We relate the total curvature and the isoperimetric deficit of a curve γ in a two-dimensional space of constant curvature with the area enclosed by the evolute of γ. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

In this paper, the invariants of Smarandache curves, which consist Frenet vectors involute curve, are calculated in terms evolute curve.

Proučava se evoluta proizvoljne krivulje u ravnini i pritom izvode formule za izračunanje parametarskih jednadžbi evolute zadane vektorskom jednadžbom, odnosno parametarskim jednadžbama, a potom ekplicitnom, implicitnom polarnom jednadžbom. Opisat će konstrukcija obrazložit njezina svojstva. Detaljnije analizirati grafički prikazati nekih ravninskih krivulja. Također, objasnit evolvente argumen...

The goal of this paper is to characterize the evolute, involute and parallel curves a Bezier curve which applicable computer graphics related subjects. Especially, these couples are investigated at endpoints. Moreover, curvatures given.

The Four-Vertex Theorem has been of interest ever since a discrete version appeared in 1813 due to Cauchy. Up until now, there have been many different versions of this theorem, both for discrete cases and smooth cases. In 2004 [16], an approach relating the discrete Four-Vertex Theorem to the evolute was published, and here we will give an overview of this paper. We then will define the notion...

We introduce a novel definition of a parabola into the framework of universal hyperbolic geometry, show many analogs with the Euclidean theory, and also some remarkable new features. The main technique is to establish parabolic standard coordinates in which the parabola has the form xz = y2. Highlights include the discovery of the twin parabola and the connection with sydpoints, many unexpected...

In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetry set, equidistants and area parallels and show that they behave quite simila...

In this study, we investigate the semi-real quaternionic curves in the semi-Euclidean space E_4_2. Firstly, we introduce algebraic properties of semi-real quaternions. Then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E42 . Finally, we give an example illustrated with Mathematica Programme.