{
  "capability_id": "math.geometry.coordinate",
  "tool_id": "coordinate-geometry",
  "engine": "coordinate-geometry",
  "calculation_version": "1.7.0",
  "generated_at": "2026-10-29T08:00:32.011Z",
  "engine_build": "2026-10-29",
  "test_count": 21,
  "passed": 21,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "passed": 8,
      "total": 8
    },
    "boundary": {
      "passed": 3,
      "total": 3
    },
    "property": {
      "passed": 10,
      "total": 10
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/coordinate-geometry",
  "api": "https://www.calculatorx.com/api/v1/calc/coordinate-geometry",
  "schema_input": "https://www.calculatorx.com/schemas/math.geometry.coordinate.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.geometry.coordinate.output.json",
  "assumptions": [
    "Euclidean plane",
    "Not 3-D / spherical"
  ],
  "limitations": [
    "Cartesian R²; all coordinates use one consistent length unit; angle outputs include radians and degrees.",
    "Strict decimal inputs; matching x0/px and y0/py aliases; unknown fields reject. Omitted mode defaults distance. Discovery entry URLs lock their documented operation.",
    "Exact dyadic predicates classify the supplied normalized binary64 values; decimal-to-binary rounding still affects nearly parallel lines and input-sensitive intersections. No absolute geometric epsilon or near-integer snapping.",
    "Nonzero inputs and published numeric fields must be finite normal binary64; NUMERIC_RANGE requires translation/rescaling. Mathematical zero is retained, never a clamp.",
    "Coincident defining points reject INVALID_INPUT for slope and all line operations; distance zero and midpoint remain valid. Vertical slope/intercept are null.",
    "Line coefficients use exact integer gcd only for exact safe integers; otherwise a common power-of-two normalization and canonical sign. Standard equations are floating presentations, not exact symbolic identities.",
    "Intersections of infinite lines (not segments): parallel distinct => empty, coincident => coincident, each with x/y null; other intersections return one point.",
    "Undirected principal line angle is atan2(abs cross, abs dot) in [0,90] degrees. Exact predicates determine status; a non-perpendicular angle can round to90 degrees.",
    "Full binary64 output without near-zero clamping. Typical well-conditioned scalar error budget relative2e-12; ill-conditioned inputs require their own sensitivity analysis. No positional bound is claimed for perturbed inputs."
  ],
  "boundary_conditions": [
    "missing coordinates → MISSING_REQUIRED_INPUT",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    {
      "title": "ISO 80000-2:2019 Quantities and units — Mathematics",
      "url": "https://www.iso.org/standard/64973.html",
      "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.",
      "kind": "standard",
      "publisher": "International Organization for Standardization",
      "reference_scope": "all"
    },
    {
      "title": "Margaret Stevens — The Maths Workbook (University of Oxford)",
      "section": "Chapter 2, §3: Linear Functions; Chapter 1, §4: Simultaneous Equations",
      "url": "https://www.some.ox.ac.uk/wp-content/uploads/2024/08/Maths-Workbook.pdf",
      "supports": "Linear-function slopes and point-slope equations, and simultaneous linear equations in two variables. Line intersection is the corresponding system solution; vertical lines use an implicit rather than finite-slope representation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "University of Oxford, Department of Economics; hosted by Somerville College"
    },
    {
      "title": "Gilbert Strang, Calculus — Chapter 11: Vectors and Matrices (MIT OpenCourseWare)",
      "section": "§§11.1–11.3 Vectors, Dot Products, Projections and Cross Products",
      "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch11.pdf",
      "supports": "Vector components and magnitudes, dot product and angle/projection relations, and cross product/determinant orientation and area. Projecting a point displacement onto the unit normal to Ax+By+C=0 gives distance |Ax₀+By₀+C|/sqrt(A²+B²); this planar expression is derived from the projection formula. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:ab91785ab2c5997d39122af2a6c922c2736434a2c3fa0d115e38e61f8c875b4a",
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 21,
      "total": 21,
      "golden": 8,
      "boundary": 3,
      "property": 10
    },
    "contract_checks": {
      "passed": 6,
      "total": 6
    }
  },
  "evidence_core_generated_at": "2026-10-29T07:58:15.177Z",
  "rendered_at": "2026-10-29T08:00:32.011Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.7.0",
    "artifacts": {
      "engine": "1.7.0",
      "page": "1.7.0",
      "capability": "1.7.0",
      "evidence": "1.7.0",
      "input_schema": "1.7.0",
      "output_schema": "1.7.0",
      "mcp_contract": "1.7.0"
    },
    "digests": {
      "input_schema_sha256": "0a4a89fc1b3b30c09ead464ce86fe1f5bab5fdc370e47940adc385742bd440bb",
      "output_schema_sha256": "b6a70c4cdb3ac0446de524a59d5cd023494869cbdaa5fb81dac31e3f5c68ca3b",
      "evidence_sha256": "96ba59375f1016d62662e8eee1c34b62248a81bc9bef6790dbd1c50cd0094d0a",
      "evidence_core_sha256": "96ba59375f1016d62662e8eee1c34b62248a81bc9bef6790dbd1c50cd0094d0a",
      "capability_sha256": "c8ae8087b3c7e7a20dacaf7ddb540891bf10a9f99b0885d4bdb70e876b050cdd",
      "engine_sha256": "c6ebd8436f08b413e1ac6317b2d48550f2e6639a90dff254244e3aa562158b8d",
      "mcp_contract_sha256": "a7e9b259b340e28a7f3f67b87f47d3ec9c44f27cd2956b43765dcb3344452a32"
    },
    "contract_checks": {
      "total": 6,
      "passed": 6,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.7.0"
        },
        {
          "id": "schema-output-required-declared",
          "status": "pass"
        },
        {
          "id": "schema-valid-input-executes",
          "status": "pass"
        },
        {
          "id": "engine-success-output-validates-schema",
          "status": "pass"
        },
        {
          "id": "schema-rejects-incomplete-input",
          "status": "pass"
        },
        {
          "id": "rest-success-payload-validates-schema",
          "status": "pass"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  },
  "tests": [],
  "disclosure": {
    "policy": "calculatorx-public-evidence-v1",
    "full_validation": "private",
    "representation": "summary"
  }
}