The Apollonius contact problem and Lie contact geometry

  title={The Apollonius contact problem and Lie contact geometry},
  author={Robert d. Knight},
  journal={Journal of Geometry},
Abstract.A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented Möbius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the… Expand
Classification topologique des solutions du Probl\`eme d'Apollonius
We give a mathematical computation of the number of solutions of Apollonius problem, by use of Lie Sphere Geometry. Unlike in higher dimensions, the number of solutions depends only on the topologyExpand
A Euclidean Area Theorem via Isotropic Projection
Abstract.We obtain a generalization of a property of the arbelos first stated as Proposition 4 in the Book of Lemmas by Archimedes, circa 250 BC. The new theorem relates the areas of a chain of fourExpand
A new solution of Apollonius’ problem based on stereographic projections of Möbius and Laguerre planes
In this paper we give a new proof of Apollonius’ problem based on the stereographic projection in spherical model of Möbius geometry and cylinder model of Laguerre geometry.
Simple and branched skins of systems of circles and convex shapes
A simple algorithm for skinning circles in the plane that allows the skin to touch a particular circle not only at a point, but also along a whole circular arc results in naturally looking skins. Expand
Geometric constructions on cycles in $\rr^n$
In Lie sphere geometry, a cycle in $\RR^n$ is either a point or an oriented sphere or plane of codimension $1$, and it is represented by a point on a projective surface $\Omega\subset \PP^{n+2}$. TheExpand
Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena
Abstract.To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamentalExpand
Dependencies of bundle forms in Laguerre planes
Several relationships between bundle forms of Laguerre planes are found, including the results (1) $${B\mathfrak{B} 0}$$BB0, $${D\mathfrak{B}0}$$DB0, $${B\mathfrak{B}1^1}$$BB11,Expand


Lie sphere geometry
In this chapter, we give Lie’s construction of the space of spheres and define the important notions of oriented contact and parabolic pencils of spheres. This leads ultimately to a bijectiveExpand
A Treatise On The Circle And Sphere
The Circle in Elementary Plane Geometry: Fundamental definitions and notation Inversion Mutually tangent circles Circles related to a triangle The Brocard figures Concurrent circles and concyclicExpand
Vorlesungen über Geometrie der Algebren
Apollonius by Inversion
Metric affine geometry
Eine Axiomatik der Kreisgeometrie und der Laguerregeometrie