Mathematical and Computational Applications
Article The Cubic α-Catmull-Rom Spline
Juncheng Li 1,* and Sheng Chen 2
1 College of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China 2 College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China; [email protected] * Correspondence: [email protected]; Tel.: +86-738-832-5585
Academic Editor: Gözde Sarı Received: 6 May 2016; Accepted: 1 August 2016; Published: 9 August 2016
Abstract: By extending the definition interval of the standard cubic Catmull-Rom spline basis functions from [0,1] to [0,α], a class of cubic Catmull-Rom spline basis functions with a shape parameter α, named cubic α-Catmull-Rom spline basis functions, is constructed. Then, the corresponding cubic α-Catmull-Rom spline curves are generated based on the introduced basis functions. The cubic α-Catmull-Rom spline curves not only have the same properties as the standard cubic Catmull-Rom spline curves, but also can be adjusted by altering the value of the shape parameter α even if the control points are fixed. Furthermore, the cubic α-Catmull-Rom spline interpolation function is discussed, and a method for determining the optimal interpolation function is presented.
Keywords: Catmull-Rom spline; interpolation spline; shape parameter; shape adjustment
1. Introduction With the development of geometric design industry, the shapes of curves often need to be changed freely. Hence, the curves with shape parameters have been paid more and more attention by many researchers in geometric modeling. Some examples are the Bézier-like curves with shape parameters [1–5], the B-spline-like curves with shape parameters [6–14], the trigonometric curves with shape parameters [15–21], and so on. Curves with shape parameters not only inherit similar or the same properties as the corresponding classical curves, but also have better performance ability because of the shape parameters. As an interpolation spline, the cubic Catmull-Rom spline [22] has been widely used in geometric design [23,24] and engineering applications [25,26] because it can automatically interpolate the given control points without solving equation systems. Generally, shapes of the standard cubic Catmull-Rom spline would be modified if the control points are changed, and the control points can be adapted during a learning procedure [25,26]. However, shapes of the standard cubic Catmull-Rom spline cannot be adjusted when the control points are fixed, which limits its applications. The main purpose of this work is to present a simple method for constructing a cubic Catmull-Rom spline with a shape parameter. A class of cubic Catmull-Rom spline basis functions with a shape parameter α, named cubic α-Catmull-Rom spline basis functions, is constructed through extending the definition interval of the standard cubic Catmull-Rom spline basis functions from r0, 1s to r0, αs pα ą 0q. Then, the corresponding cubic α-Catmull-Rom spline curves and interpolation function are defined on the basis of the cubic α-Catmull-Rom spline basis functions. The proposed cubic α-Catmull-Rom spline curves not only have the same properties as the standard cubic Catmull-Rom spline curves, but also have one degree of freedom in the interpolation curves even if the control points are fixed. The optimal cubic α-Catmull-Rom spline interpolation function can be obtained by choosing the proper value of the shape parameter.
Math. Comput. Appl. 2016, 21, 33; doi:10.3390/mca21030033 www.mdpi.com/journal/mca Math. Comput. Appl. 2016, 21, 33 2 of 14
The rest of this paper is organized as follows. In Section2, the standard cubic Catmull-Rom spline curves are introduced, and the shortcomings of the curves are pointed out. In Section3, the cubic Catmull-Rom basis functions with a shape parameter α, named cubic α-Catmull-Rom basis functions, are constructed, and the properties of the basis functions are given. In Section4, the definition and properties of the cubic α-Catmull-Rom spline curves are presented on the basis of the cubic α-Catmull-Rom basis functions. In Section5, the cubic α-Catmull-Rom spline interpolation function is discussed, and a method for determining the optimal interpolation function is presented. A short conclusion is given in Section6.
2. The Standard Cubic Catmull-Rom Spline Curves
2 3 Given control points pi pi “ 0, 1, ¨ ¨ ¨ , n; n ě 3q in R or R , for 0 ď t ď 1, the cubic Catmull-Rom spline curves can be generally expressed as follows [22]:
3 riptq “ bjptqpi`j, i “ 0, 1, ¨ ¨ ¨ , n ´ 3, (1) j“ ÿ0 where bjptq pj “ 0, 1, 2, 3q are the cubic Catmull-Rom basis functions expressed as follows:
1 2 3 b0ptq “ 2 ´t ` 2t ´ t 1 2 3 $ b1ptq “ 2 `2 ´ 5t ` 3t ˘ ’ . (2) ’ b ptq “ 1 t ` 4t2 ´ 3t3 &’ 2 2 ` ˘ 1 2 3 b3ptq “ 2 `´t ` t ˘ ’ ’ The cubic Catmull-Rom basis functions% have` the following˘ properties:
(a) partition of unity: b0ptq ` b1ptq ` b2ptq ` b3ptq ” 1; (b) symmetry: bip1 ´ tq “ b3´iptq pj “ 0, 1, 2, 3q; (c) properties at the endpoints:
b0p0q “ 0, b1p0q “ 1, b2p0q “ 0, b3p0q “ 0, # b0p1q “ 0, b1p1q “ 0, b2p1q “ 1, b3p1q “ 0.
1 1 1 1 b0p0q “ ´1{2, b1p0q “ 0, b2p0q “ 1{2, b3p0q “ 0, 1 1 1 1 # b0p1q “ 0, b1p1q “ ´1{2, b2p1q “ 0, b3p1q “ 1{2.
The cubic Catmull-Rom spline curves have the following properties:
(a) symmetry: both pi pi “ 0, 1, ¨ ¨ ¨ , nq and pn´i pi “ 0, 1, ¨ ¨ ¨ , nq define the same curves in a different parameterization; (b) geometric invariance: shapes of the curves are independent of the choice of coordinate system. An affine transformation for the curve can be performed by carrying out the same affine transformation for the control points; 1 (c) interpolation and C continuity: the curves interpolate the given control points expect p0 and pn, and the curves are C1 continuous.
