Geodesic
Reflection loops of spaces with congruence and hyperbolic incidence structure
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 2, pp. 303-320 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In an absolute space $(P, \frak L, \equiv, \alpha)$ with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that $(P, \frak L, \alpha)$ is a linear space with hyperbolic incidence structure, the addition is not associative. $(P,+)$ is a K-loop or a Bruck loop.
In an absolute space $(P, \frak L, \equiv, \alpha)$ with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that $(P, \frak L, \alpha)$ is a linear space with hyperbolic incidence structure, the addition is not associative. $(P,+)$ is a K-loop or a Bruck loop.
Classification : 20N05, 51A25, 51D99, 51E14, 51F15
Keywords: ordered space with congruence; point reflection; Bol loop; K-loop 
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     author = {Kreuzer, Alexander},
     title = {Reflection loops of spaces with congruence and hyperbolic incidence structure},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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Kreuzer, Alexander. Reflection loops of spaces with congruence and hyperbolic incidence structure. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 2, pp. 303-320. http://geodesic.mathdoc.fr/item/CMUC_2004_45_2_a11/