{
  "schema": "native-geometric-method/v1",
  "anchor": "c17-segmented-miter-layout",
  "scope": "Selected planar equal-turn segmented layout; zero-gap ideal cut-plane geometry, not full developed-shell template or service permission",
  "variables": {
    "Theta": "total axis turn, degrees",
    "m": "number of joints; pieces=m+1",
    "delta": "Theta/m axis change at each joint",
    "alpha": "Theta/(2m) cut inclination away from square end",
    "d": "actual outside diameter in same length units as R",
    "r": "d/2",
    "R": "centerline tangent-construction circle radius",
    "a": "r*tan(alpha), axial half-setback",
    "lc": "interior axial centerline length between cut planes"
  },
  "equations": {
    "delta": "Theta/m",
    "alpha": "delta/2",
    "a": "(d/2)*tan(alpha)",
    "interior_lc": "2*R*tan(alpha)",
    "interior_outer": "lc+2*a",
    "interior_inner": "lc-2*a",
    "terminal_lc": "R*tan(alpha)",
    "terminal_outer": "terminal_lc+a",
    "terminal_inner": "terminal_lc-a",
    "single_cut_long_short_difference": "d*tan(alpha)"
  },
  "angle_cases": [
    {
      "total_turn_deg": 90,
      "joints": 1,
      "pieces": 2,
      "axis_change_deg": 90.0,
      "cut_inclination_deg": 45.0,
      "radial_sectors_deg": [
        45.0,
        45.0
      ]
    },
    {
      "total_turn_deg": 90,
      "joints": 2,
      "pieces": 3,
      "axis_change_deg": 45.0,
      "cut_inclination_deg": 22.5,
      "radial_sectors_deg": [
        22.5,
        45.0,
        22.5
      ]
    },
    {
      "total_turn_deg": 90,
      "joints": 3,
      "pieces": 4,
      "axis_change_deg": 30.0,
      "cut_inclination_deg": 15.0,
      "radial_sectors_deg": [
        15.0,
        30.0,
        30.0,
        15.0
      ]
    },
    {
      "total_turn_deg": 90,
      "joints": 4,
      "pieces": 5,
      "axis_change_deg": 22.5,
      "cut_inclination_deg": 11.25,
      "radial_sectors_deg": [
        11.25,
        22.5,
        22.5,
        22.5,
        11.25
      ]
    }
  ],
  "single_joint_cases": [
    {
      "total_turn_deg": 30,
      "cut_inclination_deg": 15.0
    },
    {
      "total_turn_deg": 45,
      "cut_inclination_deg": 22.5
    },
    {
      "total_turn_deg": 60,
      "cut_inclination_deg": 30.0
    }
  ],
  "dimension_mapping": {
    "two_joint_90": {
      "A": "axial half-setback",
      "B_C": "separate projected terminal-reference intervals; not middle piece lengths",
      "D": "middle outer axial length",
      "E": "middle centerline axial length",
      "F": "middle inner axial length"
    },
    "three_joint_90": {
      "G_H_I": "configuration-specific end references",
      "J": "interior outer",
      "K": "interior centerline",
      "L": "interior inner"
    },
    "four_joint_90": {
      "M_N_P": "configuration-specific end references",
      "S": "interior outer",
      "T": "interior centerline",
      "U": "interior inner"
    }
  },
  "worked_source_row": {
    "configuration": "two-joint 90-degree layout",
    "labelled_size": 6,
    "units": "inches",
    "R": 9,
    "A": 1.375,
    "D": 10.1875,
    "E": 7.4375,
    "F": 4.6875,
    "status": "selected printed table fractions, not production dimensions",
    "actual_OD": "not printed; not inferred from nominal size",
    "checks": {
      "mean_outer_inner": 7.4375,
      "D_minus_E": 2.75,
      "E_minus_F": 2.75,
      "two_A": 2.75,
      "D_minus_F": 5.5,
      "four_A": 5.5,
      "ideal_centerline": 7.455844122715711,
      "ideal_minus_printed_E": 0.018344122715711286
    }
  },
  "companion_table_route": "Three/four-joint table does not print Size/R labels; no row-to-size/radius join made",
  "assumptions": [
    "planar turn",
    "equal axis changes",
    "straight circular pipe pieces",
    "symmetric cuts",
    "terminal faces square at first/last tangency points",
    "no additional terminal tangents included",
    "zero gap/kerf/bevel allowances in ideal geometry"
  ],
  "fabrication_interfaces": [
    "actual OD",
    "bevel/root face",
    "fit-up gap",
    "kerf and trimming",
    "inside/outside marking",
    "piece sequence",
    "rotation/reversal",
    "terminal datums"
  ],
  "not_established": [
    "pressure/service eligibility",
    "fatigue/vibration suitability",
    "qualified welding procedure",
    "all-cell numeric table certification",
    "developed circumferential cutting template"
  ]
}