We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.

We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci.126 (2016), 591–612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from θ-covers, and consider related problems.

Among the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

We describe the structure of the collineation groups of Figueroa planes, giving examples and explanations that show why the description in Dempwolff [3] is not completely accurate. We also give criteria for when Figueroa planes are isomorphic, and show that certain subplanes are Figueroa or Pappian planes.