A Method for Local Interpolation with Tension Trigonometric Spline Curves and Surfaces

Transcription

1 Applied Mathematical Sciences, Vol. 9, 015, no. 61, HIKARI Ltd, A Method for Local Interpolation with Tension Trigonometric Spline Curves and Surfaces Abdellah Lamnii Univ. Hassan 1 er, Laboratoire de Mathématiques Informatiques et Sciences de l Ingnieur, 6000, Settat, Morocco Fatima Oumellal Univ. Hassan 1 er, Laboratoire de Mathématiques, Informatiques et Sciences de l Ingénieur 6000, Settat, Morocco Copyright c 015 Abdellah Lamnii and Fatima Oumellal. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this work a family of tension trigonometric curves analogous to those of cubic Bézier curves is presented. Some properties of the proposed curves are discussed. We propose an efficient interpolating method based on the tension trigonometric splines with various properties, such as partition of unity, geometric invariance and convex hull property, etc. This new interpolating method is applied to construct curves and surfaces. Moreover, one can adjust the shape of the constructed curves and surfaces locally by changing the tension parameter, the latter is included mainly because of its importance for object visualization. To illustrate the performance and the practical value of this model as well as its accuracy and efficiency, we present some modeling examples. Mathematics Subject Classification: 65T40, 65D05, 65D17, 76B45. Keywords: Trigonometric Bézier Curves, Tension Parameter, Shape Parameter, Interpolation.

2 300 Abdellah Lamnii and Fatima Oumellal 1 Introduction Let P 1,..., P N be an ordered set of points in the plane and suppose that B 1,..., B N is a set of functions defined on the unit interval I := [0, 1] with the following properties: Then B i t 0, for all t I, 1 N B i t = 1, for all t I, i=1 C N t := N P i B i t, for t I, 3 i=1 is the parametric representation of a curve which lies in the convex hull of the points P 1,..., P N. These points are called the control points and the associated polygon C N formed by connecting the points with straight lines is called the control polygon. In general, curve 3 does not pass through the control points, it just approximates the shape of the control polygon determined by these points. Designers need a curve representation that is directly related to the control points and is flexible enough to bend, twist or change the curve shape by changing one or more control points. For these reasons, Bézier curves arepowerful tools for constructing curves with free form. 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 trigonometric functions or the blending of polynomial and trigonometric functions in [1, 8, 9, 11, 1, 15, 16]. The idea of utilizing normalized trigonometric basis that form a partition of unity to construct shape-preserving curves has been suggested by Pena [9]. It proves that a curve in trigonometric form is the linear combination of its control points and certain Bernstein-like trigonometric functions, which form a normalized totally positive B-basis of the space C m = {1, cost,..., cosmt}, with optimal shape preserving properties. This representation have no shape parameters; hence the shape of the curves or surfaces cannot be modified when their control points are determined. In order to improve the shape of a curve and adjust the extent where a curve approaches its control polygon, we included the tension parameter which is mainly important for object visualization. More recently, Lamnii et al. in [3], have presented the quartic trigonometric polynomial blending functions where they have included the tension parameter β, which is mainly important for object visualization. In [6], Lamnii and Oumellal have constructed

