Projected area and solid angle

What is projected area?

Projected area is defined as the rectilinear projection of a surface of any shape onto a plane normal to the unit vector. The differential form is dAproj = cos(b) dA where b is the angle between the local surface normal and the line of sight. We can integrate over the (perceptible) surface area to get

Some common examples are shown in the table below:

SHAPE	AREA	PROJECTED AREA
Flat rectangle	A = L×W	Aproj= L×W cos b
Circular disc	A = p r2     = p d2 / 4	Aproj = p r2 cos b          = p d2 cos b / 4
Sphere	A = 4 p r2 = p d2	Aproj = A/4 = p r2

What is solid angle?

Plane angle and solid angle are two derived units in the SI system. The following definitions are taken from NIST SP811.

"The radian is the plane angle between two radii of a circle that cuts off on the circumference an arc equal in length to the radius."

The abbreviation for the radian is rad. Since there are 2p radians in a circle, the conversion between degrees and radians is 1 rad = (180/p) degrees.

A solid angle extends the concept to three dimensions.

"One steradian (sr) is the solid angle that, having its vertex in the center of a sphere, cuts off an area on the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere."

The solid angle is thus ratio of the spherical area to the square of the radius. The spherical area is a projection of the object of interest onto a unit sphere, and the solid angle is the surface area of that projection. If we divide the surface area of a sphere by the square of its radius, we find that there are 4p steradians of solid angle in a sphere. One hemisphere has 2p steradians.

The symbol for solid angle is either w , the lowercase Greek letter omega, or W , the uppercase omega. I use w exclusively for solid angle, reserving W for the advanced concept of projected solid angle (w cosq ).

Both plane angles and solid angles are dimensionless quantities, and they can lead to confusion when attempting dimensional analysis.