Projective reference mark

A projective space of dimension $n$ is located by $n+2$ points.

(For the definitions, to see projective Geometry).

Intuitively, one wants to locate a point of projective space by giving oneself a point of the vector space of dimension $n + 1$ associated. One wants thus to choose a base $(e_1,…, e_ {n+1})$ of this space, and to consider the points $(p_1,…, p_ {n+1}) = (\ pi (e_1),…, \ pi (e_ {n+1}))$ like a reference mark of projective space. Having the coordinates $(x_1,…, x_ {n+1})$ in this reference mark, one would consider the vector then $x = x_1 \ cdot e_1 +… + x_ {n+1} \ cdot e_ {n+1}$ which defines a single point $\ pi (X)$ in projective space. The error of the argument above is that when one knows only the projective reference mark $(p_1,…, p_ {n+1})$ , one cannot find the vectors $(e_1,…, e_n)$ which had defined it, but only vectors of the form $\ tilde e_1 = \ lambda_1 \ cdot e_1,…, \ tilde e_ {n+1} = \ lambda_ {n+1} \ cdot e_ {n+1}$ . If one considers the new vector $\ tilde X = x_1 \ cdot \ tilde e_1 +… + x_ {n+1} \ cdot \ tilde e_ {n+1} =
x_1 \ lambda_1 \ cdot e_1 +… + x_ {n+1} \ lambda_ {n+1} \ cdot e_ {n+1}$ , this one does not have any raison d'être colinéaire with $x$ , and thus to give the same point of projective space after projection, except if all the $\ lambda_i$ are equal. The idea is thus then to associate at the points $(p_1,…, p_ {n+1})$ a constraint, which can be also seen like a point of projective space, obliging any choice of vectors $\ tilde e_1,…, \ tilde e_ {n+1}$ like checking $\ lambda_1 above =… = \ lambda_ {n+1}$ . For that, one imposes a constraint on the sum $\ tilde e_1 +… + \ tilde e_ {n+1}$ which must be colinéaire with the sum $e_1 +… + e_ {n+1}$ initially selected. It is then easy to see that implies the required constraint. It is thus enough to associate with the $p_i$ the point $p_ {n+2} = \ pi (e_1 +… + e_ {n+1})$ , and then any choice of $\ tilde e_1,…, \ tilde e_ {n+1}$ checking $\ pi (\ tilde e_1 +… + \ tilde e_ {n+1}) = p_ {n+2}$ makes it possible to find the point of projective space correpondant with the coordinates $(x_1,…, x_ {n+1})$ as indicated above.

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