Some engineering applications of new trigonometric cubic Bézier-like curves to free-form complex curve modeling

Abstract

The construction of free-form complex curves is very hot topic in engineering and computer aided design/computer aided manufacturing (CAD/CAM). The trigonometric Bézier-like curves got more attention in the fields of mathematics and engineering in recent years because of their useful geometric properties as compared to classical Bézier curves as well as ordinary Bernstein basis functions. The new trigonometric cubic Bézier-like (or TC Bézier-like, for short) curves along with new trigonometric cubic Bernstein-like (or TC Bernstein-like, for short) basis functions with single shape parameter with continuity conditions are presented in this paper. The necessary and sufficient constraints for C 2 and G 2 between two contiguous TC Bézier-like curves are constructed in order to remove the difficulty to composite curves which they cannot often be constructed by means of single curve. The role of shape parameter on connected curves with detailed smooth continuity steps is also part of this study. The ellipses and parabolas can also be represented exactly by using the TC Bézier-like curves. Some important applications of TC Bézier-like curves as: approximate some conic curves, font designing and free form complex curves are discussed. Some modelling examples show that the proposed TC Bézier-like curve technique is time saving, very effective and efficient which they can easily be modelled and give a powerful tool to design engineering complex curves.

$$\epsilon \kappa $$-Curves: controlled local curvature extrema

“…Quasi-cubic trigonometric Bernstein basis [30] 6. Trigonometric cubic Bernstein-like basis [24] Our method proposed in the next section can be applied for curves based not only on polynomials, but also other bases such as a trigonometric basis with degree elevation property which we will introduce in Sect. 4.…”

Curve and Surface Construction Using Hermite Trigonometric Interpolant

“…More recently, BiBi et al constructed various shapes and font designing of curves and described the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves [15]. In another very recent work, Usman et al [16] presented a new TC Bernstein-like basis functions analogous to standard Bernstein basis functions with single shape parameter to overcome the drawing of complex curves. They constructed some conic curves by choosing appropriate points and some complex curves by using continuity conditions.…”

Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

“…Yan expressed an algebraic-trigonometric mixed piecewise curve with two shape parameters and cubic trigonometric nonuniform B-spline curves with local shape parameters in [21] and [22], respectively. Hu et al [23] constructed geometric continuity constraints for H-Bézier curve of degree n. Recently, many researchers have developed the positivity-preserving rational quartic spline interpolation [24], cubic triangular patches scattered data interpolation [25], rational bi-cubic Ball image interpolation [26], quasiquintic trigonometric Bézier curve with shape parameters [27], curve modeling by new cubic trigonometric Bézier with shape parameters [28], continuity conditions for G 1 joint of S-λ curves and surfaces [29], generalized Bernstein basis functions for approximation of multi-degree reduction of Bézier curve [30], surface modeling in medical imaging by Ball basis functions [31], and geometric conditions for the generalized H-Bézier model [32] which have many applications in medicine, science, and engineering. Khalid and Lobiyal [33] presented the extension of Lupaş Bézier curves/surfaces and rational Lupaş Bernstein functions with shape parameters having all positive (p, q)-integers values.…”