The complement of a codimension-one projective subspace of $\mathbb{R}\mathbf{P}^{3}$ is identifiable in a geodesic structure preserving manner with an affine $3$-space so that the group of projective transformations acting on it is identical with the group of affine transformations of the affine $3$-space. We call this set an affine patch.


In the above sentence what does the "geodesic structure preserving manner" part mean?



complement of a codimension-one projective subspace
Hi. One can read this in M. Berger's Geometry I, Chapter 5, from Springer Verlag. This is standard classical work. An affine space can be completed to a projective space by adding a subspace of codimension-one and a real projective space becomes an affine space by removing a subspace of codimension-one. 


Also, please see my new MSJ book No. 27. One can find it in my homepage and 
<http://mathsoc.jp/publication/memoir/memoirs-e.html>. It has some explanations.


Anyway you can send me questions also. 