3 Interpolation with tension trigonometric spline curves and surfaces 301 the Trigonometric Bzier curves whith n = 5 followed by a construction of the shape preserving interpolation spline curves with local shape parameters. In this work we propose to generalize this technique for n = 4k 1 and n = 4k + 1, k 1, more precisely, the basic idea of this method is to incorporate the parameter β into the B-basis functions of C m, where the parameter β can adjust the shape of the curves without changing the control points. As an application, we present trigonometric blending functions with shape parameters; without solving a linear system, these blending functions can be applied to construct continuous shape preserving local interpolation trigonometric spline curves with local shape parameters. Recently in [6] we used tensor-product for surfaces interpolation, we would like to generalize this technique for sphere like-surfaces. Let S be a closed and bounded surface in R 3 which is topologically equivalent to a unit sphere. Given a set of scattered points P 1,..., P s located on S, along with data values r 1,..., r s corresponding to these points. In many practical applications, we wish to find a smooth function F defined on S which interpolates or approximates the given data set so that its associated closed surface S F := {F ss, s S} be of class C 1 on S. A very popular method for fitting gridded data is to use tensor-products because of their computational simplicity and their many attractive geometric properties. In this paper, we π suppose that S is the unit sphere centered at the origin. Let I = [0, ] and β π J = [0, ], by using spherical coordinates we can identify the sphere S with λ the rectangle R := I J by the mapping χ : S R cos4πβθ cosπλφ θ, φ cos4πβθ sinπλφ sinπλθ Now since the problem is defined on a rectangle R we can use the tensorproduct methods. This approach has been studied in several papers with various choices of basic functions see, for instance, [, 13] and references therein. More specifically, Schumaker and Traas [13] have chosen the periodic trigonometric B-splines of order three and the quadratic polynomial B-splines. Recently, another approach was proposed in [] and [4] using πperiodic uniform algebraic trigonometric B-splines UAT B-splines of order four generated over the space spanned by {1, t, cost, sint} and the cubic polynomial B-splines. All these methods use quasi-interpolants or approximants while our thechnique interpolates the data in a local way and gives a good shape visulisation. The remainder of this paper is organized as follows. In section, the expression of the generalized trigonometric polynomial blending functions are described and the properties of these functions are discussed. Trigonometric Bzier curves are constructed in this section. In section 3, the trigonometric parametric curve segments as well as the trigonometric polynomial interpolation are discussed. In section 4, several numerical examples are presented

4 30 Abdellah Lamnii and Fatima Oumellal in which open and closed trigonometric curves are described. We end this section by constructing tensor product trigonometric polynomial B-surfaces just like B-spline surfaces. A brief conclusion is given in section 5. Trigonometric polynomial curves with shape parameter In this section, we will define the trigonometric polynomial blending functions with shape parameter β and the corresponding trigonometric Bézier curves..1 Trigonometric basis functions Let β > 0 be an arbitrarily fixed parameter. We will produce trigonometric polynomial curves, that are described by means of the combination of control points and basis functions, therefore we need a proper basis in the space of trigonometric polynomials C n,β = {1, cosβt,..., cosβnt}. We will use a transformed version of the basis specified in Pea [9]. Let B n,β = B0,β n,..., n,β Bn be the system of functions given by B n i,β t = n i [ ] cos n i βt sin i π βt, i = 0,..., n, t 0,. 4 β Each trigonometric polynomial function B n 1 i,β t of order n 1 can be expressed as a linear combination of trigonometric polynomial function Bi,β n t of order n. More precisely we have: B n 1 i,β n i t = n B n i,β t + i + 1 n Bi+1,β n t 5 Theorem.1 The family B n,β forms a basis for the space C n,β. Proof Let us prove this by induction on n. For n = 0, this case is trivial because the family C 0,β = {1} and we have only B0,β 0 t = 1. Now suppose that B n 1 i,β t, i = 0,..., n 1 form a basis for the space C n 1,β and let us show that Bi,β n t, i = 0,..., n form a basis for the space C n,β. It is easy to see that DimC n,β = n + 1 because cosβkt, k = 0,..., n are linearly independent and since Bi,β n 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, cosβkt can be written as a linear combination of B n 1 i,β t, i = 0,..., n 1. On the other hand, by using the formula 5, cosβkt can also be written as a linear combination of Bi,β n t, i = 0,..., n. It remains only to write cosβnt as a linear combination of Bi,β n t, i = 0,..., n.

