{
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      "assumptions",
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      "sources",
      "tests"
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  "limitations": [
    "θ is counterclockwise from +x and is an input, not a solved principal angle",
    "Ix′ + Iy′ = Ix + Iy about the composite centroid",
    "Negative area is a hole",
    "Not I1 or I2, not a Mohr-circle drawing, not section modulus, and not beam deflection"
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  "boundary_conditions": [
    "no parts → MISSING_REQUIRED_INPUT",
    "missing θ → MISSING_REQUIRED_INPUT",
    "θ is not a finite number → INVALID_INPUT",
    "total area zero → INVALID_INPUT"
  ],
  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Moments of inertia for an area about inclined axes",
      "supports": "The centroidal moments transform with the double-angle formulas in θ",
      "kind": "reference"
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  "methods": [],
  "fingerprint": "sha256:50ebfacae9124ebd4c0080bd1b83f8ddcf352daa9e2458118252efbe1e302da4",
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      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "4,1,0,2,0,0",
        "theta_deg": 0
      },
      "expect": {
        "Ix": 4,
        "Iy": 1,
        "Ixy": 0,
        "Ix_prime": 4,
        "Iy_prime": 1,
        "Ixy_prime": 0,
        "theta_deg": 0,
        "model": "inclined_axis_moment"
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      "expectWarning": null,
      "actual": {
        "model": "inclined_axis_moment",
        "area": 2,
        "x_bar": 0,
        "y_bar": 0,
        "Ix": 4,
        "Iy": 1,
        "Ixy": 0,
        "theta_deg": 0,
        "Ix_prime": 4,
        "Iy_prime": 1,
        "Ixy_prime": 0,
        "n": 1,
        "solver": "principal moment",
        "convention": "θ is counterclockwise from +x. Ix′ = (Ix+Iy)/2 + (Ix−Iy)/2 cos 2θ − Ixy sin 2θ. Iy′ flips the middle sign. Ixy′ = (Ix−Iy)/2 sin 2θ + Ixy cos 2θ. Moments are about the composite centroid."
      },
      "tol": 1e-12
    },
    {
      "id": "golden-45",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "4,1,0,2,0,0",
        "theta_deg": 45
      },
      "expect": {
        "Ix_prime": 2.5,
        "Iy_prime": 2.5,
        "Ixy_prime": 1.5
      },
      "expectWarning": null,
      "actual": {
        "model": "inclined_axis_moment",
        "area": 2,
        "x_bar": 0,
        "y_bar": 0,
        "Ix": 4,
        "Iy": 1,
        "Ixy": 0,
        "theta_deg": 45,
        "Ix_prime": 2.5,
        "Iy_prime": 2.5,
        "Ixy_prime": 1.5,
        "n": 1,
        "solver": "principal moment",
        "convention": "θ is counterclockwise from +x. Ix′ = (Ix+Iy)/2 + (Ix−Iy)/2 cos 2θ − Ixy sin 2θ. Iy′ flips the middle sign. Ixy′ = (Ix−Iy)/2 sin 2θ + Ixy cos 2θ. Moments are about the composite centroid."
      },
      "tol": 1e-9
    },
    {
      "id": "golden-principal-angle",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "parts": "0,0,0,1,0,0; 0,0,0,1,2,2",
        "theta_deg": -45
      },
      "expect": {
        "Ix": 2,
        "Iy": 2,
        "Ixy": 2,
        "Ix_prime": 4,
        "Iy_prime": 0,
        "Ixy_prime": 0,
        "area": 2
      },
      "expectWarning": null,
      "actual": {
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        "area": 2,
        "x_bar": 1,
        "y_bar": 1,
        "Ix": 2,
        "Iy": 2,
        "Ixy": 2,
        "theta_deg": -45,
        "Ix_prime": 4,
        "Iy_prime": 0,
        "Ixy_prime": 1.2246467991473532e-16,
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        "solver": "principal moment",
        "convention": "θ is counterclockwise from +x. Ix′ = (Ix+Iy)/2 + (Ix−Iy)/2 cos 2θ − Ixy sin 2θ. Iy′ flips the middle sign. Ixy′ = (Ix−Iy)/2 sin 2θ + Ixy cos 2θ. Moments are about the composite centroid."
      },
      "tol": 1e-9
    },
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      "id": "boundary-empty",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "parts": "4,1,0,2,0,0"
      },
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "theta_deg is required."
    },
    {
      "id": "boundary-parts",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "theta_deg": 0
      },
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      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter at least one part as Ix,Iy,Ixy,A,x,y."
    },
    {
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      "ok": true,
      "kind": "boundary",
      "inputs": {
        "parts": "4,1,0,2,0,0",
        "theta_deg": "no"
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "theta_deg must be a finite number."
    }
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