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Irradiance measures incident radiant flux per actual receiving area. Its units $\mathrm{W}{\mathrm{m}}^{-2}$ match radiant exitance, but it counts the incoming side of the radiation exchange. Measure incoming angles from the unit normal pointing toward the incident hemisphere. ThenFor $2\mathrm{W}$ distributed uniformly over $0.010{\mathrm{m}}^2$, $E=200\mathrm{W}{\mathrm{m}}^{-2}$. If the receiving plane is tilted $60^{\circ }$ in the same broad uniform parallel beam, the incident power per actual area is halved to $100\mathrm{W}{\mathrm{m}}^{-2}$. This receiving-surface cosine factor is geometric and does not assume Lambertian reflectance. Together, irradiance and radiant exitance describe how radiation supplies and removes energy at a surface, while radiance preserves the directional information.

Lambertian reflectance is the ideal diffuse-reflection model with direction-independent outgoing radiance, the frequency-integrated specific intensity. If a fraction $0\le \rho \le 1$ of incident radiant flux is reflected, a surface receiving irradiance $E$ has reflected radiance $L=\rho E/\pi$. The reflected outward power per area is $\pi L=\rho E$ by the same hemispherical integral used for radiant exitance. Its radiant intensity still acquires the projected area factor in Lambert's cosine law.

For $E=100\mathrm{W}{\mathrm{m}}^{-2}$ and $\rho =0.80$, the reflected radiance is $80/\pi \approx 25.4648\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$. A patch of area $0.020{\mathrm{m}}^2$ reflects $1.6\mathrm{W}$ in total, and its radiant intensities at $0^{\circ }$ and $60^{\circ }$ are respectively $1.6/\pi$ and $0.8/\pi \mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. The radiance stays fixed as the view changes, provided the illumination stays fixed. Changing the incident angle can change irradiance and therefore the reflected radiance.

A spherical body of radius R observed from distance $d\gg R$ is illuminated and viewed from the same direction, with incident irradiance $F_{\star }$. For uniform Lambertian reflectance A, a patch at angle $\theta$ has outgoing radiance $A{F}_{\star }\cos \theta /\pi$. The observer receives this radiance times projected area divided by $d^2$. Since ${\int }_{\mathrm{hemisphere}}{\cos }^2\theta dS=2\pi R^2/3$, integration proves the displayed full-phase irradiance. Its geometric albedo is $2A/3$. Total reflected power alone does not specify the brightness in one direction for a non-Lambertian surface.

This idealized surface follows Lambertian reflectance, with projected-area dependence governed by Lambert's cosine law.

A Lambertian surface has direction-independent radiance. A perfectly reflecting Lambertian patch receiving normal flux $F$ emits specific intensity $I=F/\pi$ into its outward hemisphere.

For an ideal diffuse emitting or reflecting planar surface with uniform direction-independent radiance, the radiant intensity in a direction at angle $\theta$ to its unit normal is $I(\theta )=I(0)\cos \theta$. This follows immediately from $I=LA\cos \theta$ and the projected area formula. The worked radiance example and diagram show the $0^{\circ }$ and $60^{\circ }$ cases. Lambertian reflectance supplies one physical idealization realizing the law; an ideal blackbody provides an emitting example.

The historical setting is Johann Heinrich Lambert's Photometria sive de mensura et gradibus luminis, colorum et umbrae, published in Augsburg in 1760. Its Part I, §139, explicitly compares normal and oblique illumination using the sine of the source's elevation, equivalently the cosine of its angle from the receiving normal vector. This is an early receiving-surface projection calculation; the modern outgoing-patch statement above keeps radiance, radiant intensity and irradiance distinct. ^{[ref] [ref]}.

At a surface point $p$, let $\hat{N}$ be the outward unit normal vector, $\hat{V}$ the unit vector from $p$ toward the viewer, and $\hat{L}$ the unit vector from $p$ toward the light. The incoming propagation direction is therefore $-\hat{L}$. The specular reflection direction $\hat{R}$ points away from the surface along the ideal reflected ray.

Resolve $\hat{L}$ into its normal and tangential components:Reflection of the incoming direction preserves its tangential component and reverses its normal component. Equivalently, the outgoing direction has the same normal component as $\hat{L}$ and the opposite tangential component, givingThe dot product identity shows $\Vert \hat{R}\Vert =1$ when $\Vert \hat{L}\Vert =\Vert \hat{N}\Vert =1$.

In the Phong reflection model, ambient lighting approximates unmodelled indirect illumination and is independent of these directions. The diffuse term models Lambertian reflectance and varies as $\max (0,\hat{N}\cdot \hat{L})$. The specular term models a highlight around $\hat{R}$ and varies as $\max (0,\hat{R}\cdot \hat{V}{)}^n$, where $n>0$ controls its sharpness. For a visible front surface, a convenient per-colour-channel equation iswhere $I_a$ is ambient illumination, $I_L$ is the source strength at the surface including any chosen attenuation, $k_a,k_d,k_s\ge 0$ are material weights, and $\chi$ is one when $\hat{N}\cdot \hat{L}>0$ and zero otherwise. The weights can be colour-dependent. An idealized point source's incident strength decreases as inverse distance squared; the empirical equation can also use a prescribed attenuation law.

In degenerate geometric cases the effective weights switch off. A light behind or tangent to the surface contributes neither direct diffuse nor specular illumination. If $\hat{R}\cdot \hat{V}\le 0$, the specular term is zero. A viewpoint on the back side is rejected by back-face culling for a one-sided surface. A material with no highlight has $k_s=0$; with no diffuse response, $k_d=0$; without ambient illumination the ambient term vanishes. These zeros do not require renormalizing the other weights, and the empirical Phong reflection model does not impose a universal condition $k_a+k_d+k_s=1$. At coincident light/surface or eye/surface positions a direction cannot be normalized; such singular configurations require a separate convention, rather than substituting an undefined unit vector into the formula.

A bidirectional reflectance distribution function, or BRDF, describes how a surface scatters incident illumination into each outgoing direction. At a fixed surface point $p$ and wavelength, its inputs are two directions above the surface, ${\omega }_i$ toward the incident source and ${\omega }_o$ toward the observer; equivalently these are two pairs of angular coordinates. Its output is the ratio of differential reflected radiance to differential incident irradiance:Position and wavelength may be additional arguments when material properties vary. A colour implementation can return one value per colour channel. The BRDF is a material response, not the final outgoing light value: it must be combined with the incident illumination and the projected-angle factor.^[ref]

For incident radiance $L_i$, the reflection part of the rendering equation isEmission is added separately if the surface itself emits. This integral explains how a BRDF replaces the assumed material response of a local lighting model. For Lambertian reflectance, $f_r={\rho }_d/\pi$, so the cosine factor yields the diffuse term from part (a). A directional specular lobe supplies a highlight. The empirical Phong reflection model is an approximate angular-response model of this kind; its original direct specular term is not automatically a normalized, energy-conserving BRDF. The ambient term approximates missing illumination and is not itself a third direction-independent material scattering mechanism. More general BRDFs can reproduce measured surface behaviour, while illumination from other surfaces supplies indirect light in the rendering equation.

Bui Tuong Phong's empirical local lighting model combines ambient illumination, cosine-weighted Lambertian reflectance, and a specular highlight about the specular reflection direction. The exponent of the specular lobe controls highlight width. It is distinct from interpolating normal vectors across a triangle.