C 1 positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular...

JAMCJ Appl Math Comput (2011) 35: 281–293DOI 10.1007/s12190-009-0356-0

C1 positivity preserving scattered data interpolationusing rational Bernstein-Bézier triangular patch

Malik Zawwar Hussain · Maria Hussain

Received: 6 May 2009 / Published online: 4 November 2009© Korean Society for Computational and Applied Mathematics 2009

Abstract A local C1 positivity preserving scheme is developed using Bernstein-Bézier rational cubic function. The domain is triangulated by Delaunay triangulationmethod. Simple sufficient conditions are derived on the inner and boundary Bézierordinates to preserve the shape of positive data. These inner and boundary Bézierordinates involve weights in their definition. In any triangular patch if the Bézier or-dinates do not satisfy the derived conditions of positivity, then these are modified bythe weights (free parameters) involved in the construction of Bernstein-Bézier ratio-nal cubic function to preserve the shape of positive scattered data.

Keywords Positive surface · Scattered data · Bernstein-Bézier rational cubicfunction · Weights

Mathematics Subject Classification (2000) 68U05 · 65D05 · 65D07 · 65D18

1 Introduction

In Computer Graphics and Scientific Visualization the user is always in need of in-terpolatory scheme which possesses certain characteristics like shape preservation,shape control, etc. Ordinary interpolation techniques although smoother are not help-ful for the visualization of shaped data. Positive, monotone and convex are the fun-damental shapes of data. This paper is particularly concerned with the constructionof interpolating surfaces that preserve the shape of positive scattered data.

M.Z. Hussain · M. Hussain (�)Department of Mathematics, University of the Punjab, Lahore, Pakistane-mail: [email protected]

M.Z. Hussaine-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

282 M.Z. Hussain, M. Hussain

Positivity is an important shape property. There are many physical situationswhere entities only have a meaning when their values are positive such as monthlyrainfall amounts, levels of gas discharge in certain chemical reactions, progress of anirreversible process, resistance offered by an electric circuit etc.

In recent years, a good amount of work has been published [1, 2, 9–12] that fo-cuses on shape preserving positive scattered data interpolation. Asim, Mustafa andBrodlie [1] visualized positive scattered data subject to global positivity constraintsusing modified quadratic Shepard method. Positivity was achieved by scaling the ba-sis functions as far as it was necessary. Chan and Ong [2] described a local schemefor range restricted C1 interpolation of scattered data. The interpolating surface is apiecewise convex rational combination of three cubic Bézier patches. Sufficient con-ditions for the non-negativity of a cubic Bézier triangle are derived and used as lowerbounds on the Bézier ordinates. If the Bézier ordinates did not satisfy the derivedlower bounds, then non-negativity can be achieved by modifying the first order par-tial derivatives at data sites. Kong, Ong and Saw [9] discussed the problem of rangerestricted scattered data interpolation using triangles. Each triangle of the Delaunaytriangulated domain is split into three and each sub-triangle is interpolated using acubic Bézier triangular patch. The non-negativity is expressed as lower bounds toBézier ordinates. Mulansky and Schmidt [11] proposed a C1 non-negativity preserv-ing scheme by using a quadratic spline interpolant on the Powell-Sabin refinementof the Delaunay triangulation of the data sites. The non-negativity constraints giverise to a system of linear inequalities with gradients as parameters. The selection ofa suitable solution is based on minimization of thin plate energy functional. Piah,Goodman and Unsworth [12] constructed the interpolating surfaces comprising cu-bic Bézier triangular patches. They imposed sufficient conditions on the ordinates ofthe Bézier control net in each triangle to ensure positivity. The derivatives at the datapoints are determined to be consistent with these conditions. The disadvantage of thescheme is that actual derivatives cannot be supplied. In [9, 12], if the Bézier ordi-nates fail to satisfy derived lower bounds for non-negativity then these are rescaledby modifying first order partial derivatives at data sites to achieve non-negativity. In[2, 9, 11, 12], positivity preserving constraints were derived either on derivatives orpositivity was achieved by modifying the derivatives. So, whilst all these schemesare local, they will not work if the data are given with the derivatives. In [8], Han andSchumaker developed a C1 monotonicity preserving scheme for scattered data.

