A Method for Local Interpolation with Tension ...m-hikari.com/ams/ams-2015/ams-61-64-2015/p/lamniiAMS61-64-2015.pdfInterpolation with tension trigonometric spline curves and surfaces

Applied Mathematical Sciences, Vol. 9, 2015, no. 61, 3019 - 3035HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.52154

A Method for Local Interpolation with Tension

Trigonometric Spline Curves and Surfaces

Abdellah Lamnii

Univ. Hassan 1er, Laboratoire de MathematiquesInformatiques et Sciences de l’Ingnieur,

26000, Settat, Morocco

Fatima Oumellal

Univ. Hassan 1er, Laboratoire de Mathematiques,Informatiques et Sciences de l’Ingenieur

26000, Settat, Morocco

Copyright c© 2015 Abdellah Lamnii and Fatima Oumellal. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distri-

bution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this work a family of tension trigonometric curves analogousto those of cubic Bezier curves is presented. Some properties of theproposed curves are discussed. We propose an efficient interpolatingmethod based on the tension trigonometric splines with various prop-erties, such as partition of unity, geometric invariance and convex hullproperty, etc. This new interpolating method is applied to constructcurves and surfaces. Moreover, one can adjust the shape of the con-structed curves and surfaces locally by changing the tension parameter,the latter is included mainly because of its importance for object visu-alization. To illustrate the performance and the practical value of thismodel as well as its accuracy and efficiency, we present some modelingexamples.

Mathematics Subject Classification: 65T40, 65D05, 65D17, 76B45.

Keywords: Trigonometric Bezier Curves, Tension Parameter, Shape Pa-rameter, Interpolation.

3020 Abdellah Lamnii and Fatima Oumellal

1 Introduction

Let P1, . . . , PN be an ordered set of points in the plane and suppose thatB1, . . . , BN is a set of functions defined on the unit interval I := [0, 1] withthe following properties:

Bi(t) ≥ 0, for all t ∈ I, (1)

N∑i=1

Bi(t) = 1, for all t ∈ I, (2)

Then

CN(t) :=N∑i=1

PiBi(t), for t ∈ I, (3)

is the parametric representation of a curve which lies in the convex hull ofthe points P1, . . . , PN . These points are called the control points and theassociated polygon CN formed by connecting the points with straight linesis called the control polygon. In general, curve (3) does not pass throughthe control points, it just approximates the shape of the control polygondetermined by these points. Designers need a curve representation that isdirectly related to the control points and is flexible enough to bend, twist orchange the curve shape by changing one or more control points. For thesereasons, Bezier curves arepowerful tools for constructing curves with freeform. But the most popular curves in engineering applications, such as arcs,helix, cycloid and cones, etc., are only approximated by polynomial curves.To avoid their shortcomings many bases are presented using trigonometricfunctions or the blending of polynomial and trigonometric functions in [1, 8,9, 11, 12, 15, 16].

The idea of utilizing normalized trigonometric basis that form a parti-tion of unity to construct shape-preserving curves has been suggested byPena [9]. It proves that a curve in trigonometric form is the linear com-bination of its control points and certain Bernstein-like trigonometric func-tions, which form a normalized totally positive B-basis of the space Cm ={1, cos(t), . . . , cos(mt)}, with optimal shape preserving properties. This rep-resentation have no shape parameters; hence the shape of the curves or sur-faces cannot be modified when their control points are determined. In orderto improve the shape of a curve and adjust the extent where a curve ap-proaches its control polygon, we included the tension parameter which ismainly important for object visualization. More recently, Lamnii et al. in[3], have presented the quartic trigonometric polynomial blending functionswhere they have included the tension parameter β, which is mainly impor-tant for object visualization. In [6], Lamnii and Oumellal have constructed

Interpolation with tension trigonometric spline curves and surfaces 3021

the Trigonometric Bzier curves whith n = 5 followed by a construction ofthe shape preserving interpolation spline curves with local shape parame-ters. In this work we propose to generalize this technique for n = 4k − 1and n = 4k + 1, k ≥ 1, more precisely, the basic idea of this method is toincorporate the parameter β into the B-basis functions of Cm, where the pa-rameter β can adjust the shape of the curves without changing the controlpoints. As an application, we present trigonometric blending functions withshape parameters; without solving a linear system, these blending functionscan be applied to construct continuous shape preserving local interpolationtrigonometric spline curves with local shape parameters.

