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See Review policy for supplemental domain assurance." }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/math/conic-section", "api": "https://www.calculatorx.com/api/v1/calc/conic-section", "schema_input": "https://www.calculatorx.com/schemas/math.geometry.conic.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.geometry.conic.output.json", "assumptions": [ "Euclidean plane", "Axis-aligned standard forms", "Not rotated (no xy term)", "Not translated", "Not the circle-only page", "Not coordinate geometry", "Not CAS" ], "limitations": [ "Axis-aligned standard forms only; not rotated (xy term), not translated", "Not the circle-only calculator — a=b is a valid ellipse with e=0; /calc/math/circle stays the circle page", "Not coordinate geometry (two-point line / intersection / angle)", "Ellipse perimeter is Ramanujan I, not a complete elliptic integral", "a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE", "parabola p = 0 → INVALID_INPUT", "unknown mode (including cas, circle, pythagorean) → INVALID_MODE", "Decimal inputs normalize to binary64; circle uses exact a===b after normalization. Near-circle c/e are sensitive to input rounding; noncircle eccentricity may round to1.", "Nonzero input/output scalars must be finite normal binary64; NUMERIC_RANGE requires rescaling. No absolute-zero clamp. Explicit empty/null inputs reject; inactive supplied numeric fields are validated.", "Ellipse perimeter is Ramanujan I approximation, not exact arclength. Relative2e-12 verification compares the stated approximation formula, not its model error.", "Local browser calculation; shared URLs are received by the server; REST/MCP send their inputs." ], "boundary_conditions": [ "a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE", "parabola p = 0 → INVALID_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": "S. Ramanujan — Modular Equations and Approximations to π (1914)", "supports": "Ellipse perimeter approximation I: π[3(a+b)−√((3a+b)(a+3b))]; not an elliptic integral", "kind": "reference", "reference_scope": "model", "oracle_class": "O1" }, { "title": "NIST DLMF — 19.30 Lengths of Plane Curves", "section": "19.30(i), especially equation 19.30.5", "url": "https://dlmf.nist.gov/19.30", "supports": "Exact ellipse perimeter L(a,b) = 4a E(k), with k² = 1−b²/a² and a ≥ b. A displayed elementary perimeter approximation is distinct from this exact elliptic-integral definition; the reference does not establish the approximation’s error or validate CalculatorX.", "kind": "reference", "publisher": "National Institute of Standards and Technology" }, { "title": "Gilbert Strang, Calculus — Chapter 3: Applications of the Derivative (MIT OpenCourseWare)", "section": "§3.5 Parabolas, Ellipses, and Hyperbolas", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch03.pdf", "supports": "Standard equations and focal geometry of axis-aligned conic sections, including translated equations and the ellipse’s semiaxes. Ellipse area and perimeter are supported by separate integral references. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.", "kind": "reference", "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare" }, { "title": "Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare)", "section": "§15.3 Green’s Theorem", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch15.pdf", "supports": "The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. 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