Obvious question on angles and solid-angles

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In summary, the differential solid angle d\omega can be calculated by multiplying the oriented differential surface area dA by the cosine of the angle \theta between the normal of dA and the direction from the origin, divided by the squared distance r^2 between the origin and the "centre" of dA. This relationship is not always obvious, as seen in the 2D case with ordinary angles, but it can be derived through further analysis.
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mnb96
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"obvious" question on angles and solid-angles

Hello,
I have found in http://www.cg.tuwien.ac.at/hostings/cescg/CESCG97/csebfalvi/node2.html" [Broken] an important statement about solid angles. Namely, the differential solid angle [itex]d\omega[/itex] is related to an oriented differential surface area [itex]dA[/itex] placed at distance [itex]r[/itex] from the origin in the following way:

[tex]d\omega = \frac{dA \cdot cos\theta}{r^2}[/tex]

where [itex]\theta[/itex] is the angle between the normal of [itex]dA[/itex] and the direction from the origin (see the figure in the http://www.cg.tuwien.ac.at/hostings/cescg/CESCG97/csebfalvi/node2.html" [Broken]), and [itex]r^2[/itex] is the squared distance between the origin and the "centre" of dA.

The authors say that this very "obvious", but it is not for me, not even in the 2D case with ordinary angles.
How can I derive this result?
 
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1. What is the difference between angle and solid angle?

Angle is a measure of the amount of rotation needed to bring one ray into coincidence with another, while solid angle is a measure of the extent of a three-dimensional region in space.

2. How is solid angle measured?

Solid angle is measured in steradians (sr), which is the ratio of the surface area of a sphere covered by a cone or pyramid to the square of the radius of the sphere.

3. What is the formula for calculating solid angle?

The formula for calculating solid angle is Ω = A/r^2, where Ω is solid angle in steradians, A is the surface area of the cone or pyramid, and r is the radius of the sphere.

4. Can solid angle be negative?

No, solid angle cannot be negative as it is a measure of the extent of a three-dimensional region and cannot have a negative value.

5. How is solid angle related to the concept of field of view?

Solid angle is the three-dimensional equivalent of field of view, which is a measure of the angular extent of a two-dimensional region. Solid angle takes into account the entire three-dimensional space, while field of view only considers the two-dimensional plane of view.

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