Recently in [6] we used tensor-product for surfaces interpolation, we wouldlike to generalize this technique for sphere like-surfaces. Let S be a closedand bounded surface in R3 which is topologically equivalent to a unit sphere.Given a set of scattered points P1, ..., Ps located on S, along with datavalues r1, ..., rs corresponding to these points. In many practical applica-tions, we wish to find a smooth function F defined on S which interpo-lates or approximates the given data set so that its associated closed surfaceSF := {F (s)s, s ∈ S} be of class C1 on S. A very popular method for fit-ting gridded data is to use tensor-products because of their computationalsimplicity and their many attractive geometric properties. In this paper, wesuppose that S is the unit sphere centered at the origin. Let I = [0, π

2β] and

J = [0, π2λ

], by using spherical coordinates we can identify the sphere S withthe rectangle R := I × J by the mapping

χ : S → R

(θ, φ) →

(cos(4πβθ) cos(2πλφ)cos(4πβθ) sin(2πλφ)

sin(2πλθ)

Now since the problem is defined on a rectangle R we can use the tensor-

product methods. This approach has been studied in several papers withvarious choices of basic functions see, for instance, [2, 13] and referencestherein. More specifically, Schumaker and Traas [13] have chosen the pe-riodic trigonometric B-splines of order three and the quadratic polynomialB-splines. Recently, another approach was proposed in [2] and [4] using 2π-periodic uniform algebraic trigonometric B-splines (UAT B-splines) of orderfour generated over the space spanned by {1, t, cos(t), sin(t)} and the cubicpolynomial B-splines. All these methods use quasi-interpolants or approxi-mants while our thechnique interpolates the data in a local way and gives agood shape visulisation.

The remainder of this paper is organized as follows. In section 2, the ex-pression of the generalized trigonometric polynomial blending functions aredescribed and the properties of these functions are discussed. TrigonometricBzier curves are constructed in this section. In section 3, the trigonometricparametric curve segments as well as the trigonometric polynomial interpo-lation are discussed. In section 4, several numerical examples are presented


in which open and closed trigonometric curves are described. We end thissection by constructing tensor product trigonometric polynomial B-surfacesjust like B-spline surfaces. A brief conclusion is given in section 5.

2 Trigonometric polynomial curves with shape

parameter

In this section, we will define the trigonometric polynomial blending functionswith shape parameter β and the corresponding trigonometric Bezier curves.

2.1 Trigonometric basis functions

Let β > 0 be an arbitrarily fixed parameter. We will produce trigono-metric polynomial curves, that are described by means of the combinationof control points and basis functions, therefore we need a proper basis inthe space of trigonometric polynomials Cn,β = {1, cos(2βt), . . . , cos(2βnt)}.We will use a transformed version of the basis specified in Pea [9]. LetBn,β =

(Bn

0,β, . . . , Bnn,β

)be the system of functions given by

Bni,β(t) =(ni

)cos2(n−i)(βt) sin2i(βt), i = 0, . . . , n, t ∈

[0,

π

2β

]. (4)

Each trigonometric polynomial function Bn−1i,β (t) of order n − 1 can be ex-

pressed as a linear combination of trigonometric polynomial function Bni,β(t)

of order n. More precisely we have:

Bn−1i,β (t) =

(n− in

)Bni,β(t) +

( i+ 1

n

)Bni+1,β(t) (5)

Theorem 2.1 The family Bn,β forms a basis for the space Cn,β.

Proof Let us prove this by induction on n.For n = 0, this case is trivial because the family C0,β = {1} and we have onlyB0

0,β(t) = 1.

Now suppose that Bn−1i,β (t), i = 0, . . . , n− 1 form a basis for the space Cn−1,β

and let us show that Bni,β(t), i = 0, . . . , n form a basis for the space Cn,β.

