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Do there exist a way to define angle between lines at $\mathbb{P}^2(k)$?, where $k$ is an algebraically closed field of characteristic zero.

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    You have that locally $\mathbb{P}^2$ is like $k^2$; if you can define angles on $k^2$ then you can do it on $\mathbb{P}^2$. For example, if $k=\mathbb{C}$, then defining angles is the same as defining an angle between two lines in $\mathbb{R}^4$, you just find the point of intersection of the two lines, and see which affine piece it is in.2011-09-09
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    However, such angles are not preserved by projective transformations, so it is debatable whether they ought to count as angles "in" $\mathbb P^2$.2011-09-09
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    That's true, I guess it depends on what properties you're looking for.2011-09-09
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    If I wanted to know this, I might look in Norman Wildberger's writings.2011-09-09

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