Shape analysis of cubic trigonometric Bézier curves with a shape parameter (original) (raw)
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The cubic trigonometric Bézier curve with two shape parameters
Applied Mathematics Letters, 2009
A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.
A New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces
2016
A kind of quasi-cubic Bézier curves by the blending of algebraic polynomials and trigonometric polynomials using weight method is presented, named WAT Bézier curves. Here weight coefficients are also shape parameters, which are called weight parameters. The interval [0, 1] of weight parameter values can be extended to [ −2, π 2 / (π2 − 6 )] and the corresponding WAT Bézier curves and surfaces are defined by the introduced base functions. The WAT Bézier curves inherit most of properties similar to those of c Bézier curves, and can be adjusted easily by using the shape parameter λ. The jointing conditions of two pieces of curves with G2 and C4 continuity are discussed. With the shape parameter chosen properly, the defined curves can express exactly any plane curves or space curves defined by parametric equation based on{1, sint, cost, sint2t, cos2t} and circular helix with high degree of accuracy without using rational form. Examples are given to illustrate that the curves and surface...
WAT RATIONAL CUBIC TRIGONOMETRIC BEZIER CURVES AND ITS APPLICATIONS
In this paper, a kind of Rational cubic Bézier curves by combining algebraic polynomials and trigonometric polynomials, using the weight method is developed, named weight algebraic trigonometric (WAT) Rational Bézier curves. Here, weight coefficients are referred as shape parameters, which are called weight parameters. The value of weight parameters can be extended to the interval [0, 1] to [-2, 2.33], and the corresponding WAT Rational Bézier curves and surfaces are defined. The WAT Rational Bézier curves inherit most of properties similar to those of Cubic Bézier curves, and can be adjusted easily by using the shape parameter λ. The jointing conditions of two pieces of curves with the G2 and C 2 continuity are discussed. With the shape parameter chosen properly, the defined curves can express exactly in the form of plane curves or space curves defined by a parametric equation based on {1, Sint, cost, sint2t, cos2t} and circular helix and ellipse, with a high degree of accuracy without using rational form. Examples are given to illustrate that the curves and surfaces, which can be used as an efficient new model for geometric design in the fields of CAGD. It is clear that the existing techniques based on C-Bézier spline can approximate the Bézier curves only from a single side, the WAT Rational Bézier curves can approximate the Bézier curve from both the sides, and the change range of shape of the curves is wider than that of C-Bézier curves. It is important that this WAT Rational Bézier curves much be closer to the control point, compared to the other spline curves. The geometric effect in case of shape preservation of this weight parameter is also discussed.
Applied Mathematics and Computation, 2013
The rational quadratic trigonometric Bézier curve with two shape parameters is presented in this paper, which is new in literature. The purposed curve inherits all the geometric properties of the traditional rational quadratic Bézier curve. The presence of shape parameters provides a control on the shape of the curve more than that of traditional Bézier curve. Moreover the weight offers an additional control on the curve. Simple constraints for shape parameters are derived using the end points curvature so that their values always fall within the defined range. The composition of two segments of curve using G 2 and C 2 continuity is given. The new curves can accurately represent some conics and best approximates the traditional rational quadratic Bézier curve.
Mathematics, 2020
The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by ad...
The WAT Bézier Curves and Its Applications
In this paper, a kind of quasi-cubic Bézier curves by the blending of algebraic polynomials and trigonometric polynomials using weight method is presented, named WAT Bézier curves. Here weight coefficients are also shape parameters, which are called weight parameters. The interval [0, 1] of weight parameter values can be extended to [-2,π2/(π2-6)]. The WAT Bézier curves include cubic Bézier curves and C-Bézier curves () as special cases. Unlike the existing techniques based on C-Bézier methods which can approximate the Bézier curves only from single side, the WAT Bézier curves can approximate the Bézier curve from the both sides, and the change range of shape of the curves is wider than that of C-Bézier curves. The geometric effect of the alteration of this weight parameter is discussed. Some transcendental curves can be represented by the introduced curves exactly.
Quintic Trigonometric Bezier Curve with Two Shape Parameters
Sains Malaysiana, 2017
The fifth degree of trigonometric Bézier curve called quintic with two shapes parameter is presented in this paper. Shape parameters provide more control on the shape of the curve compared to the ordinary Bézier curve. This technique is one of the crucial parts in constructing curves and surfaces because the presence of shape parameters will allow the curve to be more flexible without changing its control points. Furthermore, by changing the value of shape parameters, the curve still preserves its geometrical features thus makes it more convenient rather than altering the control points. But, to interpolate curves from one point to another or surface patches, we need to satisfy certain continuity constraints to ensure the smoothness not just parametrically but also geometrically.
The Quintic Trigonometric Bézier Curve with single Shape Parameter
In this paper, we have constructed a quintic trigonometric Bézier curve with single shape parameter. The shape of the curve can be adjusted as desired, by simply altering the value of shape parameter, without changing the control polygon. The quintic trigonometric Bézier curve can be made close to the cubic Bézier curve or closer to the given control polygon than the cubic Bézier curve. Approximation property has been discussed.
QUARTIC TRIGONOMETRIC BÉZIER CURVE WITH A SHAPE PARAMETER
TJPRC, 2013
Analogous to the cubic Bézier curve, a quartic trigonometric Bézier curve with a shape parameter is presented in this paper. Each curve segment is generated by four consecutive control points. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. These curves are closer to the control polygon than the cubic Bézier curves, for all values of shape parameter. With the increase of the shape parameter, the curve approaches to the control polygon.
Shape Analysis of Cubic Bézier Curves – Correspondence to Four Primitive Cubics
Computer-Aided Design and Applications, 2014
In this paper, we show that any planar polynomial cubic Bézier curve can be described as an affine transformation of a part of four primitive cubics (x 3 + x 2 − 3y 2 = 0, x 3 − 3y 2 = 0, x 3 − x 2 − 3y 2 = 0, and x 3 − y = 0), and propose an algorithm to derive the transformation matrix. For a given cubic Bézier curve, we derive the linear moving line that follows the curve, and find the double point D of the curve. If D is not a point at infinity, we derive the quadratic parallel moving line that follows the curve. By testing whether the quadratic moving line crosses D or not, we classify the curve into three cases (crunode, cusp, or acnode) which correspond to three primitive cubics. If D is a point at infinity, the curve is classified as fourth case (explicit cubic), which requires exceptional process. For each case, the affine transformation matrix between the primitive cubic and given Bézier curve can be derived. We confirm that the proposed algorithm never fails unless the cubic Bézier curve is degree reducible or consists of four collinear control points.