Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications (original) (raw)
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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...
Curve and Surface Geometric Modeling via Generalized Bézier-like Model
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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.
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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.