Kenzi Sato, Orthocenters of simplices on spheres,
Forum Geometricorum, 17 (2017) 301--324.
Abstract. We consider orthocenters of simplices of the unit sphere of the n-dimensional Euclidean space. For n=3, orthocenters always exist for all simplices, but for n >= 4, they do not necessarily exist. Moreover, unlike the case of the Euclidean space, it is possible that there exist infinite numbers of orthocenters. In this paper, we give characterizations of the existence and the uniqueness of orthocenters.
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