5 Interpolation with tension trigonometric spline curves and surfaces 303 According to Moivre s formula, we have for any real number t and for any integer on n: cos t + i sin t n = cosnt + i sinnt. Thus, cosβnt + i sinβnt = cosβt + i sinβt n = n n s s=0 cosβt n s i s sinβt s So, cosβnt represents the real part of the complex number cosβnt = = n n cosβt n k 1 k sinβt k k k=0 n 1 k n cosβt n k sinβt k k k=0 Hence the result. The following result shows that B n,β gives the optimal shape preserving representation of C n,β. As 1 = cos βt + sin βt n, hence n i=0 n i cos n i βt sin i βt = 1. Therefore, B n,β is normalized. B n,β is equivalent to the Bernstein basis on [0, 1] by the transformation x = sin βt. Consequently, B n,β possess the following properties analogous to those of the Bernstein basis: 1. Nonnegativity : t [0, π ], β Bn i,β t 0.. Partition of unity : n Bi,βt n = 1. i=0 3. Symmetry : B n i,β t = Bn n i,β π β t. 4. Maximum : Each B n i,β has one maximum value in [0, π β ]. Fig. 1 and show the curves of the blending functions for n = 7 and n = 9. For t [0, π ] and β > 1 we observed that the blending functions start to duplicate as a function of the tension parameter value. So, we included the tension parameter in the interval to become t [0, π ], in this interval no β more duplications are observed. Let us observe that B0,β 0 = 1. For the following recurrence formulas, we shall use the following notation: B n i,βt = 0, i / {0,..., n}.

6 304 Abdellah Lamnii and Fatima Oumellal a β = 3 and t [0, π ]. b β = 3 and t [0, π β ]. Figure 1: Normalized trigonometric basis functions for n = a β = and t [0, π ]. b β = and t [0, π β ]. Figure : Normalized trigonometric basis functions for n = 9. Proposition. The trigonometric basis functions given in 4 and defined on [0, π β ] satisfy: Proof From 4, we get Bi,β n t = cosβt B n 1 i,β t + sinβt B n 1 i 1,β t, 0 i n. 6 cos βtb n 1 i,β t + sin βtb n 1 n 1 i 1,β t = cos βt sin i βt cos n 1 i βt i n 1 + sin βt sin i 1 βt cos n 1 i 1 βt = = = i 1 n 1 sin i βt cos n i βt + i n 1 + sin i βt cos n i βt n 1 i 1 i n sin i βt cos n i βt i = Bi,β n t. n 1 sin i βt cos n i βt i 1. Trigonometric Curves of Order n Given control points V i i = 0,..., n in R or R 3. Then, B β t := B β t; V 0, V 1,..., V n = n Bi,β n tv π i, t [0, ], 7 β i=0 is called a trigonometric polynomialbézier curve of order n with a global shape parameter.

7 Interpolation with tension trigonometric spline curves and surfaces 305 From the properties of the trigonometric basis functions 4, some properties of the trigonometric Bzier curve 7 can be obtained as follows. 1. Endpoint interpolation property: with a simple computation, we have: B β 0 = V n, B β 0 = 0, B β 0 = nβ V n 1 V n, B β π β = V 0, B β π β = 0, B β π β = nβ V 1 V 0.. Symmetry: V 0, V 1,..., V n and V n, V n 1,..., V 0 define the same trigonometric Bzier curve, i.e., B β t; V 0, V 1,..., V n = B β π β t; V n, V n 1,..., V Geometric invariance: since the blending functions have the properties of partition of unity, the shape of these trigonometric Bézier curves is independent of the choice of coordinates. 4. Convex hull property: the blending functions have the properties of nonnegativity and partition of unity, as a consequence, the entire trigonometric Bzier curve segment must lie inside the control polygon spanned by V 0,..., V n. 5. Variation diminishing property : no straight line intersects a Bézier curve more times than it intersects its control polygon. 6. Convexity-preserving property: the variation diminishing property means the convexity preserving property holds. Now we shall provide a de Casteljau-type algorithm for the evaluation of the curve B β t. If the V 0,..., V n points in R or R 3, are given and t [0, π ] β let us define the points B j i,β t = cos βtb j 1 i 1,β t + sin βtb j 1 i,β t, j = 1,..., n, i = j,..., n, 9 where B 0 i,β t = V i, then B n n,β t = B βt. Figure 3, illustrates two examples of evalution for the curve B β t for t = π left: n = 5 and right: n = 7. 4β 8 3 Interpolating trigonometric spline curve Our objective is to generalize the Overhauser spline curve by solving the following interpolation problem. Given the sequence of data points {P i } m i=0 with associated parameter values u 1 < u < < u m 1. Find the control points of a trigonometric spline curve that interpolates the given data points and that consists of arcs of type 7.

