Projective geometry

A point in projective -space, , is given by a -vector of coordinates
. At least one of these coordinates should differ from zero. These coordinates are called *homogeneous* coordinates. In the text the coordinate vector and the point itself will be indicated with the same symbol. Two points represented by -vectors and are equal if and only if there exists a nonzero scalar such that
, for every
. This will be indicated by
.

Often the points with coordinate are said to be *at infinity*. This is related to the affine space . This concept is explained more in detail in section 2.2.

A *collineation* is a mapping between projective spaces, which preserves collinearity (i.e. collinear points are mapped to collinear points). A collineation from to is mathematically represented by a
-matrix . Points are transformed linearly:
. Observe that matrices and
with a nonzero scalar represent the same collineation.

A *projective basis* is the extension of a coordinate system to projective geometry. A projective basis is a set of points such that no of them are linearly dependent. The set
for every
, where 1 is in the th position and
is the standard projective basis. A projective point of can be described as a linear combination of any points of the standard basis. For example:

It can be shown [31] that any projective basis can be transformed via a uniquely determined collineation into the standard projective basis. Similarly, if two set of points and both form a projective basis, then there exists a uniquely determined collineation such that for every . This collineation describes the change of projective basis. In particular, is invertible.