that
dim(VSk
4,r
)= dim S0
2
Nuovi Spazi Funzionali 63
[4] – “Quartic” splines
n = 4 ,m = 2 (the case m = 3 is also possible)
VSk4,r =
{s ∈ C2[r0, rN] : s|[ri,ri+1] ∈ IPki,ki+1
4,i
}
• For limit values of the degrees: k0, . . . ,kN →∞
VSk4,r → S0
2
• For technical reasons in data approximation
IPk0,k14,0 = span
{IP2, rk1
}, IPkN−1,kN
4,N−1 = span{(1− r)kN−1, IP2
}so that
dim(VSk
4,r
)= dim S0
2
Nuovi Spazi Funzionali 63
[4] – “Quartic” splines
n = 4 ,m = 2 (the case m = 3 is also possible)
VSk4,r =
{s ∈ C2[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
4,i
}
• For limit values of the degrees: k0, . . . ,kN →∞
VSk4,r → S0
2
• For technical reasons in data approximation
IPk0,k14,0 = span
{IP2, rk1
}, IPkN−1,kN
4,N−1 = span{(1− r)kN−1, IP2
}so that
dim(VSk
4,r
)= dim S0
2
Nuovi Spazi Funzionali 63
[4] – “Quartic” splines
n = 4 ,m = 2 (the case m = 3 is also possible)
VSk4,r =
{s ∈ C2[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
4,i
}
• For limit values of the degrees: k0, . . . , kN →∞
VSk4,r → S0
2
• For technical reasons in data approximation
IPk0,k14,0 = span
{IP2, rk1
}, IPkN−1,kN
4,N−1 = span{(1− r)kN−1, IP2
}so that
dim(VSk
4,r
)= dim S0
2
Nuovi Spazi Funzionali 63
[4] – “Quartic” splines
n = 4 ,m = 2 (the case m = 3 is also possible)
VSk4,r =
{s ∈ C2[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
4,i
}
• For limit values of the degrees: k0, . . . , kN →∞
VSk4,r → S0
2
• For technical reasons in data approximation
IPk0,k14,0 = span
{IP2, r
k1}, IPkN−1,kN
4,N−1 = span{(1− r)kN−1, IP2
}• so that
dim(VSk
4,r
)= dim S0
2
Nuovi Spazi Funzionali 64
[4] – “Quartic” splines: geometric construction
GEOMETRIC CONSTRUCTION: the polynomial control points areobtained via convex combinations of the spline control points
Di−1
Di D
i+1
Di+2
bi,0
bi,1
bi,2
bi,3
o
o
o
o
o o o
o
hi−1
hi h
i+1
hi
hi−1
hi
hi+1
hi h
i
o o
Classical example: C2 cubic splines
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si
Si+1
Si+2
Di
Di−1
D
i+1
Si−1
Example: ki = ki+1 = 6
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si−1
Si
Si+1
Si+2
Di
Di−1
D
i+1
fi+ f
i+1−
fi−1+
fi−
fi+1+
fi+2−
αi
βi
γi
δi
Example: ki = ki+1 = 6
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si−1
Si
Si+1
Si+2
Di
Di−1
D
i+1
fi+ f
i+1−
fi−1+
fi−
fi+1+
fi+2−
αi
βi
γi
δi
bi,1
bi,2
bi,4
bi,5
o
o
o
o
Example: ki = ki+1 = 6
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si−1
Si
Si+1
Si+2
Di
Di−1
D
i+1
fi+ f
i+1−
fi−1+
fi−
fi+1+
fi+2−
αi
βi
γi
δi
bi,0
bi,1
bi,2
bi,4
bi,5
bi,6
o
o
o
o
o
o
Example: ki = ki+1 = 6
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si−1
Si
Si+1
Si+2
Di
Di−1
D
i+1
fi+ f
i+1−
fi−1+
fi−
fi+1+
fi+2−
αi
βi
γi
δi
bi,0
bi,1
bi,2
bi,3
bi,4
bi,5
bi,6
o
o
o o
o
o
o
Example: ki = ki+1 = 6
REMARK: approximately a quadratic computational cost
Nuovi Spazi Funzionali 65
[4] – “Quartic” splines: geometric construction
FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r
Si−1
Si
Si+1
Si+2
Di
Di−1
D
i+1
fi+ f
i+1−
fi−1+
fi−
fi+1+
fi+2−
αi
βi
γi
δi
bi,0
bi,1
bi,2
bi,3
bi,4
bi,5
bi,6
o
o
o o
o
o
o
Example: ki = ki+1 = 6
REMARK: approximately the computational cost of quadratic splines .
