In traditional Euclidean geometry, points serve as the foundational elements for constructing and analyzing space. In contrast, Laguerre geometry, a non-Euclidean geometry, uses oriented circles (or hyperspheres, in the context of higher dimensions) and oriented lines (or hyperplanes), as fundamental objects. Here, a “point” is simply a circle with radius zero, i.e. having no special status within this geometric framework.

The goal of this talk is to elaborate the advantages of adopting this alternative, albeit seemingly complex, perspective, starting from a reevaluation of the ancient problem of Apollonius. This problem, originally solved utilizing ruler and compass, is revisited here within the context of Laguerre geometry, promising a more efficient solution. To this end, the talk will introduce the cyclographic model of Laguerre geometry, a model where we identify the oriented circles with points in a space of one dimension higher – and which is a crucial tool for obtaining the simplified solution of the problem of Apollonius. In addition, we will explore the Blaschke cylinder model, where on the other hand the oriented lines are identified with points, and which bears advantages for other applications.

Moreover, the talk will compare Laguerre geometry with another sphere geometry, Möbius geometry, outlining their analogies and differences. This shall serve as an opportunity to present some contents of my ongoing paper, which “translates” theorems of Möbius geometry to analogous theorems within the domain of Laguerre geometry. Finally, an outlook on potential future applications will be given.