8 306 Abdellah Lamnii and Fatima Oumellal B 1,β B 3 3,β B 0,β B 1 3,β B3,β B 3 B4,β 4,β B4,β 4 B5,β 5 B5,β 4 B 0 3,β B 3 5,β B 1 4,β B 0,β B 3,β B3,β 0 B4,β 1 B4,β 0 B4,β B 3 B5,β B3,β 1 B4,β 3 5,β B5,β 4 B 5 B 6,β 6,β 4 B 3 B5,β 1 6,β B 5 5,β B 6 B7,β 7 6,β B7,β 6 B 5 7,β B6,β B5,β 0 B 4 4,β B 4 7,β B 0 1,β B,β B 5,β B 0 4,β B 1,β B 3 3,β B 3 7,β B 1 6,β B 0 1,β B,β B 7,β B 0 6,β B 1 1,β B 1 5,β B 1 1,β B 1 7,β B 0 0,β B 0 5,β B 0 0,β B 0 7,β a n = 5, t = π 4β and β = 0.5. b n = 7, t = π 4β and β = 0.. 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 curve blending, 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 P i, i = 0, 1,..., n+1 and the trigonometric Bzier control points V i, i = 0,..., n. We propose the following endpoints requirements. If n = 4k + 1, we have If n = 4k 1, we have B β 0 = V n = P k, B β 0 = 0, B β 0 = nβ V n 1 V n = α 1 P k P k+1, B β π β = V 0 = α P k+1 P 0, B β π β = 0, B β π β = nβ V 1 V 0 = P k+1, 10 B β 0 = V n = P k+1, B β 0 = 0, B β 0 = nβ V n 1 V n = α 1 P k 1 P k, B β π β = V 0 = α P k+1 P 0, B β π β = 0, B β π β = nβ V 1 V 0 = P k 1, 11 where α 1, α [0, + [ are shape parameters. We noted that these end points requirements are analogous to those given in [17]. Define φ 0,β t, α 1, α = T B 0,β t, α 1, α := α 1 nβ B n 1,β t α 1 nβ B n,β t 3α 1 nβ B n 3,β t kα 1 nβ B n k,β t, φ 1,β t, α 1, α = T B k,β t, α 1, α := B n 0,β t + + Bn k,β t + kα nβ B k+1,β t + + α nβ B n n 1,β t, φ,β t, α 1, α = T B k+1,β t, α 1, α := α 1 nβ B n 1,β t + + kα 1 nβ B n k,β t + Bn k+1,β t + + Bn n,β t, φ 3,β t, α 1, α = T B k+1,β t, α 1, α := kα nβ Bn k+1,β t k 1α nβ B n k,β t α nβ Bn n 1,β t, 1

