{
  "capability_id": "math.geometry.coordinate",
  "tool_id": "coordinate-geometry",
  "engine": "coordinate-geometry",
  "calculation_version": "1.5.0",
  "generated_at": "2026-09-21T16:00:36.128Z",
  "engine_build": "2026-09-21",
  "test_count": 21,
  "passed": 21,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 8,
      "passed": 8
    },
    "boundary": {
      "total": 3,
      "passed": 3
    },
    "property": {
      "total": 10,
      "passed": 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": [
    "Not the Pythagorean-only calculator — distance of (0,0) to (3,4) is the same 5 as 3-4-5, but this page is coordinates",
    "Vertical line → slope null, vertical true",
    "Share URLs use slope as output, not query key m",
    "Two-point line is mode=line; coincident points → INVALID_INPUT",
    "Point-line distance is mode=point_line; requires x0, y0; coincident line points → INVALID_INPUT",
    "Line intersection is mode=intersect; requires x3, y3, x4, y4; parallel distinct → empty; coincident → coincident; degenerate points → INVALID_INPUT",
    "Angle between two lines is mode=angle; requires x3, y3, x4, y4; principal angle [0°, 90°]; parallel → 0°; coincident → 0°; degenerate points → INVALID_INPUT"
  ],
  "boundary_conditions": [
    "missing coordinates → MISSING_REQUIRED_INPUT",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
  ],
  "methods": [],
  "fingerprint": "sha256:bf0bdeec4b0f52e0ff0a18c02f725e129958d490367c3acb50736782c684ac2e",
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      "ok": true,
      "kind": "golden",
      "inputs": {
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        "x1": 0,
        "y1": 0,
        "x2": 3,
        "y2": 4
      },
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        "distance": 5
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      "expectWarning": null,
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        "x1": 0,
        "y1": 0,
        "x2": 3,
        "y2": 4,
        "dx": 3,
        "dy": 4,
        "distance": 5,
        "formula": "d = √((x₂−x₁)²+(y₂−y₁)²)"
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      "tol": null
    },
    {
      "id": "golden-midpoint",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "midpoint",
        "x1": 0,
        "y1": 0,
        "x2": 4,
        "y2": 6
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      "expect": {
        "mx": 2,
        "my": 3
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      "expectWarning": null,
      "actual": {
        "mode": "midpoint",
        "x1": 0,
        "y1": 0,
        "x2": 4,
        "y2": 6,
        "mx": 2,
        "my": 3,
        "formula": "M = ((x₁+x₂)/2, (y₁+y₂)/2)"
      },
      "tol": null
    },
    {
      "id": "golden-slope",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "slope",
        "x1": 0,
        "y1": 1,
        "x2": 2,
        "y2": 5
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      "expect": {
        "slope": 2,
        "intercept": 1
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      "expectWarning": null,
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        "x1": 0,
        "y1": 1,
        "x2": 2,
        "y2": 5,
        "dx": 2,
        "dy": 4,
        "slope": 2,
        "vertical": false,
        "intercept": 1,
        "formula": "slope = (y₂−y₁)/(x₂−x₁)"
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      "tol": null
    },
    {
      "id": "golden-vertical",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "slope",
        "x1": 2,
        "y1": 0,
        "x2": 2,
        "y2": 5
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      "expect": {
        "vertical": true,
        "slope": null
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      "expectWarning": null,
      "actual": {
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        "x1": 2,
        "y1": 0,
        "x2": 2,
        "y2": 5,
        "dx": 0,
        "dy": 5,
        "slope": null,
        "vertical": true,
        "intercept": null,
        "formula": "x₂ = x₁ → vertical (undefined slope)"
      },
      "tol": null
    },
    {
      "id": "golden-angle-45",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 2,
        "x3": 0,
        "y3": 0,
        "x4": 2,
        "y4": 0
      },
      "expect": {
        "angle_deg": 45,
        "status": "oblique",
        "supplement_deg": 135
      },
      "expectWarning": null,
      "actual": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 2,
        "x3": 0,
        "y3": 0,
        "x4": 2,
        "y4": 0,
        "A1": 1,
        "B1": -1,
        "C1": 0,
        "A2": 0,
        "B2": 1,
        "C2": 0,
        "standard1": "x − y = 0",
        "standard2": "y = 0",
        "status": "oblique",
        "angle_deg": 45,
        "angle_rad": 0.785398163397,
        "supplement_deg": 135,
        "formula": "θ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))"
      },
      "tol": null
    },
    {
      "id": "golden-angle-parallel",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 0,
        "y3": 1,
        "x4": 2,
        "y4": 1
      },
      "expect": {
        "angle_deg": 0,
