What is solid angle

Solid Angle , Is the angle of an object in three-dimensional space to a specific point , It is an analogy of plane angle in three-dimensional space , Common letters Ω Express . It describes the size of an object measured by an observer standing at a certain point . for example , For a particular observation point , A small object near the observation point may have the same solid angle as a large object far away .
The solid angle of the cone is defined as , Take the apex of the cone as the center of the sphere , The ratio of the area intercepted by the cone on the surface of the ball to the square of the radius of the ball , The unit is sphericity （sr）, A sphere is 4π, The hemisphere is 2π.

In the picture above $d\Omega$ Is a minimum , It can be understood as the minimal area per unit spherical area , that $d\Omega$ The definition is the ratio of the area of the minimal solid angle projection to the radius of the sphere ：
$$d\Omega =\frac{dA}{r^2}$$
there $dA$ Is the area of the shadow sphere in the above figure , The two become ：$r\sin \theta d\varphi$,$rd\theta$( Notice that an approximation is used here .), that $dA$ Satisfy ：
$$dA=r\sin \theta d\varphi \times r\theta =r^2\times \sin \theta d\varphi d\theta$$
therefore , The unit minimal solid angle is ：
$$d\Omega =\frac{dA}{r^2}=\sin \theta d\varphi d\theta$$
therefore , The solid angle is the ratio of the projected area to the square of the spherical radius , This sum “ The plane angle is the ratio of arc length to radius of a circle ” similar . The minimal solid angle can be obtained by surface integration ：
$$\Omega =\int {\int }_{S}d\Omega =\int {\int }_S\sin \theta d\varphi d\theta$$