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" the complement of a codimension-one projective subspace of RP3(real projective space) is identifiable in a geodesic-structure 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 ' .conversely , a natural completion of an affine 3-space is identified with RP3 in a geodesic preserving manner ".

what does 'geodesic preserving manner' mean? I know it means that we can embed affine 3-space in RP3 by a map which preserves geodesics , but how does this map do this?

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Do you actually want to know what the map is? I am not sure what you mean by "how does this map to this". Can you give the source from which you quoted the description? –  Willie Wong Oct 10 '12 at 7:05
    
let me ask another question? –  david Oct 11 '12 at 16:12
    
how can we embed an affine space in a vector space? –  david Oct 11 '12 at 16:12
    
is this related to your question above at all? $\mathbb{R}P^3$ is not a vector space. The point is that this embedding is not fully structure preserving: it is not preserving the affine structure of $\mathbb{E}^3$. I'm pretty sure this map is just an embedding in the sense of differential topology. It has the additional property that geodesics are preserved, but nothing is said about affine, or Riemann structure in addition. –  Willie Wong Oct 12 '12 at 9:31
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Seems david has asked this question before. The source appears to be this paper by Suhyoung Choi. –  MvG Oct 12 '12 at 19:30

1 Answer 1

The way I read this statement is the following: you can take projective 3-space, denote any plane as “the plane at infinity”, and the remainder (i.e. projective space minus that one plane) will behave as an affine space. Any projective transformation which preserves the plane at infinity will then correspond to an affine transformation.

All of the above can be readily seen when you consider coordinates. Using a simple transformation of your coordinate system, you can take the plane at infinite (your codimension-one subspace) to be $w=0$. Then any point $(x,y,z,w)^T$ in the complement will have $w\neq0$. So you can dehomogenize to $(x/w, y/w, z/w)^T$ and use that as affine coordinates.

what does 'geodesic preserving manner' mean?

The fact that they refer to geodesics instead of collinearity seems a bit strange to me, but I guess the message is the same: (projective) lines will map to (affine) lines. Which again is straight-forward given the dehomogenization interpretation above.

For the converse case, this “natural completion” refers to the union of the original affine space with elements at infinity. Expressed in coordinates, this is the step where you introduce homogenous coordinates, some of which may have $w=0$. Lines still remain lines.

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