In this paper, a rational cubic Bernstein-Bézier function is used to preserve theshape of positive scattered data. Simple sufficient conditions are derived on the innerand boundary Bézier ordinates to preserve the shape of positive data. These innerand boundary Bézier ordinates involve weights in their definition. In any triangularpatch if the Bézier ordinates do not satisfy the prescribed lower bounds, then these aremodified by the weights (free parameters) involved in the construction of Bernstein-Bézier rational cubic function to preserve the shape of positive scattered data.

The developed scheme also works if data are given with derivatives. The ratio-nal cubic Bernstein-Bézier function has many advantages over non-rational cubicBernstein-Bézier function such as it has specialty to deal with the singularity trends, itinvolves weight functions in its construction which provides more degree of freedomto the user. These properties make it possible that a rational cubic Bernstein-Bézierfunction can reconstruct all types of conics.

C1 positivity preserving scattered data interpolation using rational 283

Fig. 1 Description of verticesand edges of the �V1V2V3

The organization of the paper is: In Sect. 2, the Bernstein-Bézier rational cubicfunction [4] over a triangular patch used in this paper is rewritten. In Sect. 3, sufficientconditions for the positivity of Bernstein-Bézier rational cubic triangular patch arederived using the technique in [12]. Section 4 describes the method of constructionof positive Bernstein-Bézier rational cubic triangular surface. Section 5 demonstratesthe scheme developed in Sect. 4. Finally Sect. 6 concludes the paper.

2 Bernstein-Bézier rational cubic triangular patch

The Bernstein-Bézier rational cubic triangular patch [4] is defined as:

P(u, v,w) = P1(u, v,w)

P2(u, v,w), (1)

with

P1(u, v,w) =∑

i+j+k=3

uivjwkωijkbijk

= u3ω300b300 + v3ω030b030 + w3ω003b003 + 3u2vω210b210

+ 3uv2ω120b120 + 3u2wω201b201 + 3uw2ω102b102

+ 3v2wω021b021 + 3vw2ω012b012 + 6uvwω111b111, (2)

P2(u, v,w) =∑

i+j+k=3

uivjwkωijk

= u3ω300 + v3ω030 + w3ω003 + 3u2vω210 + 3uv2ω120 + 3u2wω201

+ 3uw2ω102 + 3v2wω021 + 3vw2ω012 + 6uvwω111. (3)

ωijk , i + j + k = 3 are the weight functions attached with the Bézier ordinates bijk ,i + j +k = 3. b300, b030, b003 are the Bézier ordinates at the three vertices of triangle,whereas, b210, b120, b102, b201, b012 and b021 are the boundary Bézier ordinates andb111 is the inner Bézier ordinate. u, v and w are the barycentric coordinates withrespect to the triangle having vertices V1, V2 and V3. Any point V of this triangle canbe expressed in the following convex combination:

V = uV1 + vV2 + wV3, (4)


where u,v and w satisfy the following condition

u + v + w = 1, u > 0, v > 0,w > 0. (5)

3 Positive Bernstein-Bézier rational cubic triangular patch

In this section, sufficient conditions are derived on the ordinates of Bernstein-Bézierrational cubic function defined in (1) to preserve the shape of positive scattered data.From (1), P(u, v,w) > 0 if

P1(u, v,w) > 0 and P2(u, v,w) > 0. (6)

Let

b300 = A, b030 = B, b003 = C (7)

(A > 0, B > 0, C > 0 from the necessary condition of positivity of scattered dataarranged over the triangular grid).

Moreover, all the boundary and inner Bézier ordinates have been assigned thesame value i.e.

b210 = b120 = b201 = b102 = b021 = b012 = b111 = −r, (8)

with r > 0 (to ease the computation).P2(u, v,w) is positive (P2(u, v,w) > 0) if we assign the following values to the

weight functions:

ω300 = a, ω030 = b, ω003 = c, (9)

ω210 = ω120 = ω201 = ω102 = ω021 = ω012 = ω111 = t, (10)

with

a > 0, b > 0, c > 0 and t > 0. (11)

From (3) it is clear that positivity of P1(u, v,w) (P1(u, v,w) > 0) depends upon theBézier ordinates bijk , ∀i, j, k s.t. i + j + k = 3. As b300 > 0, b030 > 0 and b003 > 0,so we shall derive conditions on the remaining Bézier ordinates for P1(u, v,w) > 0.