It is easy to see that Dim(Cn,β) = n+ 1 (because cos(2βkt), k = 0, . . . , n arelinearly independent) and since Bn

i,β(t), i = 0, . . . , n contains n+ 1 elements.So it suffices to show that the latter is a generating family.From the induction hypothesis, we have for all k = 0, . . . , n − 1, cos(2βkt)can be written as a linear combination of Bn−1

i,β (t), i = 0, . . . , n− 1.On the other hand, by using the formula (5), cos(2βkt) can also be writtenas a linear combination of Bn

i,β(t), i = 0, . . . , n.It remains only to write cos(2βnt) as a linear combination of Bn

i,β(t), i =0, . . . , n.


According to Moivre ’s formula, we have for any real number t and for anyinteger on n:

(cos t+ i sin t)n = cos(nt) + i sin(nt) .

Thus,

cos(2βnt) + i sin(2βnt) = (cos(βt) + i sin(βt))2n =

2n∑s=0

(2n

s

)cos(βt)2n−sis sin(βt)s

So, cos(2βnt) represents the real part of the complex number

cos(2βnt) =

n∑k=0

(2n

2k

)cos(βt)2n−2k(−1)k sin(βt)2k

=

n∑k=0

(−1)k(2n

2k

)cos(βt)2(n−k) sin(βt)2k

Hence the result. �The following result shows that Bn,β gives the optimal shape preserving rep-resentation of Cn,β. As 1 =

(cos2(βt) + sin2(βt)

)n, hence

n∑i=0

(ni

)cos2(n−i)(βt) sin2i(βt) = 1.

Therefore, Bn,β is normalized. Bn,β is equivalent to the Bernstein basis on[0, 1] by the transformation x = sin2(βt). Consequently, Bn,β possess thefollowing properties analogous to those of the Bernstein basis:

1. Nonnegativity : ∀t ∈ [0, π2β

], Bni,β(t) ≥ 0.

2. Partition of unity :n∑i=0

Bni,β(t) = 1.

3. Symmetry : Bni,β(t) = Bn

n−i,β( π2β− t).

4. Maximum : Each Bni,β has one maximum value in [0, π

2β].

Fig. 1 and 2 show the curves of the blending functions for n = 7 and n = 9.For t ∈ [0, π

2] and β > 1 we observed that the blending functions start to

duplicate as a function of the tension parameter value. So, we included thetension parameter in the interval to become t ∈ [0, π

2β], in this interval no

more duplications are observed.Let us observe that B0

0,β = 1. For the following recurrence formulas, weshall use the following notation:

Bni,β(t) = 0, i /∈ {0, . . . , n}.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) β = 3 and t ∈ [0, π2

].

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) β = 3 and t ∈ [0, π2β

].

Figure 1: Normalized trigonometric basis functions for n = 7.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) β = 2 and t ∈ [0, π2

].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) β = 2 and t ∈ [0, π2β

].

Figure 2: Normalized trigonometric basis functions for n = 9.

Proposition 2.2 The trigonometric basis functions given in (4) and de-fined on [0, π

2β] satisfy:

Bni,β(t) = cos(βt)2Bn−1i,β (t) + sin(βt)2Bn−1

i−1,β(t), 0 ≤ i ≤ n. (6)

Proof From (4), we get

cos2(βt)Bn−1i,β (t) + sin2(βt)Bn−1

i−1,β(t) = cos2(βt)(n− 1

i

)sin2i(βt) cos2(n−1−i)(βt)

+ sin2(βt)(n− 1

i− 1

)sin2(i−1)(βt) cos2(n−1−(i−1))(βt)

=(n− 1

i

)sin2i(βt) cos2(n−i)(βt) +

(n− 1

i− 1

)sin2i(βt) cos2(n−i)(βt)

=

((n− 1

i

)+(n− 1

i− 1

))sin2i(βt) cos2(n−i)(βt)

=(ni

)sin2i(βt) cos2(n−i)(βt)

= Bni,β(t). �

2.2 Trigonometric Curves of Order n

Given (control) points Vi (i = 0, . . . , n) in R2 or R3. Then,

Bβ(t) := Bβ(t;V0, V1, . . . , Vn) =

n∑i=0

Bni,β(t)Vi, t ∈ [0,π

2β], (7)

is called a trigonometric polynomialBezier curve of order n with a globalshape parameter.


From the properties of the trigonometric basis functions (4), some prop-erties of the trigonometric Bzier curve (7) can be obtained as follows.

