Affinity (mathematics)

In Mathematical, in Geometry in particular, a affinity is a Application closely connected or linear equal to the identity in a direction and a Homothétie in another.

Vectorial affinity

Vectorial affinities are the endomorphisms which are direct sum of the identity and a homothety. More precisely:

That is to say $E \,$ a vector space and two pennies additional spaces $F \,$ and $G \,$ ( $E=F \ oplus G$ );

the affinity of bases $F \,$ (or on $F \,$ ), of direction $G \,$ and of report/ratio $\ lambda \,$ is the single endomorphism $f \,$ which is restricted with $F \,$ in the identity, and with $G \,$ in the homothety of report/ratio $\ lambda \,$ :

If $x=x_F+ x_G \,$ then $f (X) = x_F + \ lambda x_G \,$ .

Characterization in finished dimension: endomorphism diagonalisable having two eigenvalues with more of which is the unit.

Affinities recover:

the identity ( $\ lambda=1 \,$ )
projections, or Projector S ( $\ lambda=0 \,$ )
the Symmetry linear S, or involutions ( $\ lambda=-1 \,$ ), being reduced to the identity if the characteristic of the bodies is 2)
the Homothétie S ( $G=E \,$ )
the dilations , or affinities hyperplanes, ( $\ dim G=1 \,$ ).

Specific affinity

Being given a subspace refines $F \,$ of a space refines $E \,$ associated with $\ overrightarrow E$ and an additional direction $\ overrightarrow G$ , the affinity of bases $F \,$ (or on $F \,$ ) of direction $\ overrightarrow G \,$ and of report/ratio $\ lambda$ is the application defined by construction:

for any point $M \,$ in $E \,$ one traces the single subspace $G_M \,$ passing by $M \,$ and of direction $\ overrightarrow G$ ;
$G_M \,$ cut $F \,$ in a single point $H \,$ ;
the image of $M \,$ by $f \,$ is then the point $M' \,$ such as $\ overrightarrow {HM'} = \ lambda \ overrightarrow {HM}$ .

The applications closely connected of linear part a vectorial affinity are specific affinities with the proviso of having at least a fixed point; in the general case, one obtains slipped affinities , composed of an affinity and a translation of vector parallel with the direction of affinity.

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