Substituting the values of Bézier ordinates and weight functions from (7)–(10)in (2), P1(u, v,w) is written as:

P1(u, v,w) = u3aA + v3bB + w3cC

− rt(

3u2v + 3uv2 + 3u2w + 3uw2 + 3v2w + 3vw2 + 6uvw)

.

(12)

By taking the cube of (5) and after some rearrangement we have:

3u2v + 3uv2 + 3u2w + 3uw2 + 3v2w + 3vw2 + 6uvw = 1 − u3 − v3 − w3. (13)


Using (13) in (12), P1(u, v,w) is rewritten as:

P1(u, v,w) = u3(aA + rt) + v3(bB + rt) + w3(cC + rt) − rt. (14)

From (14) it is clear that when r = 0, P1(u, v,w) > 0. As rt increases, P1(u, v,w)

decreases. Assume that r, a, b, c and t are fixed. At the minimum value of P1(u, v,w),the following conditions hold:

∂P1

∂u− ∂P1

∂v= 0,

∂P1

∂u− ∂P1

∂w= 0.

The above conditions can be written as:

∂P1

∂u= ∂P1

∂v= ∂P1

∂w. (15)

From (14)

∂P1

∂u= 3u2(aA + rt),

∂P1

∂v= 3v2(bB + rt),

∂P1

∂w= 3w2(cC + rt). (16)

From (15) and (16) we have

u2

v2= bB + rt

aA + rt, (17)

u2

w2= cC + rt

aA + rt. (18)

From (17) and (18) we obtain the following ratio

u2 : v2 : w2 = 1

aA + rt: 1

Bb + rt: 1

cC + rt(19)

or

u : v : w = 1√aA + rt

: 1√Bb + rt

: 1√cC + rt

. (20)

Using u + v + w = 1 and (20), the values of u,v and w are:

u =1√

aA+rt

1√aA+rt

+ 1√Bb+rt

+ 1√cC+rt

, (21)

v =1√

bB+rt

1√aA+rt

+ 1√Bb+rt

+ 1√cC+rt

, (22)

w =1√

cC+rt

1√aA+rt

+ 1√Bb+rt

+ 1√cC+rt

. (23)


Substituting these values of u,v and w from (21)–(23) in (14), P1(u, v,w) reducesto:

P1(u, v,w) = 1( 1√

aA+rt+ 1√

Bb+rt+ 1√

cC+rt

)2− rt. (24)

It is clear from (24), P1(u, v,w) > 0, when r = 0 and P1(u, v,w) decreases as r

increases. Equation (24) is rewritten as:

P1(u, v,w) = rt

(1

R− 1

), (25)

where

R =(

1√aArt

+ 1+ 1√

bBrt

+ 1+ 1√

cCrt

+ 1

)2

.

Put 1r

= m, m > 0 in (25),

P1(u, v,w) = t

m

(1

R1− 1

), (26)

where

R1 =(

1√aAm

t+ 1

+ 1√bBm

t+ 1

+ 1√cCm

t+ 1

)2

.

Now we are interested to find a root of the equation P1(u, v,w) = 0

P1(u, v,w) = t

m

(1

R1− 1

)= 0

or

F(m) = 1, (27)

where

F(m) = 1√aAm

t+ 1

+ 1√bBm

t+ 1

+ 1√cCm

t+ 1

, m > 0, (28)

F ′(m) = − 1

2t

(aA

(aAm

t+ 1

) 32

+ bB(

bBmt

+ 1) 3

2

+ cC(

cCmt

+ 1) 3

2

), (29)

F ′′(m) = 3

4t2

(a2A2

(aAm

t+ 1

) 52

+ b2B2

(bBm

t+ 1

) 52

+ c2C2

(cCm

t+ 1

) 52

). (30)

As a > 0, b > 0, c > 0, A > 0, B > 0, C > 0, t > 0 and m > 0, so from (29) and (30)we have

F ′(m) < 0 and F ′′(m) > 0. (31)


From (31) it is clear that F(m) is a convex function thus it must have a local minima.To locate the interval in which one of the roots of (27) will lie, we take

X = max

(aA

t,bB

t,cC

t

)and Y = min

(aA

t,bB

t,cC

t

). (32)

It is observed that

3√Xm + 1

≤ F(m) ≤ 3√Ym + 1

. (33)

Put m = 8X

in (33), it takes the form

1 ≤ F(m) ≤ 3√8YX

+ 1, (34)

also by putting m = 8Y

in (33), it takes the form

3√8XY

+ 1≤ F(m) ≤ 1. (35)

From (34) and (35), at least one of the roots of (27) will lie in the interval ( 8X

, 8Y). To

find this root (m0 = 1r0

say) we use the method of False-Position [5].All this discussion leads to the following theorem:

Theorem 1 The rational cubic Bernstein-Bézier triangular patch (1) preserves theshape of positive scattered data, if in each triangular patch

b210 = b120 = b201 = b102 = b021 = b012 = b111 ≥ −r0, (36)

where r0 = 1m0

is a root of (27).