1. Endpoint interpolation property: with a simple computation, we have:

Bβ(0) = Vn,B′β(0) = 0,

B′′β(0) = 2nβ2(Vn−1 − Vn),

Bβ( π2β

) = V0,

B′β( π2β

) = 0,

B′′β( π2β

) = 2nβ2(V1 − V0).

(8)

2. Symmetry: V0, V1, . . . , Vn and Vn, Vn−1, . . . , V0 define the same trigono-metric Bzier curve, i.e., Bβ(t;V0, V1, . . . , Vn) = Bβ( π

2β−t;Vn, Vn−1, . . . , V0).

3. Geometric invariance: since the blending functions have the propertiesof partition of unity, the shape of these trigonometric Bezier curves isindependent of the choice of coordinates.

4. Convex hull property: the blending functions have the properties of non-negativity and partition of unity, as a consequence, the entire trigono-metric Bzier curve segment must lie inside the control polygon spannedby V0, . . . , Vn.

5. Variation diminishing property : no straight line intersects a Beziercurve more times than it intersects its control polygon.

6. Convexity-preserving property: the variation diminishing property meansthe convexity preserving property holds.

Now we shall provide a de Casteljau-type algorithm for the evaluation ofthe curve Bβ(t). If the V0, . . . , Vn points in R2 or R3, are given and t ∈ [0, π

2β]

let us define the points

Bji,β(t) = cos2(βt)Bj−1i−1,β(t) + sin2(βt)Bj−1

i,β (t), j = 1, . . . , n, i = j, . . . , n, (9)

where B0i,β(t) = Vi, then Bnn,β(t) = Bβ(t).

Figure 3, illustrates two examples of evalution for the curve Bβ(t) fort = π

4β(left: n = 5) and (right: n = 7).

3 Interpolating trigonometric spline curve

Our objective is to generalize the Overhauser spline curve by solving thefollowing interpolation problem. Given the sequence of data points {Pi}mi=0

with associated parameter values u1 < u2 < · · · < um−1. Find the controlpoints of a trigonometric spline curve that interpolates the given data pointsand that consists of arcs of type (7).


B0

5,β

B0

4,β

B0

3,βB0

2,β

B0

1,β

B0

0,β

B1

1,β

B1

2,β

B1

3,β

B1

4,β

B1

5,β

B2

2,β

B2

3,β B2

4,β

B2

5,β

B3

3,β

B3

4,β

B3

5,β

B4

4,β B4

5,βB5

5,β

(a) n = 5, t = π4β

and β = 0.5.

B0

1,β

B0

2,β

B0

3,β B0

4,β

B0

6,β

B0

7,βB0

0,β

B0

5,β

B7

7,β

B1

1,β

B1

2,β

B1

3,β

B1

4,β

B1

6,β

B1

7,β

B2

3,β

B2

4,β B2

5,β

B2

6,β

B2

7,βB2

2,β

B3

3,β

B3

4,β

B3

5,β

B1

5,βB3

6,β

B3

7,β

B4

4,β

B5

5,β

B4

5,β B5

6,β

B5

7,β

B4

6,β

B4

7,β

B6

6,β B6

7,β

(b) n = 7, t = π4β

and β = 0.2.

Figure 3: The de Casteljau algorithm examples.

3.1 Trigonometric Parametric Curve Segments

We are going to describe an interpolation scheme that is based on curveblending, therefore first of all we specify an appropriate blending function,then we show an essential continuity property of the resulted blended curve.

Given the interpolation points Pi, i = 0, 1, . . . , n+12

and the trigonometricBzier control points Vi, i = 0, . . . , n. We propose the following endpointsrequirements.

If n = 4k + 1, we have

Bβ(0) = Vn = P2k,B′β(0) = 0,

B′′β(0) = 2nβ2(Vn−1 − Vn) = α1(Pk − P2k+1),

Bβ( π2β

) = V0 = α2(Pk+1 − P0),

B′β( π2β

) = 0,

B′′β( π2β

) = 2nβ2(V1 − V0) = Pk+1,

(10)