The standard cubic Catmull-Rom spline curves can automatically interpolate the given control points without solving equation systems, which cause them to be widely used in practical engineering. However, shapes of the standard cubic Catmull-Rom spline curves cannot be adjusted when the control points are fixed, which limits their applications. In order to alleviate the shortcomings of standard cubic Catmull-Rom spline curves in shape adjustment, the cubic Catmull-Rom spline curves with a shape parameter will be introduced. Math. Comput. Appl. 2016, 21, 33 3 of 14
3. The Cubic α-Catmull-Rom Basis Functions Firstly, the cubic Catmull-Rom basis functions with a shape parameter needed to be constructed. A simple idea is to extend the definition interval of the standard cubic Catmull-Rom basis functions from r0, 1s to r0, αs pα ą 0q. Suppose the new basis functions fiptq pi “ 0, 1, 2, 3q are expressed as follows:
2 3 f0ptq f1ptq f2ptq f3ptq “ 1 t t t M, (3) ´ ¯ ´ ¯ where 0 ď t ď α, 0 ă α ď 1, and M is an undetermined 4 ˆ 4 matrix. By taking derivation calculus to Equation (3), then
1 1 1 1 2 f0ptq f1ptq f2ptq f3ptq “ 0 1 2t 3t M. (4) ´ ¯ ´ ¯ Because the new basis functions hope to satisfy the same properties at the end points with the standard cubic Catmull-Rom basis functions, thus, let t “ 0 and t “ α in Equations (3) and (4), respectively. Then, 0 1 0 0 “ 1 0 0 0 M ´ ¯ ´ ¯ 0 0 1 0 “ 1 α α2 α3 M . (5) ´ ¯ ´ ¯ ´1{2 0 1{2 0 “ 0 1 0 0 M ´ ¯ ´ ¯ 0 ´1{2 0 1{2 “ 0 1 2α 3α2 M ´ ¯ ´ ¯ From Equation (5), we have
0 1 0 0 1 0 0 0 0 0 1 0 1 α α2 α3 ¨ ˛ “ ¨ ˛ M. (6) ´1{2 0 1{2 0 0 1 0 0 ˚ ‹ ˚ ‹ ˚ 0 ´1{2 0 1{2 ‹ ˚ 0 1 2α 3α2 ‹ ˚ ‹ ˚ ‹ ˝ ‚ ˝ ‚ By Equation (6), we obtain
0 1 0 0 1 1 ¨ ´ 2 0 2 0 ˛ M “ . (7) 1 α´6 3´α ´ 1 ˚ α 2α2 α2 2α ‹ ˚ ‹ ˚ ´ 1 4´α α´4 1 ‹ ˚ 2α2 2α3 2α3 2α2 ‹ ˝ ‚ From Equations (3) and (7), the new basis functions can be expressed as follows:
f ptq “ 1 ´ 2t ` t2 ´ t3 0 2α2 α 2α 1 3 2 2 3 $ f1ptq “ 3 `2α ` α ´ 6α t˘ ` p4 ´ αq t 2α , (8) ’ 1 3 2 2 3 ’ f2ptq “ 3 `α t ` `6α ´ 2α˘ t ` pα ´ 4q t˘ &’ 2α f ptq “ 1 ´ t2 ` t3 3 2α2 ` α ` ˘ ˘ ’ ’ ` ˘ where 0 ď t ď α, α ą 0. % The basis functions expressed in Equation (8) can be reparametrized by gipuq “ fipαuq pi “ 0, 1, 2, 3q. Then, the basis functions defined on a fixed interval [0,1] can be obtained as follows. Math. Comput. Appl. 2016, 21, 33 4 of 14
Definition 1. For 0 ď u ď 1, α ě 0, the following four functions of u are called the cubic Catmull-Rom basis functions with a shape parameter α (cubic α-Catmull-Rom basis functions for short):
1 2 3 g0puq “ 2 ´αu ` 2αu ´ αu 1 2 3 $ g1puq “ 2 `2 ` pα ´ 6q u ` p4˘´ αq u ’ . (9) ’ g puq “ 1 αu ` p6 ´ 2αq u2 ´ p4 ´ αq u3 &’ 2 2 ` ˘ 1 2 3 g3puq “ 2 `´αu ` αu ˘ ’ ’ From Equations (2) and% (9), the cubic` α-Catmull-Rom˘ basis functions and the standard cubic Catmull-Rom basis functions have the relationships as follows:
g0puq “ αb0puq
$ g1puq “ b1puq ` p1 ´ αqb3puq ’ . ’ g puq “ b puq ` p1 ´ αqb puq &’ 2 2 0 g3puq “ αb3puq ’ ’ %’ Remark 1. It is clear that the cubic α-Catmull-Rom basis functions would be the standard cubic Catmull-Rom basis functions for α “ 1.
Theorem 1. The cubic α-Catmull-Rom basis functions have the following properties:
(a) partition of unity: g0puq ` g1puq ` g2puq ` g3puq ” 1; (b) symmetry: gip1 ´ uq “ g3´ipuq pi “ 0, 1, 2, 3q; (c) properties at the endpoints:
g0p0q “ 0, g1p0q “ 1, g2p0q “ 0, g3p0q “ 0, # g0p1q “ 0, g1p1q “ 0, g2p1q “ 1, g3p1q “ 0.
1 1 1 1 g0p0q “ ´α{2, g1p0q “ 0, g2p0q “ α{2, g3p0q “ 0, 1 1 1 1 # g0p1q “ 0, g1p1q “ ´α{2, g2p1q “ 0, g3p1q “ α{2.
(d) monotonicity about the shape parameter: for fixed u P r0, 1s, g0puq and g3puq are monotonically decreasing about α, g1puq and g2puq are monotonically increasing about α.
Proof. By simple deduction, (a), (b) and (c) follow: (d) For fixed u P r0, 1s, by taking derivation calculus on α to (3), then
dg 1 dg 1 0 “ ´ up1 ´ uq2 ď 0, 1 “ αu2p1 ´ uq ě 0, dα 2 dα 2 dg 1 dg 1 2 “ up1 ´ uq2 ě 0, 3 “ ´ αu2p1 ´ uq ď 0. dα 2 dα 2
Thus, g0puq and g3puq are monotonically decreasing about α, and g1puq and g2puq are monotonically increasing about α. Theorem 1 shows that the cubic α-Catmull-Rom basis functions not only inherit the properties of the standard cubic Catmull-Rom basis functions, but also can be adjusted by the shape parameter α. Figure1 shows curves of the cubic α-Catmull-Rom basis functions for different values of α, where the value of α is taken as α “ 0.6 (marked with dotted lines), α “ 2.4 (marked with solid lines) and α “ 4.2 (marked with dashed lines), respectively. Math. Comput. Appl. 2016, 21, 33 5 of 14 Math. Comput. Appl. 2016, 21, 33 5 of 13
FigureFigure 1. 1. CurvesCurves of of the the cubic cubic αα‐-Catmull-RomCatmull‐Rom basis functions functions for for different α.
