The Lighting Equation

Lighting Equation

$I_{λ} = I_{aλ} k_{a} O_{dλ} + f_{att} I_{pλ} [k_{d} O_{dλ} N L + k_{s} O_{sλ} (R V)^{n}]$

$I$ - is the Intensity
$a$ - is the ambient component. eg $I_{aλ}$ is the ambient intensity.
$f_{att}$ - is the light-source attenuation factor. Light drops off with increased distance from the light source.
$λ$ - is the color component. ie red, green or blue.
$k$ - is the reflection coefficient, the amount of light reflected from the surface.
$p$ - is the light source point.
$O$ - can be thought of as an operator accessing a color vector for a color value. eg $O_{dλ}$ accesses the diffuse color vector with component $λ$ .
$d$ - is the diffuse color material property. In general our every day experience of color.
$s$ - is the specular reflection.

Let all the vectors be normalized and multiplication between vectors is the dot product.
$L$ - is the Light Source vector.
$N$ - is the Normal of the geometry at .

What is a point in 3D Space?

I've been thinking about this question over the last 6 months and find it very interesting. For start forget the literal view of a point as being the absolute reality. In 3D space the simplest representation of a surface is a triangle so when viewing the light equation it is being applied to a triangle before hitting the screen.

The triangle can be mapped back to a point by interpolating. eg consider an arbitrary point on the triangle. That point could have the average color of the three points. Similarly it could have the average normal of the three points.

The way that the lighting equations is written suggests how a point is to be interpreted. The given lighting equation is doing color and normal interpolation. The current graphics hardware does color interpolation but not normal interpolation easily.

Now that the point is found in 3D space it is projected to 2D space (the screen). Pixels are another type of point and for viewing it was found better to do the calculation from the pixels point of view. For each pixel go back and find the point in 3D space which it came from.

The consequences are vast - the maths is different in different directions, the hardware would need to be different to pipeline operations and so is the quality of the image produced.

Back to the point in 3D space for a discussion on the normal vector at a point. The normal is perpendicular(at right angles) to the surface. In maths this describes the change of the surface, so even when the geometry is not continuous having light reflected with the correct normal produces realistic results.

Ambient Component

$I_{aλ} k_{a} O_{dλ}$

Ambient light is an approximation for light from all directions. The background light. Contrast this with radiosity rendering which instead of assuming constant calculates the light reflected from surfaces.

Diffuse Component

$k_{d} O_{dλ} N L$

Specular Component

$k_{s} O_{sλ} (R V)^{n}$

General Formula Discussion

The lighting equation says how much light falls on a point. What is a point in 3D space? The equation is a linear sum (there are other ways to sum) and if the sum is greater than the maximum its set to the maximum. If the dot products result in a negative value the result is ignored and set to zero (summing of zero in a sum is ignoring).