If n = 4k − 1, we have

Bβ(0) = Vn = Pk+1,B′β(0) = 0,

B′′β(0) = 2nβ2(Vn−1 − Vn) = α1(Pk−1 − P2k),

Bβ( π2β

) = V0 = α2(Pk+1 − P0),

B′β( π2β

) = 0,

B′′β( π2β

) = 2nβ2(V1 − V0) = Pk−1,

(11)

where α1, α2 ∈ [0,+∞[ are shape parameters. We noted that these endpoints requirements are analogous to those given in [17].Defineφ0,β(t, α1, α2) = TB0,β(t, α1, α2) := − α1

2nβ2Bn1,β(t)− 2α1


2nβ2Bn3,β(t)− · · · − 2kα1

2nβ2Bn2k,β(t),

φ1,β(t, α1, α2) = TBk,β(t, α1, α2) := Bn0,β(t) + · · ·+Bn2k,β(t) + 2kα22nβ2B2k+1,β(t) + · · ·+ α2

2nβ2Bnn−1,β(t),

φ2,β(t, α1, α2) = TBk+1,β(t, α1, α2) := α12nβ2B

n1,β(t) + · · ·+ 2kα1

2nβ2Bn2k,β(t) +Bn2k+1,β(t) + · · ·+Bnn,β(t),

φ3,β(t, α1, α2) = TB2k+1,β(t, α1, α2) := − 2kα22nβ

Bn2k+1,β(t)− (2k−1)α22nβ

Bn2k,β(t)− · · · − α22nβ

Bnn−1,β(t),

(12)


for n = 4k + 1, and

φ0,β(t, α1, α2) = TB0,β(t, α1, α2) := − α1



2nβ2Bn3,β(t)− · · · − (2k−1)α1

2nβ2 Bn2k−1,β(t),

φ1,β(t, α1, α2) = TBk−1,β(t, α1, α2) := Bn0,β(t) + · · ·+Bn2k−1,β(t) +(2k−1)α2

2nβ2 B2k,β(t) + · · ·+ α22nβ2B

nn−1,β(t),

φ2,β(t, α1, α2) = TBk+1,β(t, α1, α2) := α12nβ2B

n1,β(t) + · · ·+ (2k−1)α1

2nβ2 Bn2k−1,β(t) +Bn2k,β(t) + · · ·+Bnn,β(t),

φ3,β(t, α1, α2) = TB2k,β(t, α1, α2) := − (2k−1)α22nβ

Bn2k,β(t)− (2k−2)α22nβ

Bn2k−1,β(t)− · · · − α22nβ

Bnn−1,β(t).

(13)

for n = 4k − 1. The rest of TBi,β(t, α1, α2) are null.Using equations (10), (11) and the blending functions, the curve segment

can be generated as follows

Proposition 3.1 For t ∈ [0, π2β

], we have

Pβ(t, α1, α2) =

n∑i=0

Bni,β(t)Vi =

n+12∑i=0

TBi,β(t, α1, α2)Pi (14)

proof Let

TB0,β(t, α1, α2) = a00Bn0,β(t) + a01Bn1,β(t) + · · ·+ a0nBnn,β(t)

TB1,β(t, α1, α2) = a10Bn0,β(t) + a11Bn1,β(t) + · · ·+ a1nBnn,β(t)

...

TBn+12,β

(t, α1, α2) = an+12

0Bn0,β(t) + an+1

21Bn1,β(t) + · · ·+ an+1

2nBnn,β(t).

(15)

From (14) and (15), we have

n∑i=0

Bni,β(t)Vi =(a00B

n0,β(t) + a01B

n1,β(t) + . . .+ a0nB

nn,β(t)

)P0 + · · ·+

(an+1

20Bn0,β(t) + an+1

21Bn1,β(t) + · · ·+ an+1

2nBnn,β(t)

)Pn+1

2.

Then, we have

Bn0,β(t)V0 = Bn0,β(t)(a00P0 + · · ·+ an+1

20Pn+1

2

),

...

Bnn,β(t)Vn = Bnn,β(t)(a0nP0 + · · ·+ an+1

2nPn+1

2

).

Furthermore, we get V0 = a00P0 + · · ·+ an+1

20Pn+1

2,

..

.

Vn = a0nP0 + · · ·+ an+12nPn+1

2.

(16)

According to (16) and using (10) and (11), we deduce that TBj,β(t, α1, α2)can be written in the form (12) and (13). �

Remark 3.2 The φi,β , i = 0, · · · , n+12

, verify the partition of unity:n+12∑i=0

φi,β = 1.