9 Interpolation with tension trigonometric spline curves and surfaces 307 for n = 4k + 1, and φ 0,β t, α 1, α = T B 0,β t, α 1, α := α 1 nβ B1,β n t α 1 nβ B,β n t 3α 1 nβ B3,β n t k 1α 1 nβ Bk 1,β n t, φ 1,β t, α 1, α = T B k 1,β t, α 1, α := B0,β n t + + Bn k 1,β t + k 1α nβ B k,β t + + α nβ Bn 1,β n t, φ,β t, α 1, α = T B k+1,β t, α 1, α := α 1 nβ B1,β n t + + k 1α 1 nβ Bk 1,β n t + Bn k,β t + + Bn n,β t, φ 3,β t, α 1, α = T B k,β t, α 1, α := k 1α B nβ k,β n t k α B nβ k 1,β n t α nβ Bn n 1,β t. 13 for n = 4k 1. The rest of T B i,β t, α 1, α are null. Using equations 10, 11 and the blending functions, the curve segment can be generated as follows Proposition 3.1 For t [0, π ], we have β P β t, α 1, α = n Bi,β n tv i = i=0 n+1 i=0 T B i,β t, α 1, α P i 14 proof Let T B 0,β t, α 1, α = a 00 B0,β n t + a 01B1,β n t + + a 0nBn,β n t T B 1,β t, α 1, α = a 10 B0,β n t + a 11B1,β n t + + a 1nBn,β n t. T B n+1,βt, α 1, α = a n+1 0Bn 0,β t + a n+1 1Bn 1,β t + + a n+1 nbn n,β t. From 14 and 15, we have 15 n Bi,β n tv i = i=0 Then, we have Furthermore, we get a 00 B n 0,β t + a 01B n 1,β t a 0nB n n,β t P a n+1 0Bn 0,β t + a n+1 1Bn 1,β t + + a n+1 nbn n,β t P n+1. B0,β n tv 0 = B0,β a n t 00 P a n+1 n+1 0P,. Bn,β n tvn = Bn n,β a t 0n P a n+1 n+1 np V 0 = a 00 P a n+1. V n = a 0n P a n+1 n+1 0P, n+1 np. According to 16 and using 10 and 11, we deduce that T B j,β t, α 1, α can be written in the form 1 and 13. n+1 Remark 3. The φ i,β, i = 0,, n+1, verify the partition of unity:. i=0 16 φ i,β = 1.

10 308 Abdellah Lamnii and Fatima Oumellal 3. Trigonometric parametric spline curves Let P i R d i = 0,..., m, d =, 3 be the interpolation points, U = u 1, u,..., u m 1 the knot vector where u 1 < u <... < u m 1 and the shape parameters α i [0, + [, i = 1,..., m 1. For t [0, π ] and i = 1,..., m, the ith trigonometric parametric β curve segment is given as a function of φ j,β, j = 0, 1,, 3, by the following expression: if n = 4k + 1, P i,β t, α i, α i+1 = φ 0,β t, α i, α i+1 P i 1 + φ 1,β t, α i, α i+1 P i+k φ,β t, α i, α i+1 P i+k + φ 3,β t, α i, α i+1 P i+k, if n = 4k 1, P i,β t, α i, α i+1 = φ 0,β t, α i, α i+1 P i 1 + φ 1,β t, α i, α i+1 P i+k 18 +φ,β t, α i, α i+1 P i+k + φ 3,β t, α i, α i+1 P i+k 1. The corresponding trigonometric parametric spline curve, composed by all of the trigonometric parametric curve segments are defined as follows: P β u = P i,β π β u u i u i, α i, α i+1, u [u i, u i+1 ], i = 1,..., m 1, 19 where u i = u i+1 u i. Theorem 3.3 The curve 19 has nd geometric continuity, i.e., it is a GC continuous curve. Proof Without loss of generality we may put n = 4k + 1, for i = 1,, m 1, we have P β u + i = P i,β0, α i, α i+1 = P i+k 1 P β u i+1 = P i,β π β, α i, α i+1 = P i+k P β u+ i = P i,β 0, α i, α i+1 = 0 P β u i+1 = P i,β π β, α i, α i+1 = 0 P β u+ i = P i,β 0, α i, α i+1 = πα 1P i 1 nβ P i +α 1 P i+k β 1 P β u i+1 = P i,β π β, α i, α i+1 = πα 1P i nβ P i+1 +α 1 P i+k+1 β 1 As described in [17], P β u interpolates the interpolation points P i, i = 0,..., m. As an example, adding two control interpolation points P 1, P m+1, two knots u 0, u m, and two shape parameters α 0, α m are sufficient to construct an open curve P β u interpolating all of the points P i, i = 0,..., m. Closed Bézier curves are generated by specifying the first and the last control points at the same position. For constructing a closed curve P β u interpolating all of the points P i, i = 0,..., m, we have to add three interpolation points P 1 = P m, P m+1 = P 0, P m+ = P 1 three knots u 0, u m, u m+1 and three shape parameters α 0, α m, α m+1.

