Solid Angle

Recall from 2D than the angle subtended by an object, O, from a point, x, is the length of the projection of O onto a unit circle centred at x. It is measured in "unit angles", or radians (rad). The full circle subtends index1.gif radians.

In 3D, the angle subtended by an object, O, from a point, x, is the area of the projection of O onto a unit sphere centred at x. It is measured in "unit solid angles", or steradians (sr). The full sphere subtends index2.gif steradians.

Measures of Illumination

Radiant power ( index3.gif ) is the rate at which light energy is transmitted, and is measured as energy (photons) per unit time, in Watts.

Power at a Point

Flux is the radiant power crossing a particular surface.

Flux density ( index4.gif ) is the radiant power per unit area of the surface, measured in Watts per unit area. At a point, index5.gif , surrounded by an infinitesimal surface, index6.gif , the flux density is


Incident flux density is the flux density of particles arriving at a surface from all directions. It is also called irradiance (E).

Exitant flux density is the flux density of particles leaving a surface in all directions. It is also called radiosity (B).

Power in a Direction

Consider a point light source. Its radiant intensity (I) is the power radiated per unit solid angle, and is measured in Watts per steradian.

In a particular direction, index8.gif , surrounded by an infinitesimal solid angle, index9.gif , the radiant intensity is


Power at a Point, in a Direction

Radiance (L) is the flux density at a point, index11.gif , in a direction, index12.gif . Sometimes, the direction is given by two angles: elevation index13.gif (from the surface normal, index14.gif ) and azimuth "phi" (from a given direction on the surface).

The flux density is measured with respect to a surface perpendicular to the direction. If the surface at index16.gif is not perpendicular to direction index17.gif , it is projected onto a surface that is.

The figure below shows the radiance at index18.gif as the flux density across the surface index19.gif in the direction index20.gif :


Since the ratio of the areas, index22.gif and index23.gif , is index24.gif (where index25.gif is the angle between index26.gif and index27.gif , the surface normal at index28.gif ), radiance is typically written as


Radiosity and Radiance

Radiosity is the exitant flux density from a point: index30.gif . How does radiosity relate to radiance?

Let index31.gif be the set of directions above the surface at index32.gif . From the radiance equation above,




Similarly, for incoming energy, the irradiance is


Intuitively, the cosine term occurs because the radiance in direction index36.gif is taken with respect to a surface, index37.gif , perpendicular to the direction, not with respect to the surface itself:

Path Convservation of Radiance

[Sillion & Puech 2.1.2]

Given two points, index38.gif and index39.gif , the radiance shot from index40.gif toward index41.gif is equal to that received by index42.gif from the direction of index43.gif . Radiance is not attenuated with distance.

As a consequence, all we need in order to render an image is the radiance, in the direction of the eye, leaving each point of the scene.

Reflection and BRDFs

[Sillion & Puech 2.1.4]

The bidirectional reflectance distribution function at a point, index66.gif , is the ratio of the radiance going out in one direction to the flux density coming in from another direction.

Let index67.gif and index68.gif be the incoming and outgoing directions, respectively.

The incoming directional flux density, index69.gif , is the flux density of energy arriving from an infinitesimal solid angle, index70.gif , around index71.gif . From the radiance equation,


The infinitesimal flux density, index73.gif , is taken with respect the the infinitesimal solid angle, index74.gif . The BRDF is then


Note that the units of the BRDF are per steradian. The reflected radiance due to radiance incoming from a particular direction is


BRDFs and Lambertian Surfaces

A Lambertian surface, or "ideal diffuse surface", has a constant BRDF:


The radiance of such a surface is independent of direction. The total outgoing radiance in a particular direction, index78.gif , is the integral of reflected radiance from all incoming directions, index79.gif :


How are radiosity and irradiance related for a Lambertian surface?


Inuitively, the quantity index82.gif is the fraction of irradiance, index83.gif , that is reflected (as radiosity, index84.gif ). This quantity is known as the albedo ( index85.gif ):




How are radiosity and outgoing radiance related for a Lambertian surface?