Trigonometric Spline Curves Research Articles

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.

A New Class of Trigonometric B-Spline Curves

We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties.

Constrained for G1 cubic trigonometric spline curve interpolation

This paper presented the G1 cubic trigonometric spline function with three shape parameters that generate a constrained curve that interpolates 2D data. The purpose of this research is to ensure the generated curve passes through all data point yet satisfied the three cases of line constraints given. The three cases are: the data must lie above line Li the data must lie below line Li and lastly, the data must lie between two lines Li,1 and Li,2. A simpler theorem is implemented involving the roles of shape parameters. Two of the shape parameters are set free, while another parameter is fixed to fulfil all the three cases stated. The results show that a smooth curve of the G1 cubic trigonometric spline function can be produced within the constrained line by using the theorem developed while the hereditary shape of the data is preserved. Numerical examples are illustrated and discussed.

A Rational Quadratic Trigonometric Spline (RQTS) as a Superior Surrogate to Rational Cubic Spline (RCS) with the Purpose of Designing

A key technique for modeling 2D objects is built using a Bézier-like rational quadratic trigonometric function with two form parameters. Since they are generated employing weights, the suggested rational quadratic trigonometric spline curve schemes are helpful for shape modeling. The established method yields a curve with the best geometric properties, such as convex hull, partition of unity, affine invariance, and diminishing variation. The parameters in the suggested method’s construction are helpful for several critical shape features such as local, global, and biased tension qualities, which give good control over the curve and allow for shape modification as desired. In addition, the recommended approach is C2. Furthermore, the comparative study of both splines is discussed which revealed the proposed method as a superior alternative to rational cubic spline.

Shape-Preserving Positive Trigonometric Spline Curves

This paper deliberates on the development of new smooth shape-preserving schemes. These schemes are also demonstrated with data in the form of shape-preserving trigonometric spline curves. For this persistence, a trigonometric spline function is developed, which also nourishes all the fundamental geometric properties of Bezier function as well. The developed trigonometric spline function comprises two parameters \( \alpha_{i} \) and \( \beta_{i} \), which ensures flexible tangents at the end points of each subinterval. Furthermore, constraints are derived on \( \beta_{i} \) to generate the shape-preserving trigonometric spline curves, whereas \( \alpha_{i} \) is any positive real number used for the modification of shape-preserving trigonometric spline curves. The error approximation of the developed function is \( O\left( {h_{i}^{3} } \right) \).

A method for local interpolation with tension trigonometric spline curves and surfaces

A method for local interpolation with tension trigonometric spline curves and surfaces

Blending of the Trigonometric Polynomial Spline Curve with Arbitrary Continuous Orders

In order to develop the theory of trigonometric polynomial spline curve, the representation of trigonometric polynomial spline curve is blended to a general form based on blending of trigonometric polynomial curves. Moreover, some properties of the blending curve are discussed in details. The research shows that the basis of the trigonometric polynomial curve is relative simple, and the blending curve includes the original trigonometric polynomial spline curve and shows much better shape-control capability than the original curve. Meanwhile, the blending curve keeps the same degree of original one. It is easy to find that the curve can be reshaping by adjusting the shape factor. At the same time, a new method of the representation of closed curve is given and can also accurately represent the whole ellipse arc and other conic curve.

A scheme for interpolation with trigonometric spline curves

We present a method for the interpolation of a given sequence of data points with Cn continuous trigonometric spline curves of order n+1 (n≥1) that are produced by blending elliptical arcs. Ready to use explicit formulae for the control points of the interpolating arcs are also provided. Each interpolating arc depends on a global parameter α∈(0,π) that can be used for global shape modification. Associating non-negative weights with data points, rational trigonometric interpolating spline curves can be obtained, where weights can be used for local shape modification. The proposed interpolation scheme is a generalization of the Overhauser spline, and it includes a Cn Bézier spline interpolation method as the limiting case α→0.

Cubic TP B-Spline Curves with a Shape Parameter

In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.

Description and Smoothing of NC Motion Path Based on the Cubic Trigonometric Interpolation Spline

In order to better describe the complex motion path of NC machining and realize the smooth transition between path segments, a kind of cubic trigonometric interpolation spline curve was put forward based on a set of special basis function. The spline curve which with adjustable shape satisfies the C1 continuity and it can accurately describe some common engineering curves such as straight line, circular arc and free curve. According to the given information of control points, different shapes of interpolation spline curve can be gotten by changing the adjustment coefficients. Through selecting proper control points and shape adjustment coefficients near the corner, insert the spline curve can realize the smooth transition at the corner of adjacent NC motion path segments, which can ensure the stability of motion path and the continuous of feed speed. Meanwhile, it also can reduce the impact to NC machine.

PIECEWISE QUARTIC TRIGONOMETRIC POLYNOMIAL B-SPLINE CURVES WITH TWO SHAPE PARAMETERS

Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

An approximating [formula omitted] non-stationary subdivision scheme

We present a 3-point C 2 approximating non-stationary subdivision scheme for designing curves. The weights of the masks of the scheme are defined in terms of some values of trigonometric B-spline basis functions. The scheme has some interesting properties and it is compared with the 2-point and 3-point schemes generating uniform trigonometric spline curves of order 3 and 5. The comparison and the performance of the scheme are demonstrated by examples.

Constrained curve drawing using trigonometric splines having shape parameters

The problem of drawing a smooth obstacle avoiding curve has attracted the attention of many people working in the area of CAD/CAM and its applications. In the present paper we propose a method of constrained curve drawing using certain C 1 -quadratic trigonometric splines having shape parameters, which have been recently introduced in [Han X. Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 2002;19:503–12]. Besides this, we have also presented a simpler approach for studying the approximation properties of the trigonometric spline curves.

A subdivision algorithm for trigonometric spline curves

In this paper we present a subdivision algorithm for the evaluation of the trigonometric spline curves with uniform knots.