The aim of this project is to construct a radiosity-based diffuse-diffuse interaction renderer based on half-remembered fifteen-plus-year-old knowledge and working it out again from first principles.
It's more-or-less worked.
We'll start with that fact that a matt wall, lit evenly, looks a single colour. It doesn't get brighter or darker depending on the angle we view it at, or the distance we view it from. Put another way, the amount of light our eye receives just depends on how much light the wall emits per area, and the angle it subtends in our vision.
What does this mean physically?
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In terms of scale independence, the light will fall off with the square of the distance, but so will the area subtended at the eye by a given surface, so the light per area remains constant.
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If light-received-per-area-seen-by-viewer remains constant, the amount of light emitted given off must drop off with angle, as the amount of surface-per-area-facing-the-viewer increases. This factor is cosine of the angle between the viewer and the normal.
We can thus write the amount of light received at a point as
L = B * A * cos(theta)
Where L is the amount of light, B is the brightness of the
emitting surface, A is its area, and theta is the angle between
the normal of the emitting surface and the ray to the receiver.
How do we find the amount of light received by an area?
Assuming the light comes from a reasonably-point-like light source, the amount of light received will be proportional to the area, as measured perpendicular to the ray from the emitter. Putting it all together, the equation is:
L_er = B_e * A_e * cos(theta_e) * A_r * cos(theta_r)
where e is the value for the emitter, r is the value for the
receiver, and the parameters are otherwise the same as previously.
Rather neatly, the equation is symmetric, demonstrating the nice symmetry between forwards and backwards ray-tracing, and how the radiosity solution represents an equilibrium where the net light flow between pairs of surfaces is zero.
If a point has a certain amount of brightness, how much light should it be giving off in each direction? How can work this out by integrating the equation above over all directions, to get the total it emits, and come up with a normalising factor.
To specify directions over the hemisphere we're integrating over,
we'll say that phi is the angle from the normal (0 to pi/2), and
theta is the angle around the normal (0 to 2 * pi). We want to
calculate the integral of L(phi) sin(phi) d phi d theta - the "sin phi" term is because there's a lot more surface around the edge of
the hemisphere than at the top. L is symmetric around the normal, so
doesn't depend on theta. We have:
L(phi) sin(phi) d phi d theta
= 2 * pi * L(phi) sin(phi) d phi
= 2 * pi * cos(phi) * cos(rho(phi)) * r(phi)^-2 * A(phi) * sin(phi) d phi
where rho is the angle on the receiving surface, r is the distance
from source to destination for the given angle, and A is the rate at
which area is swept out for small changes of phi and theta.
The cos(rho(phi)) * r(phi)^-2 * A(phi) terms all cancel out - as
radius increases, the area being swept out increases, so that the same
amount of light is received. If the receiving surface is at an angle,
more area is swept out, so the change in A term balances the change
from cos(rho(phi)). The equation simplifies to integrating over:
2 * pi * cos(phi) * sin(phi) d phi
from 0 to 2 * pi. The integral is... pi.
When I sum up the amount of light emitted by the light source in the code, and take out all the scaling factors I've added in, the total light received from sending out a "unit" light is 3.1449! The error is around one in a thousand, which seems about right since the calculations are done using transmission to a few thousand rectangles.
We have empirical evidence that the calculations are about right, and can look into bouncing the light around...
Source is in this project. :)