Cutting & Tiering

This chapter covers the core of Meetpoint Faceting: placing facets precisely where they meet.

The Sweep Operator ->

In faceting, we rarely place planes at arbitrary coordinates. We "sweep" a plane from infinity until it hits a specific target.

Plane = Normal -> Target

1. Cutting to Centerpoint (CP)

The most common starting move. All pavilion mains usually meet at the culet (tip).

// Define symmetry indices (e.g., 96-index gear, 8-fold)
indices = [96, 12, 24, 36, 48, 60, 72, 84]

// Cut Mains at 42 degrees to Centerpoint
p1 := indices |> (i) => Normal(42, i) -> Point(0,0,0)

2. Cutting to Girdle Width

Establishing the size of the stone.

// Cut Girdle facets at 90 degrees, distance 5mm
girdle := indices |> (i) => Normal(90, i) -> 5

Note: The -> operator automatically detects this as a "Set stone size" operation and adds it to the facet notes.

3. Cutting to Meet (Meetpoint Faceting)

The heart of the process. We cut a new tier until it touches a vertex formed by previous tiers.

Example: Cutting Breaks

Assume we have p1 (Mains) and girdle tiers. We want to cut "Break" facets that meet where the Main and Girdle intersect.

1. Find the Meetpoint. Use the * operator to intersect planes, or *> to find a vertex on the stone.

   // Find intersection of Main(96) and Girdle(96)
   // We access the specific planes from our arrays
   mp = p1[0] * girdle[0] * p1[1]
   // This defines a point where two mains and a girdle meet
   

   Alternatively, use Contact (*>):

   // "Look" in the direction of index 6 (between 96 and 12)
   // The stone's current corner is there.
   target_v = Normal(0, 6) *> Stone
   

2. Cut to the Meetpoint.

   // Breaks at 45 degrees
   break_indices = [6, 18, 30, 42, 54, 66, 78, 90]
   
   p2 := break_indices |> (i) => Normal(45, i) -> target_v
   

   Note: -> automatically detects the meetpoint source and adds "Meet [Plane Names]" to the notes.

Tiering

A "Tier" is a group of facets (usually with same angle) cut in a symmetry pattern.

// Helper for 8-fold symmetry
sym8 = (start_idx) => (0..7) |> (k) => (start_idx + k*12) % 96

// Tier 1: Pavilion Mains
t1 := sym8(96) |> (i) => Normal(42, i) -> Point(0,0,0)

// Tier 2: Girdle
t2 := sym8(96) |> (i) => Normal(90, i) -> 5

// Tier 3: Breaks (meeting Mains & Girdle)
// Calculate one meetpoint
ref_mp = t1[0] * t2[0]
// Cut tier
t3 := sym8(6) |> (i) => Normal(45, i) -> ref_mp

Auto-Noting

The -> operator is smart. It analyzes the Target expression to generate cutting instructions.

• -> Number: "Set stone size" or "Set girdle width" (if angle is 90).
• -> Point(0,0,0): "Cut to centerpoint".
• -> (P1 * P2): "Meet P1, P2".

This makes your code self-documenting for the final diagram generation.