Nuovi Spazi Funzionali 66
[4] – “Quartic” splines: geometric construction
WHY GEOMETRIC CONSTRUCTION IS SO IMPORTANT ?
⇒ the spline shape can be controlled using the de Boor control polygon
⇒ the asymptotic properties of the spline can be deduced from those ofthe de Boor control polygon
⇒ tool for the construction of a B-spline basis
→ applications to least squares approximation→ extensions to tensor-product and Boolean sum parametric surfaces
Nuovi Spazi Funzionali 66
[4] – “Quartic” splines: geometric construction
WHY GEOMETRIC CONSTRUCTION IS SO IMPORTANT ?
⇒ the spline shape can be controlled using the de Boor control polygon
⇒ the asymptotic properties of the spline can be deduced from those ofthe de Boor control polygon
⇒ tool for the construction of a B-spline basis
→ applications to least squares approximation→ extensions to tensor-product and Boolean sum parametric surfaces
Nuovi Spazi Funzionali 66
[4] – “Quartic” splines: geometric construction
WHY GEOMETRIC CONSTRUCTION IS SO IMPORTANT ?
⇒ the spline shape can be controlled using the de Boor control polygon
⇒ the asymptotic properties of the spline can be deduced from those ofthe de Boor control polygon
⇒ tool for the construction of a B-spline basis
→ applications to least squares approximation→ extensions to tensor-product and Boolean sum parametric surfaces
Nuovi Spazi Funzionali 66
[4] – “Quartic” splines: geometric construction
WHY GEOMETRIC CONSTRUCTION IS SO IMPORTANT ?
⇒ the spline shape can be controlled using the de Boor control polygon
⇒ the asymptotic properties of the spline can be deduced from those ofthe de Boor control polygon
⇒ tool for the construction of a B-spline basis
→ applications to least squares approximation→ extensions to tensor-product and Boolean sum parametric surfaces
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
k = {4, 4, 4, 4, 4, 4, 4, 4}
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
k = {4, 4, 4, 10, 10, 4, 4, 4}
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
k = {10, 10, 10, 10, 10, 10, 10, 10}
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
k = {40, 40, 40, 40, 40, 40, 40, 40}
Nuovi Spazi Funzionali 67
[4] – “Quartic” splines: B-splines
s ∈ VSk4,r, s =
∑N1 DiN
Di +
∑N0 SiN
Si
C0 Quadratic B-spline Basis
Nuovi Spazi Funzionali 68
[4] – “Quartic” splines: tensor-product
• r = {r0, . . . , rN}, kr = {kr0, . . . , k
rN}
• s = {s0, . . . , sM}, ks = {ks0, . . . , k
sM}
VSkr
4,r⊗
VSks
4,s
• Classical geometric construction
• Surface with the same shape of the De Boor control polygon
Nuovi Spazi Funzionali 69
[4] – “Quartic” splines: tensor-product
• B–spline basis
NDi N
Dj : kr
i = ksj = 4 kr
2 = kr3 = 40 kr
2 = kr3 = ks
2 = ks3 = 40
NSi N
Sj : kr
i = ksj = 4 kr
2 = kr3 = 40 kr
2 = kr3 = ks
2 = ks3 = 40
Nuovi Spazi Funzionali 70
[4] – “Quartic” splines: surface approximation
• Data:{(ui, vj, Ii,j); i = 1, . . . , Nu ; j = 1, . . . ,Mv},
BASIC ASSUMPTIONS
• The object can be approximated by a tensor product surface
• The shape of the object is “simple”, weak noise ⇒ it can be describedby biquadratic patches
• The parameter values (ui, vj) are given
Nuovi Spazi Funzionali 71
[4] – “Quartic” splines: surface approximation
(1) Extract significant knot sequences from the parameter valuesr = {r0, . . . , rN} ⊂ {u0, . . . , uNu} , s = {s0, . . . , sM} ⊂ {v0, . . . , vMv}
(2) Compute σ∗ ∈ S02
⊗S0
2 s.t.
