The 3D models | How we measure

Geometry and units

Each model is a closed, watertight solid of flat facets, in millimetres, with the table facing +Y and the girdle plane through the origin. Every cut except the heart is convex, so the tools describe it as an intersection of half-spaces, one per facet:

S = {x : n_{i} \cdot x \leq d_{i} for every facet i}

with n_i the facet’s unit outward normal and d_i its distance from the centre. A ray enters at the last plane it crosses going in and leaves at the first plane it meets going out. The heart is traced triangle by triangle instead.

The facet count on a cut page is the number of planar faces of the STEP solid. When the cutting design counts differently (a turned girdle, for instance, is one surface in the solid but many facets on the machine), the page gives both.

Volume and weight

Weight in carats follows from volume V in mm³ and specific gravity ρ in g/cm³:

m = \frac{V \cdot ρ}{200} ct

because a carat is 0.2 g and a cubic centimetre is 1,000 mm³. Every weight on a cut page uses this. At a given shape, weight scales with the cube of size, so the simulator scales each model by

k = \sqrt[3]{\frac{200 m}{ρ V_{ref}}}

to reach the carat weight you choose, where V_ref is the volume at the model’s reference size. The build measures every model’s volume from its mesh each time, by the divergence theorem over its triangles,

V = \frac{1}{6} \sum_{triangles} a \cdot (b \times c)

and measures its size, length to width and depth from the same mesh, with the stone turned level in plan. Rounds keep their diameter across the girdle corners. These are the figures the cut pages print. The build also checks them against the measurements of the STEP solid, and stops if a mesh is not closed, with every edge shared by exactly two triangles.

Proportions

Total depth is a percentage of the width, the shorter side in plan.
Length to width is the ratio of the plan’s bounding box, to three decimals.
Table is the table’s extent over the width. For rounds it is measured across the table’s corners: the largest of 64 directions.
Crown and pavilion angles are the cutting design’s own main tiers. Where a design names none, they come from the model’s specification or, failing that, the area-weighted most common facet angle, in 0.5° bins, weighting each facet by its projected area A |n_y|. Angles are measured from the girdle plane: a facet’s angle is arccos |n_y|.
Girdle facets are those within about 1° of vertical. Their thickness is sampled at every vertex column, and the page gives its minimum and maximum.

Cutting-file accuracy

Each cut page states how closely the cutting file matches the 3D model. Every facet in the .asc file, at angle a and index i on a gear of g teeth, becomes a normal:

n = (\sin a \cos φ, \pm \cos a, \sin a \sin φ), φ = \frac{2 π i}{g}

We then turn the model about its axis in steps of a quarter index, in both handedness, and keep the alignment with the smallest area-weighted mean angle between each model facet and its nearest file facet. The page reports the largest angle and the mean, with the girdle given on its own. Facets smaller than 0.02% of the largest facet are left out as slivers. The check compares facet directions only, not where facets meet.