Let's focus on a simpler implementation though: Planar, infinite, and with a linear transitional region. A transitional region is nice because it means the fog appears to gradually taper off instead of being conspicuously contained entirely below a flat plane.

In practice, there is one primary factor that needs to be determined: The amount of fog penetrated by the line from the viewpoint to the surface. In determining that, the transitional layer and the surface layer actually need to be calculated separately:

Transition layer

For the transition layer, what you want to do is multiply the distance traveled through the transition layer by the average density of the fog. Fortunately, due to some quirks of the math involved, there's a very easy way to get this: The midpoint of the entry and exit points of the transitional region will be located at a point where the fog density is equal to the average density passed through. The entry and exit points can be done by taking the viewpoint and target distances and clamping them to the entry and exit planes.

Full-density layer

The full-density layer is a bit more complex, since it behaves differently whether the camera is inside or outside of the fog. For a camera inside the fog, the fogged portion is represented by the distance from the camera to the fog plane. For a camera outside of the fog, the fogged portion is represented by the distance from the object to the fog plane. If you want to do it in one pass, both of these modes can be represented by dividing one linearly interpolated value by the linearly interpolated distance of the camera-to-point distance relative to the fog plane.

Since the camera being inside or outside the fog is completely determinable in advance, you can easily make permutations based on it and skip a branch in the shader. With a deferred renderer, you can use depth information and the fog plane to determine all of the distances. With a forward renderer, most of the distance factors interpolate linearly, allowing you to do some clamps and divides entirely in the shader.

Regardless of which you use, once you have the complete distance traveled, the most physically accurate determination of the amount still visible as:

min(1, e-(length(cameraToVert) * coverage * density))

You don't have to use e as the base though: Using 2 is a bit faster, and you can rescale the density coefficient to achieve any behavior you could have attained with using e.

As usual, the shader code spoilers:

// EncodeFog : Encodes a 4-component vector containing fraction components used

// to calculate fog factor

float4 VEncodeFog(float3 cameraPos, float3 vertPos, float4 fogPlane, float fogTransitionDepth)

{

float cameraDist, pointDist;

cameraDist = dot(cameraPos, fogPlane.xyz);

pointDist = dot(vertPos, fogPlane.xyz);

return float4(cameraDist, fogPlane.w, fogPlane.w - fogTransitionDepth, pointDist);

}

// PDecodeFog : Returns the fraction of the original scene to display given

// an encoded fog fraction and the camera-to-vertex vector

// rcpFogTransitionDepth = 1/fogTransitionDepth

float PDecodeFog(float4 fogFactors, float3 cameraToVert, float fogDensityScalar, float rcpFogTransitionDepth)

{

// x = cameraDist, y = shallowFogPlaneDist, z = deepFogPlaneDist (< shallow), w = pointDist

float3 diffs = fogFactors.wzz - fogFactors.xxw;

float cameraToPointDist = diffs.x;

float cameraToFogDist = diffs.y;

float nPointToFogDist = diffs.z;

float rAbsCameraToPointDist = 1.0 / abs(cameraToPointDist);

// Calculate the average density of the transition zone fog

// Since density is linear, this will be the same as the density at the midpoint of the ray,

// clamped to the boundaries of the transition zone

float clampedCameraTransitionPoint = max(fogFactors.z, min(fogFactors.y, fogFactors.x));

float clampedPointTransitionPoint = max(fogFactors.z, min(fogFactors.y, fogFactors.w));

float transitionPointAverage = (clampedPointTransitionPoint + clampedCameraTransitionPoint) * 0.5;

float transitionAverageDensity = (fogFactors.y - transitionPointAverage) * rcpFogTransitionDepth;

// Determine a coverage factor based on the density and the fraction of the ray that passed through the transition zone

float transitionCoverage = transitionAverageDensity *

abs(clampedCameraTransitionPoint - clampedPointTransitionPoint) * rAbsCameraToPointDist;

// Calculate coverage for the full-density portion of the volume as the fraction of the ray intersecting

// the bottom part of the transition zone

# ifdef CAMERA_IN_FOG

float fullCoverage = cameraToFogDist * rAbsCameraToPointDist;

if(nPointToFogDist >= 0.0)

fullCoverage = 1.0;

# else

float fullCoverage = max(0.0, nPointToFogDist * rAbsCameraToPointDist);

# endif

float totalCoverage = fullCoverage + transitionCoverage;

// Use inverse exponential scaling with distance

// fogDensityScalar is pre-negated

return min(1.0, exp2(length(cameraToVert) * totalCoverage * fogDensityScalar));

}