3.2 Trigonometric parametric spline curves

Let Pi ∈ Rd (i = 0, . . . ,m, d = 2, 3) be the interpolation points, U =(u1, u2, . . . , um−1) the knot vector where u1 < u2 < . . . < um−1 and the shapeparameters αi ∈ [0,+∞[, i = 1, ...,m− 1.

For t ∈ [0, π2β

] and i = 1, . . . ,m − 2, the ith trigonometric parametric

curve segment is given as a function of φj,β, j = 0, 1, 2, 3, by the followingexpression:if n = 4k + 1,

Pi,β(t, αi, αi+1) = φ0,β(t, αi, αi+1)Pi−1 + φ1,β(t, αi, αi+1)Pi+k−1 (17)

+φ2,β(t, αi, αi+1)Pi+k + φ3,β(t, αi, αi+1)Pi+2k,

if n = 4k − 1,

Pi,β(t, αi, αi+1) = φ0,β(t, αi, αi+1)Pi−1 + φ1,β(t, αi, αi+1)Pi+k−2 (18)

+φ2,β(t, αi, αi+1)Pi+k + φ3,β(t, αi, αi+1)Pi+2k−1.

The corresponding trigonometric parametric spline curve, composed by all of the trigonometric parametriccurve segments are de�ned as follows:

Pβ(u) = Pi,β(π

2β×u− ui�ui

, αi, αi+1), u ∈ [ui, ui+1], i = 1, . . . ,m− 1, (19)

where �ui = ui+1 − ui.

Theorem 3.3 The curve (19) has 2nd geometric continuity, i.e., it is aGC2 continuous curve.

Proof Without loss of generality we may put n = 4k + 1, for i = 1, · · · ,m− 1, we have

Pβ(u+i ) = Pi,β(0, αi, αi+1) = Pi+k−1

Pβ(u−i+1) = Pi,β( π2β, αi, αi+1) = Pi+k

P ′β(u+i ) = P ′i,β(0, αi, αi+1) = 0

P ′β(u−i+1) = P ′i,β( π2β, αi, αi+1) = 0

P ′′β (u+i ) = P ′′i,β(0, αi, αi+1) =

π(α1Pi−1−2nβ2Pi+α1Pi+k)2β∆1

P ′′β (u−i+1) = P ′′i,β( π2β, αi, αi+1) =

π(α1Pi−2nβ2Pi+1+α1Pi+k+1)2β∆1

�

As described in [17], Pβ(u) interpolates the interpolation points Pi, i =0, ...,m. As an example, adding two control interpolation points P−1, Pm+1,two knots u0, um, and two shape parameters α0, αm are sufficient to constructan open curve Pβ(u) interpolating all of the points Pi, i = 0, . . . ,m. ClosedBezier curves are generated by specifying the first and the last control pointsat the same position. For constructing a closed curve Pβ(u) interpolatingall of the points Pi, i = 0, . . . ,m, we have to add three interpolation pointsP−1 = Pm, Pm+1 = P0, Pm+2 = P1 three knots u0, um, um+1 and three shapeparameters α0, αm, αm+1.


3.3 Trigonometric parametric spline surfaces

In this section, we apply the trigonometric parametric spline to a set of datapoints on a rectangular grid for surface interpolation using the well-knowntensor product form. A surface may be defined by the tensor product oftwo curves so that the properties of the blending functions are not modified.Whereas a curve requires one tension parameter for its definition, a surfacerequires two tension parameters β > 0 and λ > 0. Similarly to the work doneby Liu et al. (see [7]), we define trigonometric parametric spline surfaces asa tensor product. More precisely we have the following definition.

