Lengths and incidences in Poisson hypersphere and spherical splitting tessellations

We investigate distributional properties of two random tessellation models on the $d$-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For \(d=2\), we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.