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

The $$\kappa $$ κ -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ ϵ κ -curves.$$\kappa $$ κ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ ϵ κ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

Head-Raising Method of Snake Robots Based on the Bézier Curve

SUMMARY For acquiring a broad view in an unknown environment, we proposed a control strategy based on the Bézier curve for the snake robot raising its head. Then, an improved discretization method was developed to accommodate the backbone curves with more complex shapes. Besides, in order to determine the condition of using the improved discretization method, energy of framed space curve is introduced originally to estimate the shape complexity of the backbone curve. At last, based on degree elevation of the Bézier curve, an obstacle avoidance strategy of the head-raising motion was proposed and validated through simulation.

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.

Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations

Approximating continuous functions by polynomials is vital to scientific computing and numerous numerical techniques. On the other hand, polynomials can be characterized in several ways using different bases, where every form of basis has its advantages and power. By a proper choice of basis, several problems will be removed; for instance, stability and efficiency can be improved, and numerous complications can be resolved. In this paper, we provide an explicit formula of the generalized shifted Chebyshev Koornwinder’s type polynomial of the first kind, T r * ( K 0 , K 1 ) ( x ) , using the Bernstein basis of fixed degree. Moreover, a Bézier’s degree elevation was used to rewrite T r * ( K 0 , K 1 ) ( x ) in terms of a higher degree Bernstein basis without altering the shapes. In addition, explicit formulas of conversion matrices between generalized shifted Chebyshev Koornwinder’s type polynomials and Bernstein polynomial bases were given.

INSAN KAMIL DALAM PERSPEKTIF MUHAMMAD IQBAL

Every being has individuality or self. Degree elevation on every being in nature depending on the level of development of each individuality. The criteria for determining the quality of every being is how far he can live up to himself effectively. Individuality is the distinguishing mark in the concept of Khudi or Iqbal ego. Iqbal's philosophy about Khudi ends on his thoughts about the insan kamil, the ideal man. Insan kamil is the highest level that can be achieved not by way of meditation, but with an original creative work, which is lawful overwhelmed by love or ishq, firmness self or faqr, courage and tolerance. Achieve this through three phases: obedience to God, self-control and the caliphate of God.