Understanding the Relationship Between Solid Angle and Plane Angles

In summary, the conversation is about the relation between the solid angle and the azimuthal and zenithal angles. The formula for this relation is d²Ω = sin(φ) dφ dθ. However, the question is whether there is another non-infinitesimal relation between these angles. It is mentioned that solid angle is the same as area, and there are standard formulas for calculating area on a unit sphere. There is also a mention of a possible algebraic and infinitesimal definition for solid angle.
  • #1
Jhenrique
685
4
Hellow!

I known an infinitesimal relation between the solid angle Ω with the azimutal angle θ and zenital φ, that's given by d²Ω = sin(φ) dφ dθ. But this is infinitesimal relation, exist other relation non infinitesimal between the solind angle with the plane angles? Thanks!
 
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  • #2
Hello Jhenrique! :smile:

Solid angle is the same as area (on a unit sphere), for which there are standard formulas.

In particular, if you know all three angles of a triangle, the area is their sum minus π.

(i don't remember the other formulas, but a quick google search will find them :wink:)
 
  • #3
I found nothing more. I thought if an area can have 2 definitions A = xy and dA = dxdy, then the solid angle could have 2 definitions too, 1 algebric definition and 1 infinitesimal definition...
 

What is solid angle and how is it different from plane angles?

Solid angle is a measure of the amount of a three-dimensional space that is enclosed by a given angle. It is measured in steradians and is different from plane angles because plane angles only measure the opening between two intersecting lines or planes, whereas solid angles measure the amount of space that is enclosed by those angles.

How is solid angle calculated?

Solid angle is calculated by dividing the area of a surface on a unit sphere by the square of the radius of that sphere. This gives the angle in steradians, which is the unit of measurement for solid angle.

Why is solid angle important in physics and mathematics?

Solid angle is important in physics and mathematics because it allows us to measure and understand the amount of three-dimensional space that is enclosed by a given angle. This is useful in various applications such as calculating radiation intensity, analyzing the behavior of electromagnetic waves, and determining the coverage area of a light source.

How does solid angle relate to plane angles in a three-dimensional space?

In a three-dimensional space, solid angle is related to plane angles by the concept of projection. The projection of a solid angle onto a plane is equal to the plane angle formed by the intersection of the plane and the solid angle. This relationship allows us to convert between solid and plane angles in certain calculations.

What are some real-world examples of solid angle and its relationship to plane angles?

Some examples of solid angle and its relationship to plane angles include the measurement of the coverage area of a security camera or a satellite dish, the calculation of the intensity of a light source, and the analysis of the radiation pattern of an antenna. These examples demonstrate how solid angle is used in various fields such as engineering, astronomy, and optics.

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