11 Interpolation with tension trigonometric spline curves and surfaces Trigonometric parametric spline surfaces In this section, we apply the trigonometric parametric spline to a set of data points on a rectangular grid for surface interpolation using the well-known tensor product form. A surface may be defined by the tensor product of two curves so that the properties of the blending functions are not modified. Whereas a curve requires one tension parameter for its definition, a surface requires two tension parameters β > 0 and λ > 0. Similarly to the work done by Liu et al. see [7], we define trigonometric parametric spline surfaces as a tensor product. More precisely we have the following definition. Definition 3.4 Given r s interpolation points P ij i = 0, 1,..., r; j = 0, 1,..., s, two knot vectors U = [u 1, u,..., u r 1 ] and V = [v 1, v,..., v s 1 ] and two shape parameters vectors α = [α 1, α,..., α r 1 ] and µ = [µ 1, µ,..., µ s 1 ]. For tension parameters β > 0 and λ > 0, the trigonometric parametric spline surface patch has the form : if n = 4k + 1, S β,λ i,j = φ 0,j,λ φ0,i,β P i,j + φ 1,i,β P i+k 1,j+k + φ,i,β P i+k+1,j+k 1 + φ 3,i,β P i+k+,j+k φ 1,j,λ φ0,i,β P i,j 1 + φ 1,i,β P i+k 1,j+k 1 + φ,i,β P i+k+1,j+k + φ 3,i,β P i+k+,j+k + φ,j,λ φ0,i,β P i,j + φ 1,i,β P i+k 1,j+k + φ,i,β P i+k+1,j+k+1 + φ 3,i,β P i+k+,j+k+1 + φ 3,j,λ φ0,i,β P i,j+1 + φ 1,i,β P i+k 1,j+k+1 + φ,i,β P i+k+1,j+k+ + φ 3,i,β P i+k+,j+k+ if n = 4k 1, S β,λ i,j = φ 0,j,λ φ0,i,β P i,j + φ 1,i,β P i+k,j+k 3 + φ,i,β P i+k+1,j+k 1 + φ 3,i,β P i+k+1,j+k 1 + φ 1,j,λ φ0,i,β P i,j 1 + φ 1,i,β P i+k,j+k + φ,i,β P i+k+1,j+k + φ 3,i,β P i+k+1,j+k 1 + φ,j,λ φ0,i,β P i,j + φ 1,i,β P i+k,j+k 1 + φ,i,β P i+k+1,j+k+1 + φ 3,i,β P i+k+1,j+k + φ 3,j,λ φ0,i,β P i,j+1 + φ 1,i,β P i+k,j+k + φ,i,β P i+k+1,j+k+ + φ 3,i,β P i+k+1,j+k+1 where u [0, π β ], v [0, π ], i = 0,..., r 1, j = 0,..., s 1 and Sβ,λ λ i,j := S β,λ i,j u, v, α i, α i+j, µ j, µ j+1, φ l,i,β := φ l,β u, α i, α i+1, φ l,j,λ := φ l,λ u, µ j, µ j+1, l = 0,..., 3. Then the trigonometric parametric spline surface is given by, S β,λ u, v = S β,λ i,j π β u u i u i u [u i, u i+1 ], v [v j, v j+1 ]., π λ v v j v j, α i, α i+1, µ j, µ j+1, 3.4 Trigonometric parametric spherical spline surfaces Let S be a sphere-like surface, i.e. a closed and bounded surface in R 3 which is topologically equivalent to a unit sphere. Given a set of scattered points P 1,..., P s located on S, along with data values r 1,...,r s corresponding to these points. In many practical applications, we wish to find a smooth function F defined on S which interpolates or approximates the given data set. 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 the approximation of a function defined on [0, π ] [0, β π λ ], but when using this

