A semi-biplane is a (thick connected) point-line geometry which satisfies the following axioms: any two points have zero or two lines joining them, and, dually, any two lines intersect in zero or two points. In an earlier paper, the authors showed that semi-biplanes can be constructed from anti-regular generalized quadrangles [Geom. Dedicata 69, No. 2, 207-221 (1998; Zbl 0898.51005)]. The authors discuss various models for the semi-biplane arising from the real orthogonal Moufang quadrangle \(Q(4,\mathbb{R})\) which are realized in \(\mathbb{R}^3\) and \(\mathbb{R}^2\). Furthermore, they discuss the connections between circle geometries, generalized quadrangles and semi-biplanes. They prove in particular that locally compact circle planes can be used to construct locally compact semi-biplanes.