Parallelisms of composed of non-regular spreads

Any continuous strictly monotonic function F: R->= 0 -> R with F(0) = 0 and F(t) -> infinity for t -> infinity gives rise to a topological rotational spread of PG (3, R); this spread is non-regular, if F is not linear. The action of the group SO3(R) on this spread yields a topological parallelism of PG (3, R). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P (72) of PG (3, R) which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein's correspondence of line geometry and the Thas-Walker construction we represent every parallel class of P (72) via two parallel half-lines being non-interior to a given sphere in R-3. The parallelism P (72) contains exactly one regular spread, all other members of P (72) are piecewise regular spreads with two segments. However, P (72) is not topological.