12 3030 Abdellah Lamnii and Fatima Oumellal approach, some periodicity conditions should be satisfied. Without loss of generality, we suppose that S is the unit sphere centered at the origin. Let π π I = [0, ] and J = [0, ]. By using spherical coordinates we can identify β λ the sphere S with the rectangle R := I J by the mapping χ : S R cos4πβθ cosπλφ θ, φ cos4πβθ sinπλφ sinπλθ Hence, to construct the function F, it suffices to find a function f := F oχ : R R satisfying fθ i, φ i = r i, i = 1,..., s, where θ i, φ i are the polar coordinates of P i. To make sure that a continuous function f defined on the rectangle R remains continuous after mapping it onto S, it is necessary that f satisfies the following conditions [10]: f0, θ = f π λ, θ, 0 θ π β, there exist constants S N, S S such that { fφ, 0 = SN, 0 φ π λ, 3 fφ, π β = S S, 0 φ π λ, Now since the problem is defined on a rectangle we can use the results of the previous section. More precisely, let r s be the interpolation points P i,j = θ ij, φ ij 0 i,j r 1,s 1. For constructing a spherical spline S β,λ interpolating all of the points P i,j = θ ij, φ ij 0 i,j r 1,s 1 and satisfies the conditions 3, we have to add three interpolation vectors P 1,j = P r 1,j, P r,j = P 0,j, P r+1,j = P 1,j and we take P 1,0 = P 0,0 =... = P r+1,0, P 1,s 1 = P 0,s 1 =... = P r+1,s 1. 4 Numerical Examples and Application In order to justify the accuracy and efficiency of our presented trigonometric functions 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 problems related to approximation theory, numerical analysis or computer graphics, for example representation of splines. In order to illustrate the performance and the practical value of this model, we will represent some curves and surfaces so that we can justify the accuracy and efficiency of our presented trigonometric functions. In order to do so we will present some modeling examples. Figures 4, 5, 6 and?? show open trigonometric polynomial planar curves generated by using the shape preserving trigonometric interpolation spline curves given in this paper. The plots of the given examples, for the same

13 Interpolation with tension trigonometric spline curves and surfaces 3031 control polygon, are obtained for different values of β and α. It can be seen that the space curve preserve nice feature for the space interpolation points. Figures??, 7 and 8 show closed trigonometric polynomial curves generated by using the shape preserving trigonometric interpolation spline curves, obtained for different values of β and α. Trigonometric polynomial B-spline model is a powerful tool for constructing free- form curves and surfaces in CAGD. We can construct tensor product trigonometric polynomial B-surfaces just like B- spline surfaces; the graph control polygon and the trigonometric parametric interpolation spline surface, for different values of the tension parameters β, λ, α and µ, are illustrated in figures 9, 10, 11 and 1. Note that, for each illustration example we took the following : u i = i h with h = α i = α and µ i = µ, i. The values of α, β, λ and µ are given in the figure captions. π, βm 3 a Control polygon. b β = 0.4, α = 0.. c β = 0.6, α = 0.. d β = 1.3, α = 0.. 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 = 1.3, α = 0.1. Figure 5: Open planar curves n = 9. a Control polygon. b β = 0.4, α = 0.5. c β = 0.6, α = 0.5. d β = 1., α = 0.5. Figure 6: Open planar curves n = 7.