4. The Cubic α‐Catmull‐Rom Spline Curves 4. The Cubic α-Catmull-Rom Spline Curves On the basis of the cubic α‐Catmull‐Rom basis functions, the corresponding cubic Catmull‐Rom On the basis of the cubic α-Catmull-Rom basis functions, the corresponding cubic Catmull-Rom spline curves with a shape parameter can be defined as follows. spline curves with a shape parameter can be defined as follows.
2 3 Definition 2. Given control points b (0,1,,;3)inn in 2 R or3 R , for 01u , 0 , the Definition 2. Given control points bi pi “i 0, 1, ¨ ¨ ¨, n; n ě 3q in R or R , for 0 ď u ď 1, α ě 0, the following followingcurves are calledcurves the are cubic called Catmull-Rom the cubic splineCatmull curves‐Rom with spline a shape curves parameter with αa (cubicshapeα -Catmull-Romparameter α (cubic spline curves for short): α‐Catmull‐Rom spline curves for short): 3 3 pipuq “ gjpuqbi`j, i “ 0, 1, ¨ ¨ ¨ , n ´ 3, (10) pbijij()ugu () , in 0,1, , 3 (10) j“ , j0 ÿ0 where g ()u (0,1,2,3)j are the cubic α‐Catmull‐Rom basis functions defined in (9). where gjpjuq pj “ 0, 1, 2, 3q are the cubic α-Catmull-Rom basis functions defined in (9).
Theorem 2. The cubic α‐-Catmull-RomCatmull‐Rom spline curves have the following properties:
(a) symmetry:symmetry: for for thethe samesame shape shape parameter parameterα, bothα, bothbi p i b“i (0,1,,)0,in 1, ¨ ¨ ¨, nq and andbn ´bi nipi “(0,1,2,,)in0, 1, 2, ¨ ¨ ¨ , nq definedefine the the same same curves curves in in a a different different parameterization: parameterization:
piiiii(;uu bbbb ,123 , , ) p i (1 ; bbbb iiii 321 , , , ), in 0,1, , 3 ; p pu; b , b , b , b q “ p p1 ´ u; b , b , b , b q, i “ 0, 1, ¨ ¨ ¨ , n ´ 3; (b) geometric invariance:i i i` shapes1 i`2 ofi` the3 curvesi are independenti`3 i`2 i` 1of thei choice of coordinate system. An affine transformation for the curves can be performed by carrying out the same affine (b) geometric invariance: shapes of the curves are independent of the choice of coordinate system. transformation for the control points; An affine transformation for the curves can be performed by carrying out the same affine (c) local property: at most, four segments of the curves would be affected if one control point is transformation for the control points; moved; (c) local property: at most, four segments of the curves would be affected if one control point (d) interpolation and C1 continuity: the curves interpolate the given control points expect b and is moved; 0 1 bn , and the curve is 1C continuous; (d) interpolation and C continuity: the curves interpolate the given control points expect b0 and bn, b (0,1,2,3)j (e) shapeand the adjustable curve is C property:1 continuous; when the control points ij are fixed, shape of the p ()u (e) curveshape adjustablei can be property: adjusted when by altering the control the value points of thebi` shapej pj “ parameter0, 1, 2, 3q are α. fixed, shape of the curve pipuq can be adjusted by altering the value of the shape parameter α. Proof. (a) From the symmetry of the cubic α‐Catmull‐Rom basis functions and Equation (10), then
pbbbbiiiii(1ugugugugu ;321 , , , ) 0 (1 ) b i 31 (1 ) b i 2 2 (1 ) b i 13 (1 ) b i gu()bbbbpbbbb gu () gu () gu () (; u , , , ) 3322110iiiiiiiii 123.
Math. Comput. Appl. 2016, 21, 33 6 of 14
Proof.
(a) From the symmetry of the cubic α-Catmull-Rom basis functions and Equation (10), then
pip1 ´ u; bi`3, bi`2, bi`1, biq “ g0p1 ´ uqbi`3 ` g1p1 ´ uqbi`2 ` g2p1 ´ uqbi`1 ` g3p1 ´ uqbi “ g3puqbi` ` g2puqbi` ` g1puqbi` ` g0puqbi “ p pu; bi, bi` , bi` , bi` q. Math. Comput. Appl. 2016, 21, 33 3 2 1 i 1 2 3 6 of 13 (b) Because Equation (10) is an affine combination of the control points, geometric invariance follows. (b) Because Equation (10) is an affine combination of the control points, geometric invariance (c) From Equation (10), a segment of the curve p puq is only related to the four control points follows. i b pj “ 0, 1, 2, 3q, thus, at most, four segments of the curves would be affected if one control (c) Fromi`j Equation (10), a segment of the curve p ()u is only related to the four control points point is moved. i b (0,1,2,3)j , thus, at most, four segments of the curves would be affected if one control (d) Fromij the properties at the endpoints of the α-Catmull-Rom basis functions and Equation (10), then point is moved. (d) From the properties at the endpoints of the α‐Catmull‐Rom basis functions and Equation (10), pip0q “ bi`1 then , i “ 0, 1, ¨ ¨ ¨ , n ´ 3, (11) # pip1q “ bi`2 pbii(0) 1 , in 0,1, , 3 , (11) 1pb(1) α pipii0q “ 2 p2bi`2 ´ biq 1 α , i “ 0, 1, ¨ ¨ ¨ , n ´ 3. (12) p p1q “ pbi`3 ´ bi`1q pbb#(0)i 2 iii2 2 , in 0,1, , 3 . (12) Equation (11) shows that the curves interpolate the given control points expect b0 and bn. From pbb(1) Equations (11) and (12), it follows iii that2 31 Equation (11) shows that the curves interpolate the given control points expect b and b . pkq pkq 0 n p p1q “ p p0qpk “ 0, 1q. (13) From Equations (11) and (12), it followsi that i`1 pp()kk(1) () (0) (0,1)k Equation (13) shows that the curvesii+ are C11 continuous.. (13) Equation (13) shows that the curves are C1 continuous. (e) Since Equation (10) contains the parameter α, the shape of the curve pipuq can be adjusted by (e) Since Equation (10) contains the parameter α, the shape of the curve pi ()u can be adjusted by altering the value of α even if the control points bi`j pj “ 0, 1, 2, 3q are fixed (see Figure2). altering the value of α even if the control points bij (0,1,2,3)j are fixed (see Figure 2).
