 Laguerre geometry

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.^[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.