Hearts & Arrows Scope | How we measure

The cuts

Four cuts carry the pattern: the T73 and T105 round brilliants, DG-Vitruvio and DG-Anteros. Each is traced on its own 3D model, the geometry we sell, as the planes of its facets. Pieces of one facet are merged and the plane refitted through all their corners. Each cutting design has exact eight-fold mirror symmetry, while a model matches its design only to its stated accuracy. So each facet is replaced by the average of its 16 mirror and turn copies, which moves no facet by more than 0.0005°. A cut as designed therefore reads exactly 100%. DG-Vitruvio’s model approximates its 96 girdle facets with 208 faces; its girdle is rebuilt from the faces in one sixteenth of the circle.

The viewer

The eye looks straight down the stone’s axis, with parallel rays, one or more per pixel. Each ray refracts in, reflects totally inside for up to 24 bounces, and leaves at the first facet that lets it out. The view colours it by the angle θ between its exit direction and the viewing axis: the direction the light came from, as the viewer sees it.

Lens port: θ < 2.5°, dark
White disk: 2.5° ≤ θ < 13.5°, white: this draws the arrows and the hearts
Red reflector: 13.5° ≤ θ ≤ 90°, red
Nothing: light heading away from the viewer, or still inside after 24 bounces, dark

The viewer is a direction-to-colour device, so the trace has no Fresnel loss, no partial reflection and no dispersion: one refractive index per gemstone, the simulator’s. The hearts are the same stone turned half a turn about a horizontal axis. While a slider moves, the views are traced with one ray a pixel; at rest, with four or nine.

Proportions and symmetry faults

Every change moves facet planes only, so the stone stays the convex solid its planes enclose.

Pavilion and crown angle tilt every sloping pavilion facet, or every crown facet but the table, by the change from the cut’s own angle. Each turns about the horizontal line where it crosses the girdle’s bottom (pavilion) or top (crown) on its own azimuth.
Table size sets the table plane at the height where the table measures that share of the diameter, corner to corner, found by bisection. It is solved again after the crown moves, so the table holds its size and the crown height changes.
Crown twist turns every crown facet, the table included, by τ about the axis.
Main angles tilt pavilion main k by a w_k, and main spacing turns it about the axis by s w′_k, with w = (0.6, −1, 0.35, 0.85, −0.45, −0.75, 1, −0.2) and w′ = (−0.5, 0.8, −1, 0.25, 0.9, −0.3, 0.6, −0.7). The patterns are fixed and peak at 1, so the slider gives the largest error.
Culet off-centre shears the pavilion sideways, which leaves the girdle in place and moves the culet by c, as a share of the diameter:

x′ = x + \frac{c}{y_{girdle} - y_{culet}} (y_{girdle} - y)

Symmetry

The figure asks whether the pattern’s 16 halves match. Points are laid on a polar grid over one half-sector, from a mirror line of the cut to 22.5° beside it: 32 angles by 96 radii, out to 0.49 of the diameter, inside every girdle. Each point is traced in all 16 of its images, eight turns of 45°, each with and without the mirror. It counts when all 16 show the same colour (white, red or dark; the port and “nothing” look alike). With r_j the point’s radius, which weights it by the area it stands for:

Symmetry = 100 \frac{\sum_{j} r_{j} [all 16 agree]}{\sum_{j} r_{j}}

It is computed in double precision and shown as a whole per cent; 100% means every point agrees, and nothing less rounds up to it. Proportions alone never lower it, because they change every half alike. They change what the pattern looks like, not whether its halves match. In the T73 in diamond, main angles off by up to 0.1° give 90% face-up and 85% face-down; a crown twist of 1° gives 92% and 88%.

The Hearts & Arrows Scope shows a round brilliant face-up (the arrows) and face-down (the hearts) through a Hearts & Arrows viewer, and gives each view one figure, its symmetry. It uses the same viewer as the Hearts & Arrows images on the gemstone pages.