Figure 2. AA segment segment of of the the cubic cubic α‐-Catmull-RomCatmull‐Rom spline curves for different α..
Theorem 2 shows that the cubic α‐Catmull‐Rom spline curves not only inherit the same Theorem 2 shows that the cubic α-Catmull-Rom spline curves not only inherit the same properties properties of the standard cubic Catmull‐Rom spline curves, but also can achieve shape adjustment of the standard cubic Catmull-Rom spline curves, but also can achieve shape adjustment by the shape by the shape parameter α. parameter α. 1 If two auxiliary points b1 and bn1 are added to the given control points, the C1 continuous If two auxiliary points b´1 and bn`1 are added to the given control points, the C continuous cubic α‐Catmull‐Rom spline curves pi ()u (1,0,1,,2)in interpolating all the given control cubic α-Catmull-Rom spline curves pipuq pi “ ´1, 0, 1, ¨ ¨ ¨ , n ´ 2q interpolating all the given control 1 points bi (0,1,,)in would be obtained. It is clear that there exists one degree of freedom in the C continuous cubic α‐Catmull‐Rom spline curves, even if the control points and auxiliary points are fixed. Different interpolation curves could be obtained by altering the value of the shape parameter α. Figure 3 shows the C1 continuous cubic α‐Catmull‐Rom spline curves adjusted by the shape parameter α, where the value of α is taken as 0.4 (marked with dotted lines), 1.0 (marked with solid lines) and 1.8 (marked with dashed lines), respectively.
Math. Comput. Appl. 2016, 21, 33 7 of 14
points bi pi “ 0, 1, ¨ ¨ ¨ , nq would be obtained. It is clear that there exists one degree of freedom in the C1 continuous cubic α-Catmull-Rom spline curves, even if the control points and auxiliary points are fixed. Different interpolation curves could be obtained by altering the value of the shape parameter α. Figure3 shows the C1 continuous cubic α-Catmull-Rom spline curves adjusted by the shape parameter α, where the value of α is taken as α “ 0.4 (marked with dotted lines), α “ 1.0 (marked with solid lines) andMath.α Comput.“ 1.8 Appl. (marked 2016, 21 with, 33 dashed lines), respectively. 7 of 13
Figure 3. The cubic α‐-Catmull-RomCatmull‐Rom spline curves for different α..
Remark 2. As we know, the cubic Hermite spline curves [27] can be expressed as follows: Remark 2. As we know, the cubic Hermite spline curves [27] can be expressed as follows: rpppp()tft () ft () gt () gt () iiiii01120112 , (14) 1 1 where riptq “ f0ptqpi`1 ` f1ptqpi`2 ` g0ptqpi`1 ` g1ptqpi`2, (14) 23 f0 ()ttt 1 3 2 where ft() 3 t23 2 t 2 3 1 f0ptq “ 1 ´ 3t ` 2t gt() t (1 t )22 3 $0 f1ptq “ 3t ´ 2t 2 . ’g ()ttt (1 ) 2 ’1 g ptq “ tp1 ´ tq &’ 0 . 2 If we let pppiii11(), pppiii232g1p()tq “ ´t p1 ´, tthenq Equation (14) could be rewritten as ’ 1 ’1 follows:If we let pi`1 “ αppi`1 ´ piq, p%i`2 “ αppi`3 ´ pi`2q, then Equation (14) could be rewritten rpppp()ththt () () ht () ht () as follows: iiiii0112233 , (15) where riptq “ h0ptqpi ` h1ptqpi`1 ` h2ptqpi`2 ` h3ptqpi`3, (15) where ht00() gt () h0ptq “ ´αg0ptq ht10() f () t g 0 () t
h ptq “ f ptq ` αg ptq $ht21()1 ft ()0 gt 1 () 0 ’ . ’ht()h ptq gt“ ()f ptq ´ αg ptq &’312 1 . 1 h3ptq “ αg1ptq Although the curves expressed ’in Equation (15) have the similar properties of the proposed ’ α‐CatmullAlthough‐Rom the spline curves curves, expressed the %’proposed in Equation α‐Catmull (15) have‐Rom the similarspline curves properties are ofconstructed the proposed by αextending-Catmull-Rom the definition spline curves, interval the of proposed the standardα-Catmull-Rom cubic Catmull spline‐Rom curves spline are from constructed [0,1] to by [0, extendingα], which theis a definitionnew idea for interval constructing of the standard cubic Catmull cubic‐Rom Catmull-Rom spline with spline a shape from parameter. [0,1] to [0, Thus,α], which the proposed is a new ideawork for provides constructing another cubic method Catmull-Rom for constructing spline with interpolation a shape parameter. curves without Thus, the solving proposed equation work providessystems. another method for constructing interpolation curves without solving equation systems. Since the shapes of the cubic α‐Catmull‐Rom spline curves are determined by the value of the shape parameter α when the control points and auxiliary points are fixed, how to determine the optimal value of α for making the cubic α‐Catmull‐Rom spline curves be the smoothest, therefore, needs to be discussed. A method for determining the optimal value of α is presented as follows. According to Ref. [28], the smoothness of a parametric curve r()u (0u 1) can be approximately measured by its energy expressed as follows: 1 2 Eut r() d (16) 0 . The lower the energy is, the smoother the curve.
