Exploring the Meaning of an Angular Sector

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In summary: So the usual formulas won't work, but instead we have to use the Riemannian metric to calculate distances and angles. The angular sector is defined by a small disk with radius ##r## centered around ##x## and with an angular aperture ##\theta##, measured in radians, around the direction ##v## as a unit vector emanating from ##x##. In summary, the article discusses the concept of an angular sector, which is a well-defined geometric shape on a curved surface, defined by a small disk with a radius and an angular aperture around a given direction. The Riemannian metric is used to calculate distances and angles in this context.
  • #1
arlesterc
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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
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
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
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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1. What is an angular sector?

An angular sector is a portion of a circle bounded by two radii and an arc. It is essentially a wedge-shaped region within a circle.

2. Why is it important to explore the meaning of an angular sector?

Understanding angular sectors is crucial in various fields such as mathematics, physics, and engineering. They are used to calculate areas, angles, and volumes in circular objects and can also be applied in real-life situations like measuring the angle of a sundial or determining the coverage of a satellite dish.

3. How do you calculate the area of an angular sector?

The formula for calculating the area of an angular sector is A = (θ/360) x πr², where θ is the central angle of the sector and r is the radius of the circle. This formula is derived from the fact that the area of a circle is πr² and the central angle of a sector is equal to the ratio of the arc length to the circumference of the circle.

4. Can an angular sector have a negative area?

No, an angular sector cannot have a negative area. The area of an angular sector is always a positive value since it represents a physical quantity such as an area or a volume.

5. What other shapes have angular sectors?

Angular sectors can be found in various shapes such as cones, pyramids, and spheres. These shapes can be cut into wedges or slices, which can be considered as angular sectors. In addition, any shape that can be inscribed within a circle can also have angular sectors.

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