Nu∑i=1
Mv∑j=1
‖σ∗(ui, vj)− Ii,j‖2 <Nu∑i=1
Mv∑j=1
‖σ(ui, vj)− Ii,j‖2, ∀σ
(3) Define the shape of data as the shape of σ∗
(4) Compute s∗ ∈ VSkr
4,r⊗
VSks
4,s s.t.
Nu∑i=1
Mv∑j=1
‖s∗(ui, vj)− Ii,j‖2 <Nu∑i=1
Mv∑j=1
‖s(ui, vj)− Ii,j‖2, ∀s
Nuovi Spazi Funzionali 71
[4] – “Quartic” splines: surface approximation
(5) Convergence Result:
- If kri , k
sj →∞ for all indices,
=⇒ ‖s∗ − σ∗‖∞ −→ 0
- For sufficient large degree sequences kr and ks,
=⇒ s∗ ∈ VSkr
4,r⊗
VSks
4,s has the same shape of σ∗
(6) Find kr and ks
• s∗ reproduces the convexity of σ∗ (“zero moment” approach)• Check on the fairness properties
Nuovi Spazi Funzionali 72
[4] – “Quartic” splines: example
Data points Grid lines selection
Nuovi Spazi Funzionali 73
[4] – “Quartic” splines: example
C0 Biquadratic Surface kr2 = 10, kr
i = 4, ksj = 10
⇒ Local shape control using weighted least squares
Nuovi Spazi Funzionali 74
[4] – “Quartic” splines: example
C0 Biquadratic Surface kr2 = 10, kr
i = 4, ksj = 10
⇒ Local shape control using weighted least squares
Nuovi Spazi Funzionali 75
[4] – “Quadratic” splinesn = 2,m = 1 VSk
2,r ={s ∈ C1[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
2,i
}B-spline Basis, tensor-product extension
Useful for the reproduction of density functions from histograms• area/volume matching conditions• co-monotonicity
Nuovi Spazi Funzionali 75
[4] – “Quadratic” splinesn = 2,m = 1 VSk
2,r ={s ∈ C1[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
2,i
}B-spline Basis, tensor-product extension
Useful for the reproduction of density functions from histograms• area/volume matching conditions• co-monotonicity
0 5 10 15−20
0
20
40
60
80
100
120
140
0 5 10 15−20
0
20
40
60
80
100
120
140
Nuovi Spazi Funzionali 75
[4] – “Quadratic” splinesn = 2,m = 1 VSk
2,r ={s ∈ C1[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
2,i
}B-spline Basis, tensor-product extension
Useful for the reproduction of density functions from histograms• area/volume matching conditions• co-monotonicity
BRONCHUSSTOMACH
COLONOVARY
BREAST
0
5
10
150
500
1000
1500
2000
2500
3000
3500
4000
SURVIVAL TIME(days)
01
23
45
02
46
810
1214
−500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Nuovi Spazi Funzionali 75
[4] – “Quadratic” splinesn = 2,m = 1 VSk
2,r ={s ∈ C1[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
2,i
}B-spline Basis, tensor-product extension
Useful for the reproduction of density functions from histograms• area/volume matching conditions• co-monotonicity
BRONCHUSSTOMACH
COLONOVARY
BREAST
0
5
10
150
500
1000
1500
2000
2500
3000
3500
4000
SURVIVAL TIME(days)
01
23
45
02
46
810
1214
−500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Nuovi Spazi Funzionali 76
[4] – “Cubic” splines
n = 3,m = 2 VSk3,r =
{s ∈ C2[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1
3,i
}B-spline Basis, tensor-product extension
All the properties and applications of cubic splines• interpolation, approximation, free-form design• NURBS