Definition:Solid Angle
Definition
A solid angle is a configuration in space formed by all the half-lines whose endpoints coincide at a single point and which pass through a closed plane curve.
There are two types of solid angle:
Smooth Curve
A solid angle described by a smooth curve is a solid angle whose closed plane curve through which the half-lines pass is a smooth curve.
Hence the solid angle so defined is the nappe of a cone.
Polygon
A solid angle described by a polygon is a solid angle whose closed plane curve through which the half-lines pass is a polygon.
In the words of Euclid:
- A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.
Otherwise: A solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.
(The Elements: Book $\text{XI}$: Definition $11$)
Vertex of Solid Angle
The common point through which pass the copunctal half-lines describing a solid angle is known as the vertex of that solid angle.
Subtend
Let $S$ be a surface oriented in space.
Let $P$ be a point in that space.
The solid angle subtended by $S$ at $P$ is equal to the surface integral:
- $\ds \Omega = \iint_S \frac {\mathbf {\hat r} \cdot \rd \mathbf S} {r^2}$
where:
- $\mathbf {\hat r} = \dfrac {\mathbf r} r$ is the unit vector corresponding to the position vector $\mathbf r$ of the infinitesimal area element $\d \mathbf S$ at $P$
- $r$ is the magnitude of $\mathbf r$
- $\mathbf {\hat n}$ represents the unit normal to $\d S$.
Also see
- Results about solid angles can be found here.
Souces
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): solid angle
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): solid angle