From Equation (16), for given data points bi (0,1,,)in and two auxiliary points b1 , bn1 , p ()u (1,0,1,,2)in the optimal value of α of the cubic α‐Catmull‐Rom spline curves i can be determined by an optimization model expressed as follows:
Math. Comput. Appl. 2016, 21, 33 8 of 14
Since the shapes of the cubic α-Catmull-Rom spline curves are determined by the value of the shape parameter α when the control points and auxiliary points are fixed, how to determine the optimal value of α for making the cubic α-Catmull-Rom spline curves be the smoothest, therefore, needs to be discussed. A method for determining the optimal value of α is presented as follows. According to Ref. [28], the smoothness of a parametric curve rpuq p0 ď u ď 1q can be approximately measured by its energy expressed as follows:
1 E “ pr2 puqq2 dt. (16) ż0 The lower the energy is, the smoother the curve. From Equation (16), for given data points bi pi “ 0, 1, ¨ ¨ ¨ , nq and two auxiliary points b´1, bn`1, the optimal value of α of the cubic α-Catmull-Rom spline curves pipuq pi “ ´1, 0, 1, ¨ ¨ ¨ , n ´ 2q can be determined by an optimization model expressed as follows:
n´2 1 2 2 min Epαq “ 0 pi puq du i“´1 r . (17) s.t. α ě 0 ř ` ˘
Set 1 1 l puq “ ´u ` 2u2 ´ u3 , l puq “ u2 ´ u3 , m puq “ 1 ´ 3u2 ` 2u3, m puq “ 3u2 ´ 2u3. 0 2 1 2 0 1 ´ ¯ ´ ¯ Then, Equation (10) can be rewritten as follows:
pipuq “ Ripuqα ` Sipuq, (18) where Ripuq “ l0puq pbi ´ bi`2q ` l1ptq pbi`1 ´ bi`3q, Sipuq “ m0puqbi`1 ` m1puqbi`2. From Equations (17) and (18) can be expressed as follows:
min Epαq “ c α2 ` 2c α ` c 1 2 3 . (19) s.t. α ě 0
n´2 n´2 n´2 1 2 2 1 2 2 1 2 2 where c1 “ 0 Ri puq du, c2 “ 0 Ri puq ¨ Si puq du, c3 “ 0 Si puq du. i“´1 r i“´1 r i“´1 r Since thereř is always` ˘c1 ě 0, the solutionř ` of Equation (19)˘ has theř following` two˘ cases:
Case 1. When c1 “ 0. If c2 ą 0, the solution of Equation (19) is α “ 0; else if c2 ă 0, the solution of Equation (19) is α “ ´ c3 ; else, the solution of Equation (19) is α to take any non-negative real number. 2c2 Case 2. When c ą 0. If ´ c2 ď 0, the solution of Equation (19)r is α “ 0; else, the solution of 1 c1 Equation (19) is rα “ ´ c2 . c1 r After the optimal value of α is determined by Equation (19), the smoothest cubic α-Catmull-Rom r spline curves can be obtained. For example, for the same control points and auxiliary points in Figure3, the optimal value of α determined by Equation (19) is α “ 1.10156. The energy curve of the cubic α-Catmull-Rom spline curves for 0 ď α ď 3 is shown in Figure4. The smoothest cubic α-Catmull-Rom spline curves are shown in Figure5. r Math. Comput. Appl. 2016, 21, 33 8 of 13
n2 1 minEuu ( ) p ( )2 d 0 i i1 (17) s.t. 0 . Set
1 23 1 23 23 23 lu0 () u 2 u u , lu1 () u u , mu0 () 1 3 u 2 u, mu1 () 3 u 2 u. 2 2 Then, Equation (10) can be rewritten as follows: pR()uu () S () u ii i, (18) where Rbbbbiiiii()ulu02113 () lt () , Sbbiii()umu 0112 () mu () . From Equations (17) and (18) can be expressed as follows: 2 minEc ( ) 123 2 cc (19) s.t. 0 . n2 1 2 n2 1 n2 1 2 where cuu R () d , cuuuRS() ()d , cuu S () d . 1 0 i 2 0 ii 3 0 i i1 i1 i1
Since there is always c1 0 , the solution of Equation (19) has the following two cases:
Case 1. When c1 0 . If c2 0 , the solution of Equation (19) is 0 ; else if c2 0 , the
c3 solution of Equation (19) is ; else, the solution of Equation (19) is α to take any 2c2 non‐negative real number.
c2 Case 2. When c1 0 . If 0 , the solution of Equation (19) is 0 ; else, the solution of c1 c Equation (19) is 2 . c1
After the optimal value of α is determined by Equation (19), the smoothest cubic α‐Catmull‐Rom spline curves can be obtained. For example, for the same control points and auxiliary points in Figure 3, the optimal value of α determined by Equation (19) is 1.10156 . The Math.energy Comput. curve Appl. of2016 the, 21cubic, 33 α‐Catmull‐Rom spline curves for 03 is shown in Figure 4.9 ofThe 14 smoothest cubic α‐Catmull‐Rom spline curves are shown in Figure 5.
Math. Comput. Appl. 2016Figure, 21, 33 4. The energy curve of the cubic α-Catmull-Rom‐Catmull‐Rom spline curves. 9 of 13
Figure 5. The smoothest cubic α-Catmull-Rom‐Catmull‐Rom spline curves.
5. The The Cubic Cubic α‐α-Catmull-RomCatmull‐Rom Spline Spline Interpolation Interpolation Function Function On the basis of thethe cubiccubic αα-Catmull-Rom‐Catmull‐Rom basisbasis functions,functions, the cubic α-Catmull-Rom‐Catmull‐Rom spline interpolation function can also be defineddefined asas follows.follows.
DefinitionDefinition 3.3. GivenGiven aa functionfunction yyrx“rp()xq p()aaxbď xď b,q ,∆: ax: a“01x0 xă x1 ă x ¨n ¨ ¨ ă b xisn a“ divisionb is a division of [,]ab of, ra, bs, y “ rpx q pi “ 0, 1, 2, ¨ ¨ ¨ , nq, setx ux “ x´xi , for x ď x ď x pi “ 1, 2, ¨ ¨ ¨ , n ´ 2q, the function, i i i xi`1´xi i i`1 yrxii ()(0,1,2,,)in , set u , for xii xx 1 (1,2,,in 2), the function, xii1 x 3 3 SxS()pxq “ guy ()g puqy in,i1,“ 2,1, 2, ,¨ ¨ ¨ 2, n ´ 2, (20) ijiji j ,i `j , (20) j0j“ ÿ0 is called the cubic Catmull‐Rom spline functions with a shape parameter α (cubic α‐Catmull‐Rom is called the cubic Catmull-Rom spline functions with a shape parameter α (cubic α-Catmull-Rom spline interpolation function for short), where g j ()u (0,1,2,3)j are the cubic α‐Catmull‐Rom spline interpolation function for short), where g puq pj “ 0, 1, 2, 3q are the cubic α-Catmull-Rom basis basis functions defined according to Equation (9).j functions defined according to Equation (9). From Theorem 2, the cubic α‐Catmull‐Rom spline interpolation function has the properties as From Theorem 2, the cubic α-Catmull-Rom spline interpolation function has the properties follows: as follows: (a) interpolation: the cubic α‐Catmull‐Rom spline interpolation function interpolate the given data
points expect (,x00y ) and (,xnny ),
Sxii() y i1 , in 0,1, , 3 ; Sxii()12 y i (b) C1 continuity: the cubic α‐Catmull‐Rom spline interpolation function is C1 continuous, viz., ()kk () Sxii()111 Sx ii (), k 0,1; in 1, 2, , 3 ;
(c) shape adjustable property: if two auxiliary data points (,x11y ) and (,xynn11 ) are added to the given data points, the C1 continuous cubic α‐Catmull‐Rom spline interpolation function
Sxi ()(0,1,,1)in interpolating all the given data points (,xiiy )(0,1,,)in would be obtained. The shape of the curve of the cubic α‐Catmull‐Rom spline interpolation function can be adjusted by altering the value of the shape parameter α.
