Incircular net

An incircular net in the Euclidean plane is formed by two indexed families of straight lines $({\ell }_i)$ and $(m_j)$ such that each elementary quadrilateral bounded by ${\ell }_i,{\ell }_{i+1},m_j,m_{j+1}$ has a common circle tangent to its four side lines. In the usual nondegenerate convex setting this is an incircle. The line families make the rows and columns straight; a mesh of unrelated tangential quadrilaterals is a more general object.^[ref]

The square grid is an example, with one incircle centred in every square. A rectangular grid of unequal side lengths fails: a circle tangent to both horizontal sides must have radius half the height, while tangency to both vertical sides requires radius half the width. This simple test illustrates that the tangency constraint carries metric information. Checkerboard variants impose the condition on alternating cells and use oriented contact in Laguerre geometry.