Remark 1 It is significant to observe that the weight functions (free parameters) areused to refine the shape of surfaces.

4 Positive surface interpolation

Given a positive scattered data {(xi, yi,Fi),Fi > 0, i = 1,2, . . . , n}, the constructionfor the positivity preserving interpolating surfaces by using the Bernstein-Bézier ra-tional cubic triangular patch (1), consists of the following steps:

4.1 Triangulation

Delaunay triangulation method [3] is used to triangulate the domain where the datapoints are arranged at the vertices Vi, i = 1,2, . . . , n of the triangles.


4.2 Derivative estimation

The partial derivatives fx(Vi) and fy(Vi) at the vertices (xi, yi), i = 1,2,3, . . . , n oftriangle are estimated by using the method proposed in [7]. These partial derivativesare used to find derivative in the direction of an edge of the triangle e.g. the derivativein the direction of the triangle edge e1 joining (x2, y2) to (x3, y3) is given by

De1S(V2) = (x3 − x2)fx(V2) + (y3 − y2)fy(V2). (37)

4.3 Boundary Bézier ordinates

Boundary Bézier ordinates are calculated by C1 continuity condition at the verticesof triangle. At each vertex there are two directional derivatives in the direction of boththe edges incident at the vertex.

The Bernstein-Bézier rational cubic triangular patch (1) is C1 at the vertices of thetriangle if the boundary Bézier ordinates are

b210 = b300 + ω300

3ω210[(x2 − x1)fx(V1) + (y2 − y1)fy(V1)], (38)

b201 = b300 − ω300

3ω201[(x1 − x3)fx(V1) + (y1 − y3)fy(V1)]. (39)

Similarly

b021 = b030 + ω030

3ω021[(x3 − x2)fx(V2) + (y3 − y2)fy(V2)], (40)

b120 = b030 − ω030

3ω120[(x2 − x1)fx(V2) + (y2 − y1)fy(V2)], (41)

b102 = b003 + ω003

3ω102[(x1 − x3)fx(V3) + (y1 − y3)fy(V3)], (42)

b012 = b003 − ω003

3ω012[(x3 − x2)fx(V3) + (y3 − y2)fy(V3)]. (43)

For an arbitrary value of weight function the boundary Bézier ordinates defined in(38)–(43) may not satisfy the positivity condition proposed in Theorem 1. From (38)–(43), it is clear that the positivity condition proposed in Theorem 1 will be satisfied bythe boundary Bézier ordinates, by increasing the values of the free parameters ω210,ω120, ω201, ω102, ω021 and ω012 for any value of free parameters ω300,ω030 and ω003.In such a way, the derived positivity conditions are satisfied i.e.

b210 ≥ −r0

or

b300 + ω300

3ω210[(x2 − x1)fx(V1) + (y2 − y1)fy(V1)] ≥ −r0.

Hence in this way weight functions are involved in the positivity preservation of data.The free parameters ω210, ω120, ω201, ω102, ω021 and ω012 are local to each triangle,thus the change in the value of free parameters will not affect the C1 continuity at thevertices.


4.4 Inner Bézier ordinates

To satisfy the C1 requirement on all edges of triangle, we determine b111 by a localscheme. We determine bi

111, i = 1,2,3 to satisfy the C1 requirement on the edges ei ,i = 1,2,3 respectively. b111 in (1) is determined by the following convex combina-tion of bi

111, i = 1,2,3:

b111 = C1b1111 + C2b

2111 + C3b

3111, (44)

where

C1 = v2w2

u2v2 + v2w2 + w2u2, C2 = w2u2

u2v2 + v2w2 + w2u2and

C3 = u2v2

u2v2 + v2w2 + w2u2.

