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

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