Apollonius curves and surfaces in the real and dual spaces

This study which is related to the Apollonius curves and surfaces in Real and Dual spaces consists of three chapters.In the first chapter, Apollonius curves and surfaces are defined in the Euclidean Spaces. Apollonius curves and surfaces are investigated in the Euclidean plane E^2 , Euclidean space E^3 and plane z=sy of Euclidean space E^3 and the related theorems are given and proved. Furthermore, the examples of Apollonius curves and surfaces are given in these spaces.In the second chapter, Apollonius curves and surfaces are investigated in the Lorentz space. By giving the basic facts about Lorentz geometry, Apollonius curves in the Lorentz plane R_1^2 , Apollonius curves and surfaces in the Lorentz space R_1^3 , Apollonius curves in the plane z=ky of Lorentz space R_1^3 and Apollonius spheres in the plane t=kz of space-time R_1^4 are investigated. Moreover, relevant examples are given.In the third chapter, Apollonius curves and surfaces are introduced in dual Euclidean space. Apollonius curves and surfaces in dual space D^3 and in the dual plane z=sy of dual space D^3 are given. The real parts of the theorems given in this chapter correspond to the theorems of Apollonius curves and surfaces in the Euclidean space.