Example 1. Given the function, 1 yrx() (5 x 5), 1 x2 set xi 5 i , yrxii () (i 0,1, ,10) . The two auxiliary data points are taken as follows:
(,x11yxyxy )2(,)(,) 00 11, (,x11yxyxy 11 )2(, 10 10 )(,) 9 9 .
Math. Comput. Appl. 2016, 21, 33 10 of 14
(a) interpolation: the cubic α-Catmull-Rom spline interpolation function interpolate the given data points expect px0, y0q and pxn, ynq,
S px q “ y i i i`1 , i “ 0, 1, ¨ ¨ ¨ , n ´ 3; # Sipxi`1q “ yi`2
(b) C1 continuity: the cubic α-Catmull-Rom spline interpolation function is C1 continuous, viz.,
pkq pkq Si pxi`1q “ Si`1pxi`1q, k “ 0, 1; i “ 1, 2, ¨ ¨ ¨ , n ´ 3;
(c) shape adjustable property: if two auxiliary data points px´1, y´1q and pxn`1, yn`1q are added to the given data points, the C1 continuous cubic α-Catmull-Rom spline interpolation function Sipxq pi “ 0, 1, ¨ ¨ ¨ , n ´ 1q interpolating all the given data points pxi, yiq pi “ 0, 1, ¨ ¨ ¨ , nq would be obtained. The shape of the curve of the cubic α-Catmull-Rom spline interpolation function can be adjusted by altering the value of the shape parameter α.
Example 1. Given the function,
1 y “ rpxq “ p´5 ď x ď 5q, 1 ` x2 set xi “ ´5 ` i, yi “ rpxiq pi “ 0, 1, ¨ ¨ ¨ , 10q. The two auxiliary data points are taken as follows: Math. Comput. Appl. 2016, 21, 33 10 of 13 px´1, y´1q “ 2px0, y0q ´ px1, y1q, px11, y11q “ 2px10, y10q ´ px9, y9q. Figure 6 shows the curve of the C1 continuous cubic α‐Catmull‐Rom spline interpolation functionFigure adjusted6 shows by the the curve shape of parameter the C1 continuous α, where cubic the valueα-Catmull-Rom of α is taken spline as interpolation 0.2 (marked function with adjusteddotted lines), by the shape 1.0 parameter (markedα ,with where solid the value lines) of αandis taken as1.8α “(marked0.2 (marked with with dashed dotted lines), lines), αrespectively.“ 1.0 (marked with solid lines) and α “ 1.8 (marked with dashed lines), respectively.
Figure 6. TheThe cubic α‐-Catmull-RomCatmull‐Rom spline interpolation function for different α..
It is clear that the cubic α‐Catmull‐Rom spline interpolation function is completely determined It is clear that the cubic α-Catmull-Rom spline interpolation function is completely determined by the shape parameter α when the data points and two auxiliary data points are fixed. Hence, how by the shape parameter α when the data points and two auxiliary data points are fixed. Hence, how to determine the optimal value of the shape parameter α is the key when constructing the cubic to determine the optimal value of the shape parameter α is the key when constructing the cubic α‐Catmull‐Rom spline interpolation function. A method for determining the optimal value of the α-Catmull-Rom spline interpolation function. A method for determining the optimal value of the shape parameter α is presented as follows. shape parameter α is presented as follows. Given a function yrx () ()axb , : ax01 xL xn b is a division of [,]ab ,
yrxii ()(0,1,2,,)in , and two auxiliary data points (,x11y ), (,xynn11 ) are added to the given data points. Then, the interpolation error of the cubic α‐Catmull‐Rom spline interpolation function
Sxi ()(0,1,2,,1)inL can be expressed as follows: n1 xi1 2 F( )Sx () rx () d x x i (21) i0 i , where 0 . In order to obtain the minimum interpolation error, the following optimal model can be obtained: n1 x minF ( )i1 Sx ( ) rx ( )2 d x x i i0 i (22) s.t. 0 .
