$$\epsilon \kappa $$-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.

A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.