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