Modern Projective Geometry pp 215-234 | Cite as

# Homogeneous Coordinates

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## Abstract

In Section 9.1 we show that if *G* is arguesian with respect to a hyperplane *H*, then the homothety group **H**_{ H } ^{ o } is actually the multiplicative group of a field **K**_{ H } ^{ o } whose underlying set is **H**_{ H } ^{ o } ∪*α* _{ o } where *α* _{ o } is the endomorphism of *G* having *H* as kernel (and hence as axis) and *o* as constant value. One can consider *α* _{ o } as a degenerate homothety. Since the abelian group **T**_{ H } operates simply transitively on *G*\*H* one obtains on *G*\*H* an addition such that the map ev_{ o }: **T**_{ H }→*G*\*H*, the evaluation at *o*, becomes an isomorphism of groups. The set **K**_{ H } ^{ o } also operates on *G*\*H* and one shows that the addition in *G*\*H* induces an addition in **K**_{ H } ^{ o } , characterized by the equality (*λ* + *μ*)(*x*) = *λ*(*x*) + *μ*(*x*) for every *x* ∈ *G*\*H*. With this addition and the composition as multiplication **K**_{ H } ^{ o } becomes a field. Furthermore, the set *G*\*H* with its addition and the scalar multiplication defined by *λ* · *x : = λ* (*x*) becomes a vector space over **K**_{ H } ^{ o } . We denote this vector space by **V**_{ H } ^{ o } or shortly by * V*.

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