{ "openapi": "3.1.1", "info": { "title": "Equation Art Studio API", "version": "1.0.0", "summary": "Turn math into art: parametric, polar, Lissajous, rose and spirograph curves.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://equationartstudio.usefulweb.site" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Turn math into art: parametric, polar, Lissajous, rose and spirograph curves.", "description": "Type x(t) and y(t), or r(θ), or start from a preset (Lissajous, rose, spirograph, butterfly, heart), add rotational symmetry and a colour palette, and get a crisp SVG artwork plus a PNG download. Expressions are parsed safely (no code execution) and support + − × ÷ ^, sin, cos, tan, sqrt, abs, exp, log, pi and e.\n\nMethodology:\n- Expressions are tokenised and parsed by a small recursive-descent parser into a syntax tree, which is compiled into plain arithmetic closures — no eval or Function, and only the listed functions, constants and variables are accepted.\n- Polar curves are converted with x = r(θ)·cos θ, y = r(θ)·sin θ. Presets: Lissajous x = sin(a·t + d), y = sin(b·t); rose r = cos(a/b·θ); spirograph (hypotrochoid) x = (a−b)cos t + c·cos((a−b)/b·t), y = (a−b)sin t − c·sin((a−b)/b·t); Temple Fay’s butterfly; the classic 16 sin³t heart. Roses and spirographs with whole-number a, b run exactly one full period.\n- t is sampled evenly from start to end. Undefined samples (division by zero, sqrt of a negative number) leave gaps, and very large jumps (asymptotes) are not connected.\n- The drawing is fitted into a 600 × 600 square with equal x/y scale, optionally rotated into k symmetric copies, and coloured along t with a gradient from the palette. Points closer than 0.3 px are dropped to keep the SVG small.\n- The SVG uses only plain shapes (no scripts or styles). The PNG is rendered from the same SVG with ImageMagick when it is installed; both downloads stay available for about 2 hours.", "tags": [ "fun" ], "x-mcp-tool": "equation_art_studio", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/EquationArtStudioInput" }, "examples": { "example1": { "summary": "Spirograph, sunset palette", "value": { "curve": "spirograph", "a": 7, "b": 4, "c": 3, "samples": 2000, "symmetry": 1, "palette": "sunset", "strokeWidth": 2 } }, "example2": { "summary": "Rose with 3-fold symmetry", "value": { "curve": "rose", "a": 7, "b": 3, "samples": 1500, "symmetry": 3, "palette": "ocean", "strokeWidth": 1.5, "png": false } }, "example3": { "summary": "Own parametric curve", "value": { "curve": "parametric", "xExpr": "cos(t) + cos(6t)/2 + sin(14t)/3", "yExpr": "sin(t) + sin(6t)/2 + cos(14t)/3", "tMin": "0", "tMax": "2*pi", "samples": 2500, "symmetry": 1, "palette": "neon", "strokeWidth": 2, "png": false } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/EquationArtStudioResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "EquationArtStudioInput": { "type": "object", "additionalProperties": false, "properties": { "curve": { "title": "Curve", "type": "string", "enum": [ "parametric", "polar", "lissajous", "rose", "spirograph", "butterfly", "heart" ], "default": "spirograph" }, "xExpr": { "title": "x(t)", "description": "Used for “my own parametric curve”. Variables: t, a, b, c, d. Implicit multiplication works: 2t, 3sin(t).", "type": "string", "maxLength": 400, "default": "cos(t) + cos(6t)/2 + sin(14t)/3" }, "yExpr": { "title": "y(t)", "type": "string", "maxLength": 400, "default": "sin(t) + sin(6t)/2 + cos(14t)/3" }, "rExpr": { "title": "r(θ)", "description": "Used for “my own polar curve”. Write θ as theta or t.", "type": "string", "maxLength": 400, "default": "1 + 0.5*cos(5theta)" }, "tMin": { "title": "t from", "description": "A number or constant expression such as 0 or -pi. Empty = preset default (0).", "type": "string", "maxLength": 60 }, "tMax": { "title": "t to", "description": "E.g. 2*pi. Empty = preset default (2π, or a full period for roses and spirographs).", "type": "string", "maxLength": 60 }, "a": { "title": "a", "description": "Parameter for presets and your expressions. Empty = preset default.", "type": "number", "minimum": -1000, "maximum": 1000 }, "b": { "title": "b", "type": "number", "minimum": -1000, "maximum": 1000 }, "c": { "title": "c", "type": "number", "minimum": -1000, "maximum": 1000 }, "d": { "title": "d", "type": "number", "minimum": -1000, "maximum": 1000 }, "samples": { "title": "Samples", "description": "More samples = smoother curves. Samples × symmetry copies is limited to 24,000.", "type": "integer", "minimum": 50, "maximum": 10000, "default": 2000 }, "symmetry": { "title": "Rotational symmetry (copies)", "type": "integer", "minimum": 1, "maximum": 12, "default": 1 }, "palette": { "title": "Palette", "type": "string", "enum": [ "sunset", "ocean", "forest", "neon", "rainbow", "mono" ], "default": "sunset" }, "strokeWidth": { "title": "Line width", "type": "number", "minimum": 0.25, "maximum": 12, "default": 2, "x-unit": "px" }, "png": { "title": "Also render a PNG download", "type": "boolean", "default": true } }, "required": [] }, "EquationArtStudioResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "curve", "curveName", "kind", "x", "y", "r", "tMin", "tMax", "params", "samples", "symmetry", "palette", "invalidSamples", "drawnPoints", "closed", "arcLength", "bbox", "svgBytes", "svgArtifact", "pngArtifact" ], "properties": { "curve": { "type": "string" }, "curveName": { "type": "string" }, "kind": { "type": "string", "enum": [ "parametric", "polar" ] }, "x": { "type": [ "string", "null" ] }, "y": { "type": [ "string", "null" ] }, "r": { "type": [ "string", "null" ] }, "tMin": { "type": "number" }, "tMax": { "type": "number" }, "params": { "type": "object", "additionalProperties": false, "required": [ "a", "b", "c", "d" ], "properties": { "a": { "type": "number" }, "b": { "type": "number" }, "c": { "type": "number" }, "d": { "type": "number" } } }, "samples": { "type": "integer" }, "symmetry": { "type": "integer" }, "palette": { "type": "string" }, "invalidSamples": { "type": "integer" }, "drawnPoints": { "type": "integer" }, "closed": { "type": "boolean" }, "arcLength": { "type": "number", "description": "Length of one copy of the curve in curve units." }, "bbox": { "type": "array", "items": { "type": "number" }, "description": "[minX, minY, maxX, maxY] in curve units, including symmetry copies." }, "svgBytes": { "type": "integer" }, "svgArtifact": { "type": "object", "additionalProperties": false, "required": [ "artifactId", "path", "bytes", "sha256" ], "properties": { "artifactId": { "type": "string" }, "path": { "type": "string" }, "bytes": { "type": "integer" }, "sha256": { "type": "string" } } }, "pngArtifact": { "type": [ "object", "null" ], "additionalProperties": false, "required": [ "artifactId", "path", "bytes", "sha256" ], "properties": { "artifactId": { "type": "string" }, "path": { "type": "string" }, "bytes": { "type": "integer" }, "sha256": { "type": "string" } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }