  We study geometric structures arising from Hermitian forms on linear spaces
over real algebras beyond the division ones. Our focus is on the dual numbers,
the split-complex numbers, and the split-quaternions. The corresponding
geometric structures are employed to describe the spaces of oriented geodesics
in the hyperbolic plane, the Euclidean plane, and the round $2$-sphere. We also
introduce a simple and natural geometric transition between these spaces.
Finally, we present a projective model for the hyperbolic bidisc, that is, the
Riemannian product of two hyperbolic discs.
