Editing Midpoint (section)

==Generalizations==

The [[#Formulas|abovementioned]] formulas for the midpoint of a segment implicitly use the lengths of segments. However, in the generalization to [[affine geometry]], where segment lengths are not defined,<ref>{{citation|last=Fishback|first=W.T.|title=Projective and Euclidean Geometry|edition=2nd|publisher=John Wiley & Sons|year=1969|page=214|isbn=0-471-26053-3}}</ref> the midpoint can still be defined since it is an affine [[invariant (mathematics)|invariant]]. The [[Synthetic geometry|synthetic]] affine definition of the midpoint {{mvar|M}} of a segment {{mvar|AB}} is the [[projective harmonic conjugate]] of the [[point at infinity]], {{mvar|P}}, of the line {{mvar|AB}}. That is, the point {{mvar|M}} such that {{math|H[''A'',''B''; ''P'',''M'']}}.<ref>{{citation|last=Meserve|first=Bruce E.|title=Fundamental Concepts of Geometry|year=1983|orig-year=1955|publisher=Dover|page=156|isbn=0-486-63415-9}}</ref> When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide.<ref>{{citation|last=Young|first=John Wesley|title=Projective Geometry|year=1930|publisher=Mathematical Association of America|series=Carus Mathematical Monographs #4|pages= 84–85}}</ref>

The midpoint is not naturally defined in [[projective geometry]] since there is no distinguished point to play the role of the point at infinity (any point in a [[projective range]] may be projectively mapped to any other point in (the same or some other) projective range). However, fixing a point at infinity defines an affine structure on the [[projective line]] in question and the above definition can be applied. 

The definition of the midpoint of a segment may be extended to [[curve segment]]s, such as [[geodesic]] [[arc (geometry)|arcs]] on a [[Riemannian manifold]]. Note that, unlike in the affine case, the ''midpoint'' between two points may not be uniquely determined.