Definition 3.4 Given r × s interpolation points Pij (i = 0, 1, . . . , r; j =0, 1, . . . , s), two knot vectors U = [u1, u2, . . . , ur−1] and V = [v1, v2, . . . , vs−1]and two shape parameters vectors α = [α1, α2, . . . , αr−1] and µ = [µ1, µ2, . . . , µs−1].For tension parameters β > 0 and λ > 0, the trigonometric parametric splinesurface patch has the form :if n = 4k + 1,

Sβ,λi,j = φ0,j,λ

(φ0,i,βPi−2,j−2 + φ1,i,βPi+k−1,j+k−2 + φ2,i,βPi+k+1,j+k−1 + φ3,i,βPi+2k+2,j+2k−1

)(20)

+ φ1,j,λ

(φ0,i,βPi−2,j−1 + φ1,i,βPi+k−1,j+k−1 + φ2,i,βPi+k+1,j+k + φ3,i,βPi+2k+2,j+2k

)+ φ2,j,λ

(φ0,i,βPi−2,j + φ1,i,βPi+k−1,j+k + φ2,i,βPi+k+1,j+k+1 + φ3,i,βPi+2k+2,j+2k+1

)+ φ3,j,λ

(φ0,i,βPi−2,j+1 + φ1,i,βPi+k−1,j+k+1 + φ2,i,βPi+k+1,j+k+2 + φ3,i,βPi+2k+2,j+2k+2

)if n = 4k − 1,

Sβ,λi,j = φ0,j,λ

(φ0,i,βPi−2,j−2 + φ1,i,βPi+k−2,j+k−3 + φ2,i,βPi+k+1,j+k−1 + φ3,i,βPi+2k+1,j+2k−2

)(21)

+ φ1,j,λ

(φ0,i,βPi−2,j−1 + φ1,i,βPi+k−2,j+k−2 + φ2,i,βPi+k+1,j+k + φ3,i,βPi+2k+1,j+2k−1

)+ φ2,j,λ

(φ0,i,βPi−2,j + φ1,i,βPi+k−2,j+k−1 + φ2,i,βPi+k+1,j+k+1 + φ3,i,βPi+2k+1,j+2k

)+ φ3,j,λ

(φ0,i,βPi−2,j+1 + φ1,i,βPi+k−2,j+k + φ2,i,βPi+k+1,j+k+2 + φ3,i,βPi+2k+1,j+2k+1

)where u ∈ [0, π

2β], v ∈ [0, π

2λ], i = 0, ..., r−1, j = 0, ..., s−1 and Sβ,λi,j := Sβ,λi,j (u, v, αi, αi+j , µj , µj+1),

φl,i,β := φl,β(u, αi, αi+1), φl,j,λ := φl,λ(u, µj , µj+1), l = 0, . . . , 3.

Then the trigonometric parametric spline surface is given by,

Sβ,λ(u, v) = Sβ,λi,j (π

2β×u− ui�ui

,π

2λ×v − vj�vj

, αi, αi+1, µj , µj+1), (22)

u ∈ [ui, ui+1], v ∈ [vj , vj+1].

3.4 Trigonometric parametric spherical spline surfaces

Let S be a sphere-like surface, i.e. a closed and bounded surface in R3

which is topologically equivalent to a unit sphere. Given a set of scatteredpoints P1, . . . , Ps located on S, along with data values r1,. . . ,rs correspondingto these points. In many practical applications, we wish to find a smoothfunction F defined on S which interpolates or approximates the given dataset. A very popular method for fitting gridded data is to use tensor-products.It is possible to reduce the approximation of a function defined on S to theapproximation of a function defined on [0, π

2β]× [0, π

2λ], but when using this


approach, some periodicity conditions should be satisfied. Without loss ofgenerality, we suppose that S is the unit sphere centered at the origin. LetI = [0, π

2β] and J = [0, π

2λ]. By using spherical coordinates we can identify

the sphere S with the rectangle R := I × J by the mapping

χ : S → R

(θ, φ) →

(cos(4πβθ) cos(2πλφ)cos(4πβθ) sin(2πλφ)

sin(2πλθ)

Hence, to construct the function F , it suffices to find a function f := Foχ :R → R satisfying

f(θi, φi) = ri, i = 1, . . . , s,

where (θi, φi) are the polar coordinates of Pi.To make sure that a continuous function f defined on the rectangle R re-mains continuous after mapping it onto S, it is necessary that f satisfies thefollowing conditions [10]:

f(0, θ) = f( π

2λ, θ), 0 ≤ θ ≤ π

2β,

there exist constants SN ,SS such that

{f(φ, 0) = SN , 0 ≤ φ ≤ π

2λ,

f(φ, π2β

) = SS , 0 ≤ φ ≤ π2λ,

(23)

Now since the problem is defined on a rectangle we can use the results of theprevious section. More precisely, let r × s be the interpolation points Pi,j =(θij , φij)0≤i,j≤r−1,s−1. For constructing a spherical spline Sβ,λ interpolating allof the points Pi,j = (θij , φij)0≤i,j≤r−1,s−1 and satisfies the conditions (23), we haveto add three interpolation vectors P−1,j = Pr−1,j , Pr,j = P0,j , Pr+1,j = P1,j and wetake P−1,0 = P0,0 = . . . = Pr+1,0, P−1,s−1 = P0,s−1 = . . . = Pr+1,s−1.

