Extensions of Uniform Cubic B-Spline Curve with Local Shape Parameters

In order to construct B-spline curves with local shape control parameters, a class of polynomial basis functions with two local shape parameters is presented. Properties of the proposed basis functions are analyzed and the corresponding piecewise polynomial curve is constructed with two local shape control parameters accordingly. In particular, the G1 continuous and the shapes of other segments of the curve can remain unchangeably during the manipulation on the shape of each segment on the curve. Numerical examples illustrate that the constructed curve fit to the control polygon very well. Furthermore, its applications in curve design is discussed and an extend application on surface design is also presented. Modeling examples show that the new curve is very valuable for the design of curves and surfaces.

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

International Journal of Image and Graphics ◽

10.1142/s0219467812500283 ◽

2012 ◽

Vol 12 (04) ◽

pp. 1250028

Author(s):

MRIDULA DUBE ◽

REENU SHARMA

Keyword(s):

Trigonometric Polynomial ◽

Shape Parameters ◽

B Spline ◽

Spline Curve ◽

Control Polygon ◽

Spline Curves ◽

Spline Surfaces ◽

Polynomial Curve ◽

Trigonometric Spline ◽

The Given

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.

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Two Extensions of the Quadratic Nonuniform B-Spline Curve with Local Shape Parameter Series

Mathematical Problems in Engineering ◽

10.1155/2021/9980320 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Xiang Kong ◽

Jun Chen

Keyword(s):

Line Segment ◽

Shape Parameter ◽

Linear Functions ◽

Local Shape ◽

B Spline ◽

Numerical Examples ◽

Spline Curve ◽

Control Polygon ◽

Tangent Direction ◽

Special Case

Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.

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