Solid Angle

In Mathematical, Physical Geometry and , a solid angle is the three-dimensional analog of the Angle plane or two-dimensional. The plane Angle is the report/ratio of the Length of the arc on the ray. The solid angle is in space the report/ratio of the surface of part of a sphere on the ray squared. Its unit is the Stéradian noted Sr.

It is often noted Ω (Oméga capital). It measures the surface on which an object is projected radially on a sphere of ray unit.

Calculation

To calculate the solid angle under which one sees an object starting from a point, one projects the object on a sphere of ray R centered in this point.

If surface that this projection makes on the sphere is S , the solid angle under which the observer sees the object is by definition:

\ Omega \, = \, \ frac {S} {R^2}.

The concept of angle solid intervenes in particular in the definition of the Luminosité.

Some examples

The solid angle under which one sees, of the center of the sphere, a segment of a sphere of surface S whose apparent diameter extends on an angle from 2θ writes:

\ Omega \, = \, \ frac {S} {R^2} = \, 2 \ pi \ cdot \ left (1 - \ cos \ theta \ right)

The angles θ = π/3, θ = π/2 and θ = 2π/3 delimit a cutting of the sphere on the four equal surfaces of π steradians. Thus, complete space is seen under a solid angle of 4π Sr.

Of a point of terrestrial surface one can see only half of space, i.e. a half-sphere or a hemisphere, is a solid angle of 2π Sr.

a Trihedral formed by two perpendicular walls and the ground describes a solid angle of π/2 steradians, that is to say a eighth of sphere.

Between the latitude of San Diego, in California, in the Northern hemisphere and that of Sydney, in Australia, in the southern hemisphere, it there with half of terrestrial surface.

Between the polar circle and the Tropic of Cancer in the Northern hemisphere there is the quarter of terrestrial surface, that is to say a solid angle of π steradians.

the solid angles under which one sees a part of one centime euro 1,80 away m, the the Moon and the Sun are very close (≈ 6.10^-5 Sr).

a “square of sky” of 0,923… radians on side is seen under a solid angle of a steradian.

See too

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