Polar Curve Plotter

Inspectable records

Ledger state

The current ledger marks this instrument verified with errata that must be applied at binding time.

Area

Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

Tranche

tranche-8

Evidence ID

t8-027

Ledger rows

1

No ledger note is attached.

Review findings

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.

Formula and scope findings

• 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.

Unit assumptions

• 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.

Boundary hardening

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.

Errata and corrections

Tranche errata

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "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."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

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

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