(45)

b111 defined in (44) will satisfy the C1 condition on all edges of triangle. To deter-mine the value of bi

111, i = 1,2,3, we use the linear value of normal derivatives ∂P∂ni

,

i = 1,2,3 at the edges ei , i = 1,2,3, where, n1 = (1,− 12 ,− 1

2 ), n2 = (− 12 ,1,− 1

2 )

and n3 = (− 12 ,− 1

2 ,1) are the inward normal directions to the edges ei , i = 1,2,3respectively. (A fuller explanation of inward normal at the edges can be found in [6].)

The derivative of P in the direction λ = (λ1, λ2, λ3) = λ1V1 + λ2V2 + λ3V3, λ1 +λ2 + λ3 = 0 is given by

∂P

∂λ= λ1

∂P

∂u+ λ2

∂P

∂v+ λ3

∂P

∂w. (46)

Using (1) and (46), the normal derivative at edge e3 is computed as:

∂P

∂n3= Au5 + Bu4v + Cu3v2 + Du2v3 + Euv4 + Fv5

(u3ω300 + 3u2vω210 + 3uv2ω120 + v3ω030)2, (47)

where

A = (1.5ω210ω300 − 3ω201ω300)b300 − 1.5ω210ω300b210 + 3ω201ω300b201,

B = (−1.5ω300ω210 + 3ω300ω120 − 6ω300ω111)b300

+ (1.5ω300ω210 − 9ω201ω210)b210 − 3ω300ω120b120 + 9ω201ω210b201

+ 6ω111ω300b111,

C = (−3ω300ω120 + 1.5ω300ω030 − 3ω300ω021)b300

+ (3ω120ω300 − 4.5ω210ω120 − 9ω201ω120)b120

− 1.5ω030ω300b030 + 4.5ω210ω120b210 + 9ω201ω120b201

+ 3ω021ω300b021 + 18ω111ω210b111 − 18ω210ω111b210,


D = (1.5ω300ω030 − 3ω030ω210 − 3ω201ω030)b030

+ (−4.5ω120ω210 + 3ω210ω030 − 9ω210ω021)b210

+ (4.5ω120ω210 − 18ω120ω111)b120 + 3ω201ω030b201 + 9ω021ω210b021

+ 18ω111ω120b111,

E = (3ω030ω210 − 6ω030ω111 − 1.5ω030ω120)b030

+ (1.5ω030ω120 − 9ω120ω021)b120 − 3ω210ω030b210 + 9ω021ω120b021

+ 6ω111ω030b111,

F = −1.5ω120ω030b120 + 3ω021ω030b021 + (1.5ω030ω120 − 3ω030ω021)b030.

From (47), it is observed that the normal derivative will be linear along the edge e3 ifthe following relation holds:

B + C + D + E + F = 31A. (48)

From (48), the value of b3111 for linear normal derivative along the edge e3

b3111 = 31A − A1b300 − A2b210 − A3b120 − A4b201 − A6b030 − A7b021

A5, (49)

where

A1 = ω300(−1.5ω210 − 6ω111 − 3ω021),

A2 = ω210(1.5ω300 − 9ω201 − 18ω111 − 9ω021),

A3 = ω120(−9ω201 − 18ω111 + 1.5ω030 − 9ω021),

A4 = ω201(9ω210 + 9ω120 + 3ω030),

A5 = ω111(6ω300 + 18ω210 + 18ω120 + 6ω030),

A6 = ω030(−3ω201 − 6ω111 − 1.5ω120),

A7 = ω021(3ω300 + 9ω210 + 9ω120).

b1111 and b2

111 are computed by the same procedure. If an inner Bézier ordinate (b1111

or b2111 or b3

111) fails to satisfy the positivity condition proposed in Theorem 1 then itis modified by changing the value of free parameter ω111. The final value of ω111 isthe minimum value for which bi

111 ≥ −r0, i = 1,2,3.

5 Demonstration

In this section, we shall illustrate the positivity preserving interpolating scheme de-veloped in Sects. 3 and 4 using the following test function.