4 Numerical Examples and Application

In order to justify the accuracy and efficiency of our presented trigonometric func-

tions we consider some graphical examples.

4.1 Interpolation Spline Curves and surfaces

B-spline functions have a wide range of applications, the properties of B-spline functions mentioned above can be useful in solving some problemsrelated to approximation theory, numerical analysis or computer graphics,for example representation of splines. In order to illustrate the performanceand the practical value of this model, we will represent some curves andsurfaces so that we can justify the accuracy and efficiency of our presentedtrigonometric functions. In order to do so we will present some modelingexamples. Figures 4, 5, 6 and ?? show open trigonometric polynomial planarcurves generated by using the shape preserving trigonometric interpolationspline curves given in this paper. The plots of the given examples, for the same


control polygon, are obtained for different values of β and α. It can be seenthat the space curve preserve nice feature for the space interpolation points.Figures ??, 7 and 8 show closed trigonometric polynomial curves generated byusing the shape preserving trigonometric interpolation spline curves, obtainedfor different values of β and α. Trigonometric polynomial B-spline model isa powerful tool for constructing freeform curves and surfaces in CAGD. Wecan construct tensor product trigonometric polynomial B-surfaces just like B-spline surfaces; the graph control polygon and the trigonometric parametricinterpolation spline surface, for different values of the tension parameters β,λ, α and µ, are illustrated in figures 9, 10, 11 and 12. Note that, for eachillustration example we took the following : ui = i × h with h = π

2β(m−3),

αi = α and µi = µ, ∀ i. The values of α, β, λ and µ are given in the figurecaptions.

(a) Control polygon. (b) β = 0.4, α = 0.2. (c) β = 0.6, α = 0.2. (d) β = 1.3, α = 0.2.

Figure 4: Open planar curves n = 7.






(a) Control polygon. (b) β = 0.4, α = 0.1. (c) β = 0.6, α = 0.1. (d) β = 1, α = 0.1.

Figure 7: Closed planar curves n = 9.


Figure 8: Closed planar curves n = 7.


Figure 9: Trigonometric parametric spline surface of face Nerfertiti with dif-ferent values of shape and tension parameters for n = 7.

5 Conclusion

The trigonometric polynomial blending functions constructed in this paperhave the properties analogous to those of the quintic Bernstein basis func-tions and the trigonometric Bezier curves are also analogous to the quinticBezier ones. In this basis we included the tension parameter which is mainlyimportant for object visualization. The trigonometric Bezier curves are closeto the control polygon. Therefore, these trigonometric Bezier curves can pre-serve the shape of the control polygon. For any shape parameters satisfyingthe shape preserving conditions, the obtained shape preserving trigonomet-ric interpolation spline curves are all continuous. There is no need to solve



Figure 10: Trigonometric parametric spline surface of face Nerfertiti withdifferent values of shape and tension parameters for n = 9.


Figure 11: Trigonometric parametric spline surface of human head with dif-ferent values of shape and tension parameters for n = 7.


Figure 12: Trigonometric parametric spline surface of human head with dif-ferent values of shape and tension parameters for n = 9.

a linear system and the changes of a local shape parameter will only affecttwo curve segments. Numerical examples indicate that our method can be


applied to generate nice features preserving space curves and surfaces. Gen-eralizing the idea to quasi-interpolation with trigonometric spline curve andtensor product surfaces will be reported in a future paper.

Acknowledgements. The authors are grateful to the University Hassan1st for their support.

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Received: March 6, 2015; Published: April 14, 2015

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A Method for Local Interpolation with Tension ...m-hikari.com/ams/ams-2015/ams-61-64-2015/p/lamniiAMS61-64-2015.pdfInterpolation with tension trigonometric spline curves and surfaces