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I'm learning the very basics of projective geometry, and I read the following in Stan Birchfield's notes, An Introduction to Projective Geometry:

Retrieved from http://robotics.stanford.edu/~birch/projective/node4.html Why is it that "overall scaling is unimportant"? In his notes, Birchfield uses this to build the idea that what we think of as points in Euclidean geometry can be thought of as lines in projective geometry?

on-pasta
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    It’s not. A line in euclidean space becomes a projective point. That’s why the scaling doesn’t matter. – Randall Dec 18 '17 at 23:39
  • That makes sense, but then I'm confused about this statement that comes up later in his notes: "We have just seen that, in going from Euclidean to projective, a point in $R^2$ becomes a set of points in $R^3$ which are related to each other by means of a nonzero scaling factor. Therefore, a point $p=(X,Y,W)$ in $P^2$ can be visualized as a "line" in three-dimensional space..." (from: http://robotics.stanford.edu/~birch/projective/node5.html). – on-pasta Dec 18 '17 at 23:48
  • The lines through $(0,0,0)$ in $\Bbb R^3$ can be considered to be the points of a $2$-dimensional projective plane. – DanielWainfleet Dec 19 '17 at 00:47

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Observe that it is a point in the affine plane (dimension 2) which corresponds to a line in a vector space of dimension 3. The points of a projective plane are just vector lines in this vector space.

As a vector line is defined by a single vector, which be chosen as you like, provided it is non-zero, it explains why ‘the (non-zero) scaling is unimportant’. So a point in the projective plane is defined by the ratios of its three coordinates, $(X:Y:T)$, for which not all three coordinates are $0$. If $t\ne 0$, one gets back a points of the affine plane with the map $\;(X:Y:T) \longmapsto \Bigl(x=\dfrac XT, y=\dfrac YT\Bigr)$. Conversely, a point $(x,,y)$ in the affine plane corresponds to the points $(x:y:1)$ in the projective plane. The point with projective coordinates $(x:y:0)$ are considered as the ‘point at infinity’ in the direction $(x,y)$.

Bernard
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