14 303 Abdellah Lamnii and Fatima Oumellal 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. a Control polygon. b β = 0.4, α = 0.5. c β = 0.6, α = 0.5. d β = 1.5, α = 0.5. Figure 8: Closed planar curves n = 7. a Control polygon. b β = 0.4, α = 0.8. c β = 0.45, α = 0.6. d β = 0.7, α = Figure 9: Trigonometric parametric spline surface of face Nerfertiti with different values of shape and tension parameters for n = 7. 5 Conclusion The trigonometric polynomial blending functions constructed in this paper have the properties analogous to those of the quintic Bernstein basis functions and the trigonometric Bézier curves are also analogous to the quintic Bézier ones. In this basis we included the tension parameter which is mainly important for object visualization. The trigonometric Bézier curves are close to the control polygon. Therefore, these trigonometric Bézier curves can preserve the shape of the control polygon. For any shape parameters satisfying the shape preserving conditions, the obtained shape preserving trigonometric interpolation spline curves are all continuous. There is no need to solve

15 Interpolation with tension trigonometric spline curves and surfaces 3033 a Control polygon. b β = 0.4, α = 0.8. c β = 0.45, α = 0.6. d β = 0.7, α = Figure 10: Trigonometric parametric spline surface of face Nerfertiti with different values of shape and tension parameters for n = 9. a Control polygon. b β = 0., α = c β = 0.7, α = d β = 0.85, α = 0.1. Figure 11: Trigonometric parametric spline surface of human head with different values of shape and tension parameters for n = 7. a Control polygon. b β = 0., α = c β = 0.75, α = d β = 0.85, α = 0.1. Figure 1: Trigonometric parametric spline surface of human head with different values of shape and tension parameters for n = 9. a linear system and the changes of a local shape parameter will only affect two curve segments. Numerical examples indicate that our method can be

16 3034 Abdellah Lamnii and Fatima Oumellal applied to generate nice features preserving space curves and surfaces. Generalizing the idea to quasi-interpolation with trigonometric spline curve and tensor product surfaces will be reported in a future paper. Acknowledgements. The authors are grateful to the University Hassan 1 st for their support. References [1] P. Alfeld, M. Neamtu, and L. L. Schumaker, Circular Bernstein-Brzier polynomials, in: M. Daehlen, T. Lyche, and L. L. Schumaker,eds., Mathematical Methods for Curves and Surfaces, Vanderbilt University Press, Nashville, 1995, [] E.B. Ameur, D. Sbibih, A. Almhdie, C. Lger, New spline quasiinterpolant for fitting 3D data on the sphere, IEEE Signal Process. Lett. 145, 007, [3] A. Lamnii, F. Oumellal, J. Dabounoua, Tension Quartic Trigonometric Bzier Curves Preserving Interpolation Curves Shape, International Journal of Mathematical Modelling & Computations, in press. [4] A. Lamnii, H. Mraoui, D. Sbibih, A. Zidna, A multiresolution method for fitting scattered data on the sphere,bit Numer Math, 49, 009, [5] A. Lamnii, H. Mraoui, D. Sbibih, A. Zidna, Uniform tension algebraic trigonometric spline wavelets of class C and order four, Mathematics and Computers in Simulation, 87, 013, [6] A. Lamnii and F. Oumellal, Tension Interpolation Spline Bzier Curves,Journal of Advanced Research in Applied Mathematics, in press. [7] H. Liu, Lu Li, D. Zhang, H. Wang, Cubic Trigonometric Polynomial B- spline Curves and Surfaces with Shape Parameter,Journal of Information & Computational Science, 9 4, 01, [8] H. Liu, Lu Li, D. Zhang, Blending of the Trigonometric Polynomial Spline Curve with Arbitrary Continuous Orders, Journal of Information & Computational Science 11 1, 014,

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