Polar Curve Plotter mechanism

Model scope

The stated polar area integrand is mathematically correct under radians and a properly selected, non-overcounting tracing interval. The instrument needs an explicit angle-unit contract and a distinction between swept area and geometric enclosed area before receiving a PASS verdict.

• The polar area formula A=∫(1/2)r(θ)^2 dθ is standard for swept area over a correctly chosen interval, but it does not always equal the geometric area enclosed by a self-intersecting or multiply traced curve; the specification must define whether multiplicity is counted or overlapping regions are unioned.
• Sampling over θ is insufficiently specified: the implementation needs an adaptive or resolution-controlled sampling strategy to avoid aliasing narrow loops, cusps, rapidly oscillating curves, and discontinuities.
• The specification does not state whether θ is measured in radians or degrees. The calculus formula requires θ in radians unless an explicit degree-to-radian conversion is applied.
• The specification does not define how the enclosed-area interval is selected when the supplied θ range contains only part of a curve, multiple revolutions, or retraced portions.

Units and conventions

• All θ values in the formulas and test vectors are radians unless the input explicitly declares degrees.
• r has the same distance unit as the requested plotted coordinate system; area is therefore in squared distance units.
• For an explicitly degree-based range, the implementation must convert dθ using dθ_radians=(π/180)dθ_degrees before applying the polar area formula.
• The area results below assume the supplied interval traces the intended curve exactly once, except where the interval is explicitly identified as a partial trace.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "r_theta": "2",
    "theta_range": [
      0,
      "2π"
    ],
    "theta_units": "radians",
    "area_units": "square distance units"
  },
  "expected_outputs": {
    "plot": "Circle centered at the pole with radius 2",
    "enclosed_area_exact": "2π",
    "enclosed_area_numeric": 6.283185307179586,
    "numeric_quadrature_area": 6.283185307179586
  },
  "reference_data": {
    "identity": "A=1/2∫_0^(2π)4 dθ=2π"
  },
  "working": "A=1/2∫_0^(2π)(2)^2 dθ. First, (2)^2=4. Then A=1/2·4·(2π)=4π/2=2π=6.283185307179586."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The singular expression r(theta)=1/(theta-pi/2) produces an undefined plot value and divergent area when the requested range includes theta=pi/2. The degree-versus-radian convention probe also produces materially different numeric output unless units are enforced explicitly.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json