Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width

The curves of constant width are special curves used in engineering, architecture and technology. In the literature, these curves are considered according to different roofs in different spaces and some integral characterizations of these curves are obtained. However, in order to examine the geometric properties of curves of constant width, more than characterization is required. In this study, firstly differential equations characterizing quaternionic space curves of constant width are obtained. Then, the approximate solutions of the differential equations obtained are calculated by the Morgan-Voyce polynomial approach.The geometric properties of this curve type are examined with the help of these solutions.

On a curve and a system

Curves of constant width, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, a system of differential equations characterizing the curves of constant width is examined. This is the system of the first order homogenous differential equations with variable coefficients in the normal form. Approximate solutions of the system, by means of two different polynomial approaches, are calculated and error analysis is made. The obtained results are analyzed on a numerical sample and the best method of approach is determined. This system can also constitute a characterization for different types of curves according to different frames in different spaces. Therefore, this study is important not only for this curve type but also for the geometry of all curves that can be expressed in a similar system.

THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.