Angular sector

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  • Thread starter arlesterc
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  • #1
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In this article https://terrytao.files.wordpress.com/2008/03/ricci1.pdf the following is written:
"Consider now an angular sector A(x, r, θ, v) inside a small disk B(x, r) of small angular aperture θ (measured in radians) centred around some direction v (a unit vector) emanating from x; this sector is well-defined, basically because the Riemannian metric gives us the notions of distance and angle."

I have googled and not found any reference to angular sector but have found references to circular sector. is there a difference and if there is what is an angular sector?

Thanks in advance.
 

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  • #2
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I assume he says angular instead of circular, as the underlying geometry isn't necessarily Euclidean. It has also the advantage to denote it by an ##A## as in area, whereas a ##C## is occupied by too many other meanings. But the main difference is probably, because ##A(x,r,\theta,v)## isn't flat. One could now object that the ball isn't either, but this would lead too far. In the end it's some sector on a curved surface.
 
  • #3
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Thanks. So what is the formula for calculating A using the parameters referenced? A picture or illustration would be great so I can see what exactly the variables refer to.
 
  • #4
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sector.png


Now you just have to imagine, that ##A(x,r,\theta,v)## isn't part of a flat plane but part of a somehow curved surface.
 

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