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An [[Incidence system|incidence system]] to axiomatically describe chain geometries (cf. [[Chain geometry|Chain geometry]]).

A weak chain space is an incidence system  $  {\mathcal I} = ( P, \mathfrak C,I ) $
satisfying the three conditions below. Here, the elements of  $  \mathfrak C $
are called chains and two different points (i.e., elements of  $  P $) 
are called distant if they are incident with a common chain.

i) any three pairwise distant points are contained in exactly one chain;

ii) any chain contains at least three points;

iii) any point lies in at least one chain.

For a point  $  p $, 
let  $  D _ {p} $
be the set of all points distant to  $  p $
and let  $  ( p ) = \{ {C \in \mathfrak C } : {pIC } \} $. 
Then the incidence system  $  {\mathcal I} _ {p} = ( D _ {p} , ( p ) ,I ) $
is called the residual space of  $  {\mathcal I} $
at  $  p $.

A partial parallel structure  $  ( P, \mathfrak B,I, \| ) $
is an incidence system  $  ( P, \mathfrak B,I ) $
together with an [[Equivalence|equivalence]] relation  $  \| $
on  $  \mathfrak B $
satisfying the two conditions below. Here, the elements of  $  \mathfrak B $
are called lines.

a) two different points are incident with at most one line;

b) for a line  $  L $
and point  $  p $, 
there is exactly one line,  $  L  ^  \prime  $, 
incident with  $  p $
and such that  $  L \| L  ^  \prime  $. 
Condition b) is the Euclid parallel axiom.

A partial parallel structure  $  ( P, \mathfrak B,I, \| ) $
is called a partial affine space if there is an [[Affine space|affine space]]  $  {\mathcal A} $
such that  $  P $
is the set of points of  $  {\mathcal A} $, 
$  \mathfrak B $
is the set of straight lines of  $  {\mathcal A} $
and  $  \| $
is the natural parallelism on  $  {\mathcal A} $.

A weak chain space is called a chain space if all residual spaces of it are partial affine spaces.

Every proper [[Chain geometry|chain geometry]] is a chain space. Conversely, the chain spaces that are proper chain geometries can be characterized by suitable automorphism groups [[#References|[a1]]].

A contact space  $  {\mathcal C} = ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P }  ) $
is a weak chain space  $  ( P, \mathfrak B,I ) $
together with a family  $  ( \rho _ {p} ) _ {p \in P }  $, 
where  $  \rho _ {p} $
is an equivalence relation on  $  ( p ) $
with the following properties:

1) if  $  C \rho _ {p} C  ^  \prime  $, 
then  $  p $
is the only point common to  $  C $
and  $  C  ^  \prime  $;

2) if  $  pIC $
and  $  q $
is a point distant to  $  p $, 
then there is a unique chain  $  C  ^  \prime  $
incident with  $  p $
and  $  q $
for which  $  C \rho _ {p} C  ^  \prime  $.

Clearly, for a contact space  $  ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P }  ) $
any residual space of the incidence system  $  ( P, \mathfrak B,I ) $
gives rise to a partial parallel structure  $  ( D _ {p} , ( p ) ,I, \rho _ {p} ) $. 
Conversely, any chain space  $  {\mathcal I} $
is a contact space (taking for  $  \rho _ {p} $
the natural parallelism of the affine space underlying  $  {\mathcal I} $). 
One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [[#References|[a3]]].

An affine chain space  $  \mathfrak A = ( P, \mathfrak C,I ) $
is a contact space, where  $  P $
is the point set of an affine space  $  {\mathcal A} $. 
The elements of  $  \mathfrak C $
are called affine chains and are normal rational curves in  $  {\mathcal A} $, 
i.e., affine parts of curves which are a Veronese variety (cf. [[Veronese mapping|Veronese mapping]]). For the set  $  \mathfrak L $
of all affine chains that are straight lines, the structure  $  ( P, \mathfrak L,I, \| ) $
is a partial affine space. Affine chain spaces can be constructed by means of Jordan algebras [[#References|[a2]]] (cf. also [[Jordan algebra|Jordan algebra]]).

A classical example is the quadric model of a chain space, constructed on a [[Quadric|quadric]]  $  Q $
by means of plane sections. Moreover, the [[Stereographic projection|stereographic projection]] from a simple point  $  p $
of  $  Q $(
to a hyperplane different from the tangent plane of  $  Q $
at  $  p $) 
then gives rise to an affine chain space (cf. also [[Benz plane|Benz plane]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Herzer,   "Chain geometries"  F. Buekenhout (ed.) , ''Handbook of Incidence Geometry'' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Herzer,   "Affine Kettengeometrien über Jordan-Algebren"  ''Geom. Dedicata'' , '''59'''  (1996)  pp. 181–195</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Meuren,   A. Herzer,   "Ein Axiomsystem für partielle affine Räume"  ''J. Geom.'' , '''50'''  (1994)  pp. 124–142</TD></TR></table>