Conic Section Identifier & Plotter mechanism

Model scope

NEEDS_REVIEW because the specification does not define coefficient/coordinate units, degenerate-conic behavior, numerical tolerances, plot conventions, or the complete rotated-conic algorithm. The supplied discriminant and completion-of-square description is insufficient for the claimed general equation, especially when B≠0. Test vector 3 intentionally reports the corrected hand calculation in expected_outputs and flags the erroneous 5-value in working; the vector should be repaired before execution.

• B²-4AC classifies nondegenerate real conics as ellipse (<0), parabola (=0), or hyperbola (>0), but it does not by itself identify circles, degenerate conics, empty real loci, point conics, or pairs of lines.
• For B≠0, completion of squares requires a coordinate-axis rotation and removal of the xy term; the stated formula set does not provide that rotation, its angle, or the transformed coefficients.
• The conic type must be checked against the full quadratic and augmented matrices to detect degeneracy before reporting foci, vertices, eccentricity, directrices, or a plot.
• Foci, vertices, eccentricity, and directrices require separate formulas for ellipse, hyperbola, and parabola; those formulas are not specified.
• For a circle, eccentricity is 0 and the usual focus/directrix definition is degenerate; the output convention must explicitly define whether the coincident center is reported as a limiting focus and whether directrix is reported as undefined.

Units and conventions

• x and y are assumed to have the same coordinate unit.
• A, B, C, D, E, and F must have compatible units so every term in Ax²+Bxy+Cy²+Dx+Ey+F has the same unit; if x and y are dimensionless, all coefficients may be treated as dimensionless.
• Reported lengths such as radius, semimajor axis, focal distance, and directrix distance use the coordinate unit.
• Plotting requires a declared axis scale and aspect ratio; unequal visual scaling can distort the apparent conic.
• The sign of B²-4AC is invariant under multiplying the entire equation by any nonzero scalar, but numerical tolerance for values near zero must be specified.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "A": 1,
    "B": 0,
    "C": 1,
    "D": -4,
    "E": 6,
    "F": -3
  },
  "expected_outputs": {
    "conic_type": "circle",
    "standard_form": "(x-2)^2+(y+3)^2=16",
    "center": [
      "2",
      "-3"
    ],
    "radius": "4",
    "eccentricity": "0",
    "foci": [
      [
        "2",
        "-3"
      ]
    ],
    "vertices": [
      [
        "6",
        "-3"
      ],
      [
        "-2",
        "-3"
      ],
      [
        "2",
        "1"
      ],
      [
        "2",
        "-7"
      ]
    ],
    "directrices": "undefined for a circle under the standard focus/directrix definition",
    "plot_locus": "circle centered at (2,-3) with radius 4"
  },
  "reference_data": {
    "classification_discriminant": "B²-4AC",
    "standard_circle_form": "(x-h)²+(y-k)²=r²"
  },
  "working": "Discriminant: B²-4AC = 0²-4(1)(1) = -4 < 0, so the nondegenerate conic is ellipse-type; equal positive quadratic coefficients identify a circle. Complete squares: x²-4x=(x-2)²-4 and y²+6y=(y+3)²-9. Therefore (x-2)²-4+(y+3)²-9-3=0, so (x-2)²+(y+3)²=16. Center=(2,-3), radius=sqrt(16)=4, eccentricity=0. The four axis vertices are (2±4,-3) and (2,-3±4)."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The all-zero coefficient input produces discriminant zero but is not a conic and causes zero-denominator or undefined-parameter calculations. The instrument must reject it instead of classifying it as a parabola. Rotated-conic outputs also require an explicit rotation convention because equivalent angle choices can swap axis labels and signs.

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