### Ambient The ambient term represents the "left-over" light that has scattered about the entire scene and that essentially comes from everywhere. It is an approximation to light energy at equilibrium after it has interacted with the entire environment. In many ways, it acts as a correction term that adds more light energy until the result "looks right:" ${tex.block`I_a = \color{${d3.schemeCategory10[4]}}c_d\color{${d3.schemeCategory10[1]}}I_m`} where ${tex`\color{${d3.schemeCategory10[1]}}I_m`} is the intensity of the ambient light, a property of the scene and ${tex`\color{${d3.schemeCategory10[4]}}c_d`} is the (diffuse) color of the object, one of its material properties. ### Diffuse Light directly hitting an object reflects back light that it does not scatter away with the apparent color of the object. This reflection is strongest for light surfaces if the light is directly above the surface, and decreases as the light makes a larger angle (until the point it is at right angle to the surface, and no light strikes). We represent that intensity by the scalar product of the normal and the vector towards the light ${tex.block`I_d = \color{${d3.schemeCategory10[4]}}c_d\color{black} \sum_i \color{${d3.schemeCategory10[2]}}I_{l_i}\color{black} \mathbf{n} \cdot \color{${d3.schemeCategory10[2]}}\mathbf{l}_i`} If the dot product is negative, the light is behind the object. Normally, we just do not light the object, though some approaches will allow the object to be lit in both directions. ### Specular There is a specific direction with respect to an object where the light reflection is the strongest for particularly shining objects. By [the Law of Reflection](https://en.wikipedia.org/wiki/Reflection_%28physics%29#Laws_of_reflection), the direction of maximum reflection is the ray with the same angle as the incident light with respect to the surface/normal, but at right angles to the incident light: ${tex.block`\mathbf{r}_i=2\left(\mathbf{n}\cdot\color{${d3.schemeCategory10[2]}}\mathbf{l}_i\color{black}\right)\mathbf{n}-\color{${d3.schemeCategory10[2]}}\mathbf{l}_i`} The maximum reflection occurs when this reflected ray is aligned with the vector towards the eye, which we can measure via the dot product again. This reflection strength is attenuated by a specular factor ${tex`\alpha`}—more matte objects have a low specular factor, while shiny objects have a larger factor. Combined with a reflection color, we can determine the specular intensity: ${tex.block`I_s = \color{${d3.schemeCategory10[4]}}c_s\color{black} \sum_i \color{${d3.schemeCategory10[2]}}I_{l_i}\color{black} \left(\color{${d3.schemeCategory10[1]}}\mathbf{e}\color{black} \cdot \mathbf{r}_i\right)^\alpha`} This value is treated as zero if the light is behind the object (unless two-sided lighting is performed). For lights and view positions that are very far away, we can approximate the reflection via the *halfway vector*— the average of the light vector and eye vector respectively. ${tex.block`\mathbf{h}_i=\frac{\color{${d3.schemeCategory10[1]}}\mathbf{e}\color{black} + \color{${d3.schemeCategory10[2]}}\mathbf{l}_i\color{black}}{\left\|\color{${d3.schemeCategory10[1]}}\mathbf{e}\color{black} + \color{${d3.schemeCategory10[2]}}\mathbf{l}_i\color{black}\right\|}`} If this vector aligns with the surface normal, than the eye is aligned with the maximum reflection: ${tex.block`I_s = \color{${d3.schemeCategory10[4]}}c_s\color{black} \sum_i \color{${d3.schemeCategory10[2]}}I_{l_i}\color{black} \left(\mathbf{n} \cdot \mathbf{h}_i\right)^{\alpha'}`} The halfway vector is somewhat cheaper to calculate as it does not differ per-object, only per-light. ### Emissive The emissive component just represents light the object itself contributes. Headlight in a car, a bulb in a light figure—all have emissive components. ${tex.block`I_e = \color{${d3.schemeCategory10[4]}}c_e`} ### Final Calculation Each component is calculated and added together to determine the final illuminated color; the diffuse and specular components are repeated per light for each position. Without doing high-dynamic range lighting, colors above 1 are just clamped to 1 as a maximum value. We can instead do a simple [Reinhard tone mapping](https://64.github.io/tonemapping/#reinhard) operation to pull all the colors within the range. ~~~glsl vec3 tone_mapped = color / (1.0f + color); ~~~ assuming `color` stores the calculated intensity. Reinhard will wash things out a bit, so other tone mapping might be preferred. The version I do here is [the luminance corrected Reinhard](https://64.github.io/tonemapping/#extended-reinhard-luminance-tone-map "Extened Reinhard") with the obligatory [sRGB correction](https://unlimited3d.wordpress.com/2020/01/08/srgb-color-space-in-opengl/ "sRGB color space in OpenGL").