## Geometry

### Plane angle $a$

$$a = s / r$$

Arc length ($s$) over radius ($r$).

### Solid angle

$$\omega = \frac{A}{r^2}$$

Area subtended ($A$) over squared radius ($r^2$).

### $(\theta, \phi)$-parameterization

$$dA = r^2 \sin(\theta) d\theta d\phi$$

$$d\omega = \frac{dA}{r^2} = \sin(\theta) d\theta d\phi$$

#### Sphere surface area

\begin{align} A & = \int_{\phi = 0}^{2 \pi} \int_{\theta = 0}^{\pi} r^2 \sin(\theta) d\theta d\phi \\ & = r^2 \int_{\phi = 0}^{2 \pi} \int_{\theta = 0}^{\pi} \sin(\theta) d\theta d\phi \\ & = r^2 \int_{\phi = 0}^{2 \pi} [-\cos(\theta)]_{0}^{\pi} d\phi \\ & = r^2 \int_{\phi = 0}^{2 \pi} [-\cos(\pi) + \cos(0)] d\phi \\ & = r^2 \int_{\phi = 0}^{2 \pi} 2 d\phi \\ & = r^2 (4 \pi) = 4 \pi r^2 \end{align}

### Radiant flux $\Phi$

$$\Phi = \frac{dQ}{dt}$$

Energy per unit time.

Unit: Watt $[W]$

Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".[1]

### Radiant intensity $I_\Omega$

$$I_\Omega = \frac{d\Phi}{d\omega}$$

Radiant flux per unit solid angle.

Unit: Watt per steradian $[W * sr^{-1}]$

Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.[1]

### Irradiance $E$

$$E = \frac{d\Phi}{dA}$$

Radiant flux per unit projected area.

Unit: Watt per square meter $[W * m^{−2}]$

### Radiance $L_\Omega$

$$L_\Omega = \frac{d^2 \Phi}{d\omega dA \cos(\theta)}$$

Radiant flux per unit solid angle per unit projected area.

Unit: Watt per steradian per square meter $[W * sr^{-1} * m^{−2}]$

Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".[1]

\begin{align} E = \int_{H^2} L_\Omega cos(\theta) d\omega \\ \frac{dE}{d\omega} = L_\Omega cos(\theta) \\ L_\Omega = \frac{dE}{d\omega cos(\theta)} \\ L_\Omega = \frac{d(\frac{d\Phi}{dA})}{d\omega cos(\theta)} \\ L_\Omega = \frac{d^2 \Phi}{d\omega dA cos(\theta)} \end{align}