        "status": "parallel",
        "supplement_deg": 180
      },
      "expectWarning": null,
      "actual": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 0,
        "y3": 1,
        "x4": 2,
        "y4": 1,
        "A1": 0,
        "B1": 1,
        "C1": 0,
        "A2": 0,
        "B2": 1,
        "C2": -1,
        "standard1": "y = 0",
        "standard2": "y − 1 = 0",
        "status": "parallel",
        "angle_deg": 0,
        "angle_rad": 0,
        "supplement_deg": 180,
        "formula": "θ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))"
      },
      "tol": null
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    {
      "id": "golden-angle-coincident",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 1,
        "y3": 0,
        "x4": 3,
        "y4": 0
      },
      "expect": {
        "angle_deg": 0,
        "status": "coincident",
        "supplement_deg": 180
      },
      "expectWarning": null,
      "actual": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 1,
        "y3": 0,
        "x4": 3,
        "y4": 0,
        "A1": 0,
        "B1": 1,
        "C1": 0,
        "A2": 0,
        "B2": 1,
        "C2": 0,
        "standard1": "y = 0",
        "standard2": "y = 0",
        "status": "coincident",
        "angle_deg": 0,
        "angle_rad": 0,
        "supplement_deg": 180,
        "formula": "θ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))"
      },
      "tol": null
    },
    {
      "id": "golden-angle-perpendicular",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 0,
        "y3": 0,
        "x4": 0,
        "y4": 2
      },
      "expect": {
        "angle_deg": 90,
        "status": "perpendicular",
        "supplement_deg": 90
      },
      "expectWarning": null,
      "actual": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 0,
        "y3": 0,
        "x4": 0,
        "y4": 2,
        "A1": 0,
        "B1": 1,
        "C1": 0,
        "A2": 1,
        "B2": 0,
        "C2": 0,
        "standard1": "y = 0",
        "standard2": "x = 0",
        "status": "perpendicular",
        "angle_deg": 90,
        "angle_rad": 1.57079632679,
        "supplement_deg": 90,
        "formula": "θ = arccos(|A₁A₂ + B₁B₂| / (√(A₁²+B₁²) √(A₂²+B₂²)))"
      },
      "tol": null
    },
    {
      "id": "boundary-coord-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "pythagorean",
        "x1": 0,
        "y1": 0,
        "x2": 1,
        "y2": 0
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be distance, midpoint, slope, line, point_line, intersect, or angle. Not Pythagorean, not CAS."
    },
    {
      "id": "boundary-angle-degenerate-line1",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "angle",
        "x1": 1,
        "y1": 1,
        "x2": 1,
        "y2": 1,
        "x3": 0,
        "y3": 0,
        "x4": 2,
        "y4": 0
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Two distinct points are required for a unique first line."
    },
    {
      "id": "boundary-angle-degenerate-line2",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "angle",
        "x1": 0,
        "y1": 0,
        "x2": 2,
        "y2": 0,
        "x3": 3,
        "y3": 1,
        "x4": 3,
        "y4": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "Two distinct points are required for a unique second line."
    },
    {
      "id": "property-coord-345",
      "ok": true,
      "kind": "property",
      "detail": "3-4-5 distance",
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      "actual": 5
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    {
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      "kind": "property",
      "detail": "midpoint",
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      "actual": 2
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      "detail": "slope 2",
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      "kind": "property",
      "detail": "shift 3-4-5",
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    },
    {
      "id": "property-angle-endpoint-reversal",
      "ok": true,
      "kind": "property",
      "detail": "endpoint reversal",
      "expect": 45,
      "actual": 45
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    {
      "id": "property-angle-line-order",
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      "kind": "property",
      "detail": "line order",
      "expect": 45,
      "actual": 45
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    {
      "id": "property-angle-translation",
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      "kind": "property",
      "detail": "translation",
      "expect": 45,
      "actual": 45
    },
    {
      "id": "property-angle-scale",
      "ok": true,
      "kind": "property",
      "detail": "scale",
      "expect": 45,
      "actual": 45
    },
    {
      "id": "property-angle-near-parallel",
      "ok": true,
      "kind": "property",
      "detail": "near-parallel not 0/NaN",
      "expect": true,
      "actual": 5.72957795131e-7
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 21,
      "total": 21,
      "golden": 8,
      "boundary": 3,
      "property": 10
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