Parametric Curve Plotter mechanism

Model scope

The core parametric-curve and arc-length formulas are standard. Review is required because parameter units and the meaning of velocity are unspecified, numerical quadrature tolerances and domain handling are absent, and the honest_label field is empty despite the instrument potentially presenting parameter derivatives as physical velocities.

• The arc-length formula is correct for differentiable x(t) and y(t), provided both coordinates use the same length unit and the parameter interval is properly oriented.
• The specification calls x'(t), y'(t) a velocity vector, but these are physical velocities only when t is time. Otherwise they are parameter derivatives or tangent vectors.
• Numeric quadrature requires a stated tolerance, convergence policy, and handling for discontinuities, cusps, singular derivatives, or expressions outside their domains.

Units and conventions

• x(t) and y(t) must have the same length unit.
• Arc length is returned in that coordinate length unit.
• If t is measured in seconds, x'(t) and y'(t) are velocities in length-per-second and speed is length-per-second.
• If t is dimensionless or an angular parameter, x'(t), y'(t) are derivatives with respect to the parameter, not physical velocities.
• The circle test uses t in radians; radians are treated as dimensionless for differentiation.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "x(t)": "2t m",
    "y(t)": "3t m",
    "t_range": "[0,3] s"
  },
  "expected_outputs": {
    "arc_length_exact": "3√13 m",
    "arc_length_numeric_quadrature": "10.81665382639197 m",
    "speed": "√13 m/s",
    "tangent_or_velocity_at_t_1": "(2,3) m/s",
    "curve_endpoints": {
      "at_t_0": "(0,0) m",
      "at_t_3": "(6,9) m"
    },
    "integral_check": {
      "antiderivative_difference": "3√13 m",
      "numeric_quadrature": "10.81665382639197 m"
    }
  },
  "reference_data": {
    "method": "Exact integral and numerical quadrature",
    "derivatives": {
      "x'(t)": "2 m/s",
      "y'(t)": "3 m/s"
    }
  },
  "working": "1. x'(t)=2 and y'(t)=3. 2. Speed=√(2²+3²)=√(4+9)=√13 m/s. 3. Arc length=∫_0^3 √13 dt=√13·(3−0)=3√13 m. 4. Numerically, √13=3.605551275463989, so 3√13=10.81665382639197 m. 5. At t=1, the velocity/tangent vector is (x'(1),y'(1))=(2,3) m/s."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification exposes a singular-input failure at t=1, where the curve and arc-length integral are undefined, and it does not specify trigonometric angle units, allowing a 90-versus-pi/2 arc-length discrepancy.

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