What is "solid angle" and how does it relate to photography?

Question

So, I was hanging out in the chat room, and hear mention of something called "Solid Angle". What is this, and how can it be important?

Answer 1

The solid angle is the extension of the concept of angle from two to three dimension. So let's start from 2d: consider a circle and pick two rays starting from the center. They will divide the circumference in two parts, called arcs. The length of each arc divided by the length of the radius will be the measure of the angle subtended by the arc itself.

Extend this to three dimensions: instead of a circle take a sphere, and instead of picking two rays pick a cone centered in the center of the sphere.The cone will cross the surface of the sphere: and now to define the solid angle measure the area of the surface delimited by the cone, divided by the square of the length of the radius (so that we have an area divided by an area).

The key point is that - since they are ratios - angles (and the solid ones make no exception) are dimensionless quantities: a small object as seen from a short distance can cover the same angle as a large object as seen from a long distance.

Why does this matter ? Because we live in 3 spatial dimensions ( :-) ). For instance consider a single light point source radiating (a star seen from very far?) By symmetry there is no reason for it to radiate more in one direction than in the other. So all the photons will be equally spread out in the space. Now you decide to look at how much light arrives in a given region of space: trace a "cone" from the region of space of your interest (the subject of your photo) with the vertex on the star, and you will have "measured" the solid angle. Now the ratio of photons will be equal to the ratio of the solid angle to the total (which is, by the way, 4*pi, similar to 2*pi in two dimensions): if the star is very far, this will be a very small number.

Now from stars move to flash units. These are not really point like (neither stars are, after all :) ) and not radiate isotropically (they are usually oriented so that all the light goes somewhere useful) but the same reasoning applies since they are usually much smaller than the subjects we are photographing.

This kind of computations underlies the so called inverse square law effect (basically you are spreading a fixed amount of light in a given solid angle: the area of the sphere subtended by the same solid angle grows with the square of the distance from the source, and so if you double the distance the area will be squared).

Answer 2

A solid angle is a fairly abstract concept of geometry, but hopefully easy enough to understand once the concept is grasped. One simple way to think of it is to expand the concept of a normal angle from one dimension (the length of an arc) to two dimensions (the area of a circle). An angle is defined by the arc that "subtends" two rays extending from the center point of a unit circle. The formula for an angle is:

θ = s/r

(Where s is the length of the arc between the two rays, and r is the radius the circle)

In the same way, a solid angle is defined by the area of a "circle" that subtends two rays extending from the center point of a unit sphere. Where the rays intersect with the surface of the sphere, an arc between the two rays is created at the sphere's surface...your angle. However, that same arc can be drawn at any orientation on the surface of the sphere. Assuming you spun the arc around its center point on the surface of the sphere, you would create a circle on the surface of the sphere. Another way to look at it would be to say the area of a circle on the surface of a sphere created by the projection of a cone created by the same angle from the center of the sphere. The area of that circle is a solid angle. The formula for a solid angle is:

Ω = A/r^2

(Where A is the area of the circle as subtended by the two rays, and r is the radius of the sphere)

Given the units of both equations, both angles and solid angles are unitless and independent of the actual size of the unit circle or sphere they are based on.

---

Solid angles have useful application in photography, namely in the area of calculating luminance from a light source and deriving the necessary exposure value to properly expose a scene lit by a given luminance. The standard unit of solid angles is the steradian, a unitless value that represents the solid angle of area r^2. The solid angle of a whole sphere is 4π sr. The favored unit for measurements of illumination when calculating exposure value is lux, and it so happens that one lux is the equivalent of one candela (a measurement of luminous intensity) steradian per meter squared:

1 lux = 1 cd sr/m^2

A lux is a measurement of light of a certain intensity (cd) emitted from a certain geometry (steradians) per specific area (m^2). Solid angles are important to photography as they help bring specific geometry into the equation. This is all well and good when one needs to be highly specific in regards to exposure, such as when performing scientific tests of camera equipment for the purposes of comparing one piece of gear to another.

From a practical standpoint, solid angles don't have much real-world application. One generally doesn't spend time running the math when setting up studio lighting...such things are best learned by experimentation, building up a body of experience and understanding from actual use of lighting apparatuses. Only then can all the nuances of illumination, shading, and light in general be understood in a practical sense.

For a detailed explanation of exactly how solid angles are important to calculating exposure value given a specific illumination, see my answer to the following question:

What is the difference between luminance and illuminance?