# Affine Space

## A set in which geometric operations make sense, but in which there is no distinguished
point

### (i.e. no origin such as in a vector space)

#### A affine space consists of a set, called the points of the affine space, an
associated vector space and two operations which connect the affine and the
vector space

#### Given 2 Point P and Q, we can form the difference of P and Q which lies in
the vector Space

P - Q = **v**

#### Given a point P and a vector u, we can add u to P and get another point in
the affine space

P = Q + **v**

### Properties to satisfy

(P + **v**) + **u** = P + (**v** + **u**)

P +** u** = P if and only if **u** = **0**

## Affine combination of the points P and Q by the real number t : a point such
as :

P + t (Q-P) ....... if a + b = 1 ... new notation : aP + bQ <=>
P + b (Q-P)

### NB : convex combination <=> 0 <= t <= 1

if t1 + t2 + ... tn = 1

t1 P1 + t2 P2 + ... tn Pn <=> P1
+ t2 (P2 - P1) + ... tn (Pn - P1)

NB : convex affine combination <=>
0 <= ti <= 1