The four light models
Each path is followed back from the eye, through the stone, to the direction its light came from. Only that direction’s elevation e above the horizon matters. The three dome models give it a weight:
and W = 0 below the horizon or within h degrees of the zenith, where h is the head shadow you set. ISO, COS and SC2 follow the definitions in Robert W. Strickland’s GemRay user guide; the implementation is our own. The fourth window, Light Return, uses the zones in Light Return. None of the four depends on a lighting scene: they are properties of the geometry.
The figures
Rays start on a 132 × 132 grid, straight down, one per cell centre. Of those that hit the stone, each contributes the light reflected off the crown plus the light that passes through:
R is Schlick’s approximation, R(θ) = F0 + (1 − F0)(1 − cos θ)5, at the entry facet and at the exit facet. Bounces in between are total internal reflections and lose nothing. A figure of 100% would mean every ray returned all its light at full weight. The table-only figure is the same sum over the rays that enter through the table.
Leakage counts paths, not energy: the share of rays that leave below the horizon, find no exit, or are still inside after 12 bounces.
It does not depend on the head shadow. The “vs T57” column is a relative difference, 100 (a/b − 1), and for leakage a difference in percentage points.
Because partial internal reflections are not counted, these figures sit above those of a full physical simulation, and the dip near face-up is shallower than GemRay shows.
Tilt
The stone rotates about one horizontal axis while the dome and the viewer stay fixed, as on a hand. The chart traces 13 tilts from −30° to 30° in 5° steps, on an 84 × 84 grid. The windows draw a reduced set of planes for T105 and DG-Vitruvio, because the graphics card holds 96 at most; the figures always use the full set. Only cuts whose planes agree with their STEP solid to within 1 percentage point on every light model are included.
The Brightness & Tilt Analysis measures how much of a dome’s light a cut returns to the viewer, and how that changes as the stone tilts. It uses the same tracing for its pictures and its figures, in two separate passes: the graphics card draws the four windows, and the figures are computed on the processor.