Circle Chains Inside a Circular Segment (original) (raw)

On the generation of discrete circular objects and their properties

Graphical Models /graphical Models and Image Processing /computer Vision, Graphics, and Image Processing - CVGIP, 1985

A method is proposed for the generation of a digital circle when the center and integer radius or three pels belonging to the circle are given. The properties of such circle are studied and it is shown that the rings and discs can be conveniently drawn by this method of circle generation. Furthermore, some geometric drawings involving the circle and digital straight line are investigated. 9

Ellipse Chains and Associated Sequences

2020

We define circle and ellipse chains tangent to the branches of a hyperbola and the terms of the chains are mutually tangent to each other. Our goal is to derive recurrence relations for the parameters of chains elements and to establish some connections between integer sequences and chains.

A theorem on circle configurations

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.

Discrete circles: an arithmetical approach with non-constant thickness

Vision Geometry XIV, 2006

In the present paper, we introduce an arithmetical definition of discrete circles with a non-constant thickness and we exhibit different classes of them depending on the arithmetical discrete lines. On the one hand, it results in the characterization of regular discrete circles with integer parameters as well as J. Bresenham's circles. As far as we know, it is the first arithmetical definition of the latter one. On the other hand, we introduce new discrete circles, actually the thinnest ones for the usual discrete connectedness relations.

Spherical Chains Inside A Spherical Segment

2016

The present paper deals with a spherical chain whose centers lie on a horizontal<br> plane which can be drawn inside a spherical fragment and we display some geometric propertiesrelated to the chain itself. Here, we also grant recursive and non recursive formulas for calculating the coordinates of the centers and the radii of the spheres.

Connecting Cycles for Concentric Circles

2019

We study perimeters of connecting cycles for concentric circles. More precisely, we are interested in characterization of those connecting cycles which are critical points of perimeter considered as a function on the product of given circles. Specifically, we aim at showing that, generically, perimeter is a Morse function on the configuration space, and computing Morse indices of critical configurations. In particular, we prove that the diametrically aligned configurations are critical and their indices can be calculated from an explicitly given tridiagonal matrix. For four concentric circles, we give examples of non-generic collections of radii and describe a pitchfork type bifurcation of stationary connecting cycles.

Directed packings of circles in the plane

Proceedings of the 2nd Croatian Combinatorial Days, 2019

We consider sequential packings of families of circles in the plane whose curvatures are given as members of a sequence of non-negative real numbers. Each such packing gives rise to a sequence of circle centers that might diverge to infinity or remain bounded. We examine the behavior of the sequence of circle centers as a function of the growth rate of the sequence of curvatures. In several special cases we obtain explicit formulas for the coordinates of the limit, while in other cases we obtain accurate estimates.

Stabbing Circles for Sets of Segments in the Plane

Lecture Notes in Computer Science, 2016

Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to two cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute a representation of all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a O(n log 2 n) time and O(n) space algorithm. We also observe that the stabbing circle problem for S can be solved in worst-case optimal O(n 2) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. Finally we show that the problem of computing the stabbing circle of minimum radius for a set of n parallel segments of equal length has an Ω(n log n) lower bound.