Source : ct|01.06.13

< Mathematics and physics

Elements of surface, angles and solid angles

The dictionary gives the following definition of angle : "corner" or "figure formed by two half-lines or two half-planes that intersect". However the term solid angle (which is used in mathematics and physics) does not appear in the words of common usage .
What is the precise definition of this concept and what is it exactly ?

Element of surface

A surface element dS of a surface S is shown (at a point M) with a vector dS=dS.n (the vector n is perpendicular to dS at the point M)

For the direction of n, there are two cases :

• If the surface is closed (it completely encloses a volume) n is outwardly directed;
• If the surface is not closed but is based on a contour, this contour is assigned a positive direction, chosen anticlockwise, and n is oriented in the direction of the direct trihedron (direction of corkscrew).

Angle

Definition

An angle is a dimensionless variable that defines the space between two line segments that intersect at a point. In other words, an angle is the space portion bounded by the center of a circle and a part of the circumference of this circle.

Given a circle with center 0 and radius R, and two points A and B very close to each other on this circle. The angle dθ formed by the line segments 0A and 0B is defined by

The angle dθ is expressed in radians (rad), dimensionless number.
dθ is infinitely small and dL is equivalent to an infinitely small segment.

If the points A and B are diametrically opposed, the arc AB is equal to 2πR/2 and in this case θ=2πR/2R=π (rad)

Generalization

We take any arc infinitely small dL' (assimilated to a straight segment infinitely small) inclined at an angle β with respect to dL. We have dL=dL’cos β=dL’ u.n.
and

u= 0M/0M unit vector directed in the direction 0M
n unit vector which is perpendicular to dL' at the point M

Solid angle

Definition

From similar way to an angle, which is a dimensionless quantity defining the portion of space delimited by the center of a circle and a portion of the circumference of this circle, a solid angle is a dimensionless quantity which defines the portion of the space bounded by the center of a sphere and a part of the surface of this sphere.

Are considered : a sphere with center 0 and radius R, and a surface element dS on the sphere.
The solid angle dΩ which defines the portion of space delimited by the center of the sphere and the surface element dS is defined by

The angle dΩ is expressed in steradian (sr), a dimensionless number.
If we consider the portion of space bounded by a quarter of a sphere of radius R, the corresponding solid angle is Ω = 4πR²/8R²=π/2 (sr).

Generalization

If one takes any infinitesimal surface element dS' (considered as surface of an infinitely small circle around a point M) inclined at an angle β relative to dS we have dS=dS’cos β=dS’ u.n
and

u = 0M/0M unit vector directed in the direction 0M
nunit vector which is perpendicular to dS' at the point M

Therefore the solid angle which defines the portion of space between a point 0 and an elementary surface dS =dS.n (located around a point M and having a given orientation) is (with R = OM)

In particular, if the surface element dS is perpendicular to 0M, we find

Example 1: solid angle defined by a disc

What is the solid angle defined by a disc and a point P situated on the axis of this disc?

Answer

The disc is cut into a succession of elementary circular rings (nested one inside the other) having a width equal to dr and the same axis (0P). The surface elements constituting each circular ring are oriented at the same angle φ with the axis of the disk (see diagram below).
The surface of an elementary ring is dS =2πrdr avec r=h tanφ =>dr=(h/cos²φ)dφ
and dS=2πh tanφ (h/cos²φ)dφ = 2πh²(tanφ/cos²φ)dφ
Furthermore, dΩ = dS cosφ/a²(φ) avec a(φ)=h/cosφ
and
dΩ= dS cosφ cos²φ/h²
dΩ = 2πh²(tanφ/cos²φ) cosφ cos²φ/h²dφ
dΩ = 2πsinφdφ
To cover the entire disk, we include an infinite number of elementary circular rings (φ varying between 0 and θ/2). We deduce Ω=2π (1-cos θ/2).

• if θ -> π, cos θ/2 -> 0 et Ω->2π
• if θ -> 0, cos θ/2 -> 1 et Ω->0

Example 2: solid angle defined by a rectangle

What is the solid angle defined by a rectangle of sides a and b (such as the sensor of a camera or of a digital camcorder ) and a point P located at a distance h of the center of the rectangle on its central axis ?

Answer

The calculations are longer than in the case of a disk, because in this case there is no cylindrical symmetry.
Assume, for example, the rectangle is centered at 0 and placed in the xy plane.
We calculate the fourth of the solid angle of the rectangle (side p=a/2 and q=b/2) then, by symmetry, we mulitplie the result by 4 to obtain the solid angle relative to the complete rectangle.

and

We first integrate with respect to x to obtain a function i (y)

Indeed,

We have y =h tanα => dy=(h/cos²α)dα avec 1/cos²α = 1+tan²α

Indeed, the Arcsin function has the following definition and properties

from wich

We deduce the total solid angle, defined by the entire rectangle (sides a=2p and b=2q) by multiplying the preceding result by 4.

• If h is very less than p and than q, we have Ω ~ 4 Arcsin 1 = 4π/2 = 2π
• If h is much greater than p and q we have Ω ~ 4 Arcsin 0 = 0
• For a square, p=q et Ωsquare = 4 Arcsin (p²/p²+h²)
• For a square with h>>p Ωsquare ~ 4 Arcsin (p²/h²)