Fig. 2 Delaunay triangulation of domain for F(x, y)

The positive data points are generated from the following function with a slightchange:

F̃ (x, y) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1.0 if (y − x) ≥ 0,

2(y − x) if 0 ≤ (y − x) ≤ 0.5,

cos(4π√

(x−1.5)2+(y−0.5)2)+12 if 1

16 ≥ (x − 1.5)2 + (y − 0.5)2,

0 otherwise

which is quoted from [12]. We have changed the function F̃ (x, y) to F(x, y) in thefollowing way

if F̃ (x, y) �= 0F(x, y) = F̃ (x, y)

elseF(x, y) = 0.0201

end

The domain is restricted to [0,2] × [0,1].The Delaunay triangulation of the domain of the positive scattered data points is

presented in Fig. 2. Figure 3 is the linear interpolation of the function F(x, y). Fig-ure 4 is generated by the positivity preserving scheme developed in Sect. 3 withweight functions ω300 = 0.002, ω030 = 0.003, ω003 = 0.003 and ω120 = ω210 =ω102 = ω201 = ω012 = ω021 = ω111 = 2. It is clear from the Fig. 4, the positive shapeof scattered data generated by function F(x, y) is preserved.


Fig. 3 Linear interpolation of the data generated from F(x, y)

Fig. 4 The positive Bernstein-Bézier rational cubic triangular surface


6 Conclusion

This paper is concerned to the problem of shape preservation of the positive scat-tered data. The domain is triangulated by Delaunay triangulation method with datapoints at the vertices of triangles. The rational cubic Bernstein Bézier function is usedto interpolate the positive scattered data at each vertex of triangle. Simple sufficientconditions are derived on the inner and boundary Bézier ordinates of each triangularpatch to preserve the shape of positive scattered data by using the method proposedin [12]. These sufficient conditions act as lower bound of inner and boundary Bézierordinates. The inner and boundary Bézier ordinates are calculated in such a way thatC1 continuity is assured at the vertices as well as along the edges of triangle. Theseinner and boundary Bézier ordinates involve weights in their definition. In any trian-gular patch if the Bézier ordinates do not satisfy the derived conditions of positivitythen these are modified by changing the weights that involved in the definition ofthese Bézier ordinates, to assure the positivity of triangular patches.

The presented scheme in this paper has many advantageous features over alreadydeveloped schemes [1, 2, 9, 11, 12]. The presented scheme in this paper is local whereas [1] was global. The methods developed in [2, 9, 11, 12] are failed to preserve theshape of the positive scattered data if data are given with derivatives where as thepresented scheme of this paper works if data are given with derivatives.

References

1. Asim, M.R., Mustafa, G., Brodlie, K.: Constrained visualization of 2D positive data using modifiedquadratic Shepard method. In: Proceedings of WSCG, Plzen, Czech Republic, Feb. 2–6, 2004. UnionAgency–Science Press (2004)

2. Chan, E.S., Ong, B.H.: Range restricted scattered data interpolation using convex combination ofcubic Bézier triangles. J. Comput. Appl. Math. 136, 135–147 (2001)

3. Fang, T.P., Piegl, L.A.: Algorithms for Delaunay triangulation and convex-hull computation using asparse matrix. Comput. Aided Des. 24, 425–436 (1992)

4. Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann, San Mateo(2002)

5. Gerald, C.F., Wheatly, P.O.: Applied Numerical Analysis, 6th edn. Addison Wesley, Reading (1998)6. Goodman, T.N.T., Said, H.B.: A C1 triangular interpolant suitable for scattered data interpolation.

Commun. Appl. Numer. Methods 7, 479–485 (1991)7. Goodman, T.N.T., Said, H.B., Chang, L.H.T.: Local derivative estimation for scattered data interpola-

tion. Appl. Math. Comput. 68, 41–50 (1995)8. Han, L.U., Schumaker, L.L.: Fitting monotone surfaces to scattered data using C1 piecewise cubics.

SIAM J. Numer. Anal. 34(2), 569–585 (1997)9. Kong, V.P., Ong, B.H., Saw, K.H.: Range restricted interpolation using cubic Bézier triangles. In:

Proceedings of WSCG, Plzen, Czech Republic, Feb. 2–6, 2004. Union Agency–Science Press (2004)10. Luo, Z., Peng, X.: A C1-rational spline in range restricted interpolation of scattered data. J. Comput.

Appl. Math. 194(2), 255–266 (2006)11. Mulansky, B., Schmidt, J.W.: Nonnegative C1 interpolation of scattered data using Powell-Sabin

splines. MATH-NM-02-1993, Technical University Dresden (1993)12. Piah, A.R.Mt., Goodman, T.N.T., Unsworth, K.: Positivity-preserving scattered data interpolation. In:

Lecture Notes in Computer Science, vol. 3604, pp. 336–349. Springer, Berlin (2005)

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