Apollonius Points and Anharmonic Ratios

Fixed Points and Stability of an Additive Functional Equation of n-Apollonius Type in C-Algebras

Apollonius Solutions in Rd

Voronoi and related diagrams have technological applications, for example, in motion planning and surface reconstruction, and also find significant use in materials science, molecular biology, and crystallography. Apollonius diagrams arguably provide the most natural division of space for many materials and technology problems, but compared to Voronoi and power diagrams, their use has been limi...

Cross Ratios of Points and Lines in Moufang Planes

In this paper, first we extend the known definition of cross-ratio of collinear points to whole Moufang plane. Later we introduce the cross-ratios for lines and the known results about the cross-ratios of points which are adapted to crossratios of lines without using the principle of duality. Finally, we give a theorem which describes the relation between the cross-ratios of points and lines. M...

Conic tangency equations and Apollonius problems in biochemistry and pharmacology

The Apollonius Circle Problem dates to Greek antiquity, circa 250 BC. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Vieta [19] (or Viète) solved the problem using circle inversions before 1580. Two generations later, Descartes considered a special case in which all four circles are mutually ...

The predicates of the Apollonius diagram: Algorithmic analysis and implementation

We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient implementation of the algorithm, that handle...