x xi Let ui , and set xii1 x
1 23 1 23 23 23 Lxiiii,0 () u 2 u u, Lxiii,1 () u u, Miii,1 ()xuu 1 3 2 , M iii,2 ()xuu 3 2 . 2 2 Then, Equation (20) can be rewritten as follows: Sx() Gx () Hx () ii i, (23) where Gxiiiiiii() L,0 () x y y 2 L ,1 () x y 1 y 3 , Hxiiiii() M,1 () xy 1 M ,2 () xy 2 . From Equations (22) and (23) can be expressed as follows: minF ( ) CCC2 2 112 s.t. 0 (24) ,
Math. Comput. Appl. 2016, 21, 33 11 of 14
Given a function y “ rpxq pa ď x ď bq, ∆ : a “ x0 ă x1 ă ¨ ¨ ¨ ă xn “ b is a division of ra, bs, yi “ rpxiq pi “ 0, 1, 2, ¨ ¨ ¨ , nq, and two auxiliary data points px´1, y´1q, pxn`1, yn`1q are added to the given data points. Then, the interpolation error of the cubic α-Catmull-Rom spline interpolation function Sipxq pi “ 0, 1, 2, ¨ ¨ ¨ , n ´ 1q can be expressed as follows:
n´1 xi`1 2 Fpαq “ pSipxq ´ rpxqq dx, (21) i“ xi ÿ0 ż where α ě 0. In order to obtain the minimum interpolation error, the following optimal model can be obtained:
n´1 min Fpαq “ xi`1 pS pxq ´ rpxqq2dx xi i i“0 r . (22) s.t. α ě 0 ř
Let u “ x´xi , and set i xi`1´xi
1 1 L pxq “ ´u ` 2u2 ´ u3 , L pxq “ u2 ´ u3 , M pxq “ 1 ´ 3u2 ` 2u3, M pxq “ 3u2 ´ 2u3. i,0 2 i i i i,1 2 i i i,1 i i i,2 i i ´ ¯ ´ ¯ Then, Equation (20) can be rewritten as follows:
Sipxq “ Gipxqα ` Hipxq, (23) where Gipxq “ Li,0pxq pyi ´ yi`2q ` Li,1pxq pyi`1 ´ yi`3q, Hipxq “ Mi,1pxqyi`1 ` Mi,2pxqyi`2. From Equations (22) and (23) can be expressed as follows:
min Fpαq “ C α2 ` 2C α ` C 1 1 2 , (24) s.t. α ě 0
n´1 n´1 n´1 where C “ xi`1 G2pxqdx, C “ xi`1 G pxq pH pxq ´ rpxqqdx, C “ xi`1 pH pxq ´ rpxqq2dx. 1 xi i 2 xi i i 3 xi i i“0 r i“0 r i“0 r Since thereř is always C1 ě 0, theř solution of Equation (24) has the followingř two cases:
Case 1. When C1 “ 0. If C2 ą 0, the solution of Equation (24) is α “ 0; else if C2 ă 0, the solution of Equation (24) is α “ ´ C3 ; else, the solution of Equation (24) is α to take any non-negative 2C2 real number. r Case 2. When C ą 0.r If ´ C2 ď 0, the solution of Equation (24) is α “ 0; else, the solution of 1 C1 C2 Equation (24) is α “ ´ C . 1 r After the optimal value of the shape parameter α is determined by Equation (24), the optimum r cubic α-Catmull-Rom spline interpolation function could be naturally obtained from Equation (20).
Example 2. Given the function
y “ rpxq “ 4 ` sinpxqp0 ď x ď 5πq,
5π set xi “ n i, yi “ rpxiq pi “ 0, 1, 2, ¨ ¨ ¨ , nq. The two auxiliary data points are taken as follows:
px´1, y´1q “ 2px0, y0q ´ px1, y1q, pxn`1, yn`1q “ 2pxn, ynq ´ pxn´1, yn´1q.
For different n (where n + 1 is the number of the given data points), the optimal value of the shape parameter α solved by Equation (24), the interpolation errors of the optimum cubic α-Catmull-Rom Math. Comput. Appl. 2016, 21, 33 12 of 14 spline interpolation function and the interpolation error of the standard cubic Catmull-Rom spline interpolation function calculated by Equation (21) are shown in Table1.
Table 1. Interpolation results for different n.
The Interpolation Error The Interpolation Error The Optimal Value of n of the Optimum Cubic of the Standard Cubic the Shape Parameter α α-Catmull-Rom Spline Catmull-Rom Spline 10 1.63405 0.14 ˆ 10´3 0.60 ˆ 10´1 20 1.11735 0.15 ˆ 10´5 0.49 ˆ 10´3 30 1.04889 0.84 ˆ 10´7 0.31 ˆ 10´4 40 1.02677 0.97 ˆ 10´8 0.48 ˆ 10´5
Table1 shows that the interpolation result of the optimum cubic α-Catmull-Rom spline interpolation function is better than the standard cubic Catmull-Rom spline interpolation function, which is due to the fact that the shape parameter α is taken as the optimal value. The curves of the optimum cubic α-Catmull-Rom spline interpolation function for n = 10 (marked with solid lines) and the standard cubic Catmull-Rom spline interpolation function (i.e., the shape parameterMath. Comput. is Appl. taken 2016 as, 21α, “33 1, marked with dashed lines) are shown in Figure7. 12 of 13
Figure 7. The optimum cubic α-Catmull-Rom‐Catmull‐Rom splinespline interpolationinterpolation functionfunction ((nn == 10).
6. Conclusions 6. Conclusions A cubic α‐Catmull‐Rom spline with a shape parameter is presented in this paper. The spline A cubic α-Catmull-Rom spline with a shape parameter is presented in this paper. The spline curves not only have the same properties as the standard cubic Catmull‐Rom spline curves, but can curves not only have the same properties as the standard cubic Catmull-Rom spline curves, but also achieve shape adjustment by altering the value of the shape parameter, even if the control can also achieve shape adjustment by altering the value of the shape parameter, even if the control points are fixed. The corresponding cubic α‐Catmull‐Rom spline interpolation function has the same points are fixed. The corresponding cubic α-Catmull-Rom spline interpolation function has the characteristics with the spline curves, and the optimal interpolation function can be obtained by same characteristics with the spline curves, and the optimal interpolation function can be obtained by choosing proper values of the shape parameter. The examples illustrate the feasibility of those choosing proper values of the shape parameter. The examples illustrate the feasibility of those methods. methods. Furthermore, the corresponding spline surfaces and binary interpolation function with shape Furthermore, the corresponding spline surfaces and binary interpolation function with shape parameters could be constructed by using tensor products. The spline surfaces and binary interpolation parameters could be constructed by using tensor products. The spline surfaces and binary functions would enjoy the same characteristics with the one-dimensional models. Some results in this interpolation functions would enjoy the same characteristics with the one‐dimensional models. area will be presented in the following study. Some results in this area will be presented in the following study.
Acknowledgments: This work was supported by the Scientific Research Fund of the Hunan Provincial Education Department of China under the grant number 14B099 and the Nanjing University of Posts and Telecommunications Startup Foundation under the grant number NY215050. The authors are also very grateful to the Hunan Provincial Key Construction Discipline “Computer Application Technology” of Hunan University of Humanities, Science and Technology of China.
Author Contributions: Juncheng Li conceived and designed the manuscript; Juncheng Li and Sheng Chen performed the numeric experiments; Juncheng Li wrote the paper.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1. Mainar, E. A general class of Bernstein‐like bases. Comput. Math. Appl. 2007, 53, 1686–1703. 2. Han, X.A.; Ma, Y.C.; Huang, X.L. A novel generalization of Bézier curve and surface. J. Comput. Appl. Math. 2008, 217, 180–193. 3. Yang, L.Q.; Zeng, X.M. Bézier curves and surfaces with shape parameters. Int. J. Comput. Math. 2009, 86, 1253–1263. 4. Yan, L.L.; Liang, Q.F. An extension of the Bézier model. Appl. Math. Comput. 2011, 218, 2863–2879. 5. Qin, X.Q.; Hu, G.; Zhang, N.J.; Shen, X.L.; Yang, Y. A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters. Appl. Math. Comput. 2013, 223, 1–16. 6. Han, X.L. Piecewise quartic polynomial curves with a local shape parameter. J. Comput. Appl. Math. 2006, 23, 34–45.
