= Laguerre geometry
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Classical <Laguerre geometry> studies oriented <hyperplanes>, oriented <spheres> and their oriented tangency in <Euclidean space>. Its transformations preserve that contact relation. In the plane the objects are oriented <straight lines> and oriented <circles>; points can be represented as radius-zero circles. The orientation and signed radius are part of the data, so ordinary unoriented tangency alone loses information.<Laguerre geometry in space forms and incircular nets by Alexander I. Bobenko and others>[Abstract, introduction and Appendix B.1]{ref}
For example, fix a unit normal $u$ and an oriented line $u\cdot x=h$. Under a consistent signed-radius convention, a circle with centre $c$ and signed radius $r$ is in oriented contact when $u\cdot c-h=r$. The absolute-distance equation $|u\cdot c-h|=|r|$ forgets which side and orientation participate. This linear signed-contact relation motivates coordinate models for tangency and the use of <Laguerre geometry> in studying <incircular nets>.
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