Witold Mozgawa

• Maria Curie-Skłodowska University

In the present paper we consider two families of curves, namely the family of all ellipses and the family of all Cassini ovals. We introduce the notion of a chordal condition of curve at a fixed point. The chordal condition is a generalization of a well-known property of ellipses, from their foci to other points. This paper is based on Stewart’s th...

In the present paper, we introduce two important modifications to the recurrence relation for the Fuss relations derived in Cieślak (Comput Aided Geom Des 66:19–30, 2018). These modifications allow for a simpler and unified method for determining the relations, and additionally explain why their derivation is extremely difficult. We introduce two c...

For a given plane curve, consider a one-parameter family of curves consisting of those points at which two support lines to the initial curve intersect at a constant angle. Such curves are well known in differential and convex geometry and called isoptics. In this paper, we describe parametrizations of orthogonal trajectories to isoptics of ovals....

While the standard (outer) isoptics of ellipses have been studied in depth, we derive a number of parametric forms of inner isoptics of ellipses.

In the present paper we describe the family of all closed convex plane curves of class C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} which have...

While the standard (outer) isoptics of ellipses have been studied in depth, we derive implicit forms of inner isoptics of ellipses. We also demonstrate that the outer and inner isoptics are projective duals of each other.

We derive a special formula for the total mean curvature of an ovaloid. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, an integral formula is proved.

For a given curve C and a given angle θ, the θ-isoptic curve of C is the geometric locus of points through which passes a pair of tangents to C making an angle equal to θ. If the curve C is smooth and convex, isoptics exist for any angle, and through every point exterior to the curve, there is exactly one pair of tangents. The isoptics of conics ar...

In this paper we present a certain modification of the Holditch construction. This construction allows to consider a geometric family of pairs of ring domains. It is proved that the ratio of areas of ring domains of each pair belonging to this family is constant. Problems on extremal chords of constant length sliding around a given oval with both e...

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic app...

In this article we present a new formula for the length of a closed curve involving a double integral of a certain potential function. This formula is based on the construction of pedal curves, and it turns out that the integral taken over the interior of a pedal curve does not depend on the choice of a pedal point. First, using the notion of suppo...

A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.

In this paper we give a proof of Poncelet's closure theorem for ring domains using elementary functions and a certain differential equation which has a solution with suitable geometric properties. We give a necessary and sufficient condition of existence of a constant solution of the equation which explains the phenomenon of the Poncelet porism. In...