Math. Comput. Appl. 2016, 21, 33 13 of 14
Acknowledgments: This work was supported by the Scientific Research Fund of the Hunan Provincial Education Department of China under the grant number 14B099 and the Nanjing University of Posts and Telecommunications Startup Foundation under the grant number NY215050. The authors are also very grateful to the Hunan Provincial Key Construction Discipline “Computer Application Technology” of Hunan University of Humanities, Science and Technology of China. Author Contributions: Juncheng Li conceived and designed the manuscript; Juncheng Li and Sheng Chen performed the numeric experiments; Juncheng Li wrote the paper. Conflicts of Interest: The authors declare no conflicts of interest.
References
1. Mainar, E. A general class of Bernstein-like bases. Comput. Math. Appl. 2007, 53, 1686–1703. [CrossRef] 2. Han, X.A.; Ma, Y.C.; Huang, X.L. A novel generalization of Bézier curve and surface. J. Comput. Appl. Math. 2008, 217, 180–193. [CrossRef] 3. Yang, L.Q.; Zeng, X.M. Bézier curves and surfaces with shape parameters. Int. J. Comput. Math. 2009, 86, 1253–1263. [CrossRef] 4. Yan, L.L.; Liang, Q.F. An extension of the Bézier model. Appl. Math. Comput. 2011, 218, 2863–2879. [CrossRef] 5. Qin, X.Q.; Hu, G.; Zhang, N.J.; Shen, X.L.; Yang, Y. A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters. Appl. Math. Comput. 2013, 223, 1–16. [CrossRef] 6. Han, X.L. Piecewise quartic polynomial curves with a local shape parameter. J. Comput. Appl. Math. 2006, 23, 34–45. [CrossRef] 7. Brinks, R. On the convergence of derivatives of B-splines to derivatives of the Gaussian function. Comput. Appl. Math. 2008, 27, 79–92. [CrossRef] 8. Juhász, I.; Hoffmann, M. On the quartic curve of Han. J. Comput. Appl. Math. 2009, 223, 124–132. [CrossRef] 9. Cao, J.; Wang, G.Z. Non-uniform B-spline curves with multiple shape parameters. J. Zhejiang Univ. Sci. C 2011, 12, 800–808. [CrossRef] 10. Allasia, G.; Cavoretto, R.; de Rossi, A. Lobachevsky spline functions and interpolation to scattered data. Comput. Appl. Math. 2013, 32, 71–87. [CrossRef] 11. Cavoretto, R.; Fasshauer, G.E.; McCourt, M. An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels. Numer. Algorithms 2015, 68, 393–422. [CrossRef] 12. Schumaker, L.L. Spline Functions: Computational Methods; SIAM: Philadelphia, PA, USA, 2015. 13. Allasia, G.; Cavoretto, R.; de Rossi, A.; Quatember, B.; Recheis, W.; Mayr, M.; Demertzis, S. Radial basis functions and splines for landmark-based registration of medical images. In Proceedings of the 8th International Conference on Numerical Analysis and Applied Mathematics (ICNAAM10), AIP Conference Proceedings, Rhodes, Greece, 19–25 September 2010; Simos, T.E., Psihoyios, G., Tsitouras, C., Eds.; Melville: New York, NY, USA, 2010; Volume 1281, pp. 716–719. 14. Allasia, G.; Cavoretto, R.; de Rossi, A. A class of spline functions for landmark-based image registration. Math. Methods Appl. Sci. 2012, 35, 923–934. [CrossRef] 15. Han, X.L. Quadratic trigonometric polynomial curves with a shape parameter. Comput. Aided Geom. Des. 2002, 19, 503–512. [CrossRef] 16. Han, X.L. Cubic trigonometric polynomial curves with a shape parameter. Comput. Aided Geom. Des. 2004, 21, 535–548. [CrossRef] 17. Juhász, I.; Róth, Á. Closed rational trigonometric curves and surfaces. J. Comput. Appl. Math. 2010, 234, 2390–2404. [CrossRef] 18. Han, X.L.; Zhu, Y.P. Curve construction based on five trigonometric blending functions. BIT Numer. Math. 2012, 52, 953–979. [CrossRef] 19. Yan, L.L. An algebraic-trigonometric blended piecewise curve. J. Inf. Comput. Sci. 2015, 12, 6491–6501. [CrossRef] 20. Han, X.L. Normalized B-basis of the space of trigonometric polynomials and curve design. Appl. Math. Comput. 2015, 251, 336–348. [CrossRef] 21. Yan, L.L. Cubic trigonometric nonuniform spline curves and surfaces. Math. Probl. Eng. 2016, 2016, 1–9. [CrossRef] Math. Comput. Appl. 2016, 21, 33 14 of 14
22. Derose, T.D.; Barsky, B.A. Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines. ACM Trans. Gr. 1988, 7, 1–41. [CrossRef] 23. Maggini, M.; Melacci, S.; Sarti, L. Representation of facial features by Catmull-Rom splines. Lecture Notes Comput. Sci. 2007, 4673, 408–415. 24. Yuksel, C.; Schaefer, S.; Keyser, J. Parameterization and applications of Catmull-Rom curves. Comput.-Aided Des. 2011, 43, 747–755. [CrossRef] 25. Uncini, A.; Piazza, F. Blind signal processing by complex domain adaptive spline neural networks. IEEE Trans. Neural Netw. 2003, 14, 399–412. [CrossRef][PubMed] 26. Scarpiniti, M.; Comminiello, D.; Parisi, R.; Uncini, A. Nonlinear spline adaptive filtering. Signal Process. 2013, 93, 772–783. [CrossRef] 27. Frederic, P.; Agnes, F.; John, M. Cubic Hermite Spline; Alphascript Publishing: Beau Bassin-Rose Hill, Mauritius, 2010. 28. Poliakoff, J.F. An improved algorithm for automatic fairing of non-uniform parametric cubic splines. Comput.-Aided Des. 1996, 28, 59–66. [CrossRef]
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