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:

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

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:

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

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

• Euclidean Geometry

• Solid geometry
• apparent Diameter

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