Equation.io

Equation.io

[equation.io](https://equation.io) — a graphing calculator with a built-in CAS. Type equations; they compile to GPU shaders and render as 2D curves, 3D surfaces, vector fields, ODE phase portraits, probability densities, and more. Every graph lives entirely in its URL, so the address bar is the share button.

The graph.tk story

This is the successor to graph.tk, which started in this repository in May 2010 as an HTML5-canvas grapher and picked up 400+ stars over the years. The site ran on a free .tk domain — which turned out to be the fatal flaw: the registrar (Freenom) eventually seized the domain to serve ads on it, and after Meta sued Freenom the whole .tk registry collapsed and the domain stopped resolving entirely.

The lesson was learned and the grapher was rebuilt from scratch — new parser, new CAS, WebGL rendering instead of canvas — on a domain that's actually owned: equation.io. The original code is preserved on the legacy branch (tag graph.tk-final) under its original LGPL-3.0 terms; everything on main is a clean-room rewrite, MIT licensed. The old UI remains usable at graph.equation.io.

Architecture

Deployed as a Cloudflare Worker. The page is web/; the mathematics is lib/.

• lib/expr.ts is the tokenizer, the shunting-yard parser, and the expression tree. lib/defs.ts resolves names and definitions. Resolve-time expansions live with the mathematics they implement: vector-calculus.ts (gradient, divergence, curl), sums.ts (Σ and Π), and the integral, curve, and surface passes in integrate.ts, curves.ts, and surface-geometry.ts.
• Distributions are lib/dist.ts. It re-exports base laws (dist-base.ts), continuous sampling (dist-sample.ts), discrete laws (dist-discrete.ts), and the random-variable system (dist-system.ts). The family table is dist-families.ts.
• Geometry, tensors, measures, lists, and automata make up the rest of lib/: geom.ts, pga.ts, clifford.ts, tensor.ts, measure.ts, list.ts, automaton.ts, and the modules beside them. compiler.ts turns a classified row into a CPU plan and a GLSL plan (glsl.ts).
• web/main.ts is the equation editor and the boot. web/session.ts owns the document, the panels, and the shared WebGL renderers. web/scene.ts builds one frame and hands it to the 2D and 3D renderers. Mutable state those three share lives on web/app-state.ts.
• Every equation is compiled to a GLSL scalar field F whose zero set is the graph:
• 2D: fullscreen-quad fragment shader; the curve is drawn where the distance estimate |F|/|∇F| is under a pixel, with a two-scale consistency test rejecting fake lines at poles/asymptotes (e.g. y=tan(x)).
• 3D (automatic when z appears): raymarched implicit surface — sign-change detection along each ray, bisection refinement, finite-difference normals, gl_FragDepth so multiple surfaces intersect correctly. Equations without z extrude to their true locus in R³.

The whole graph state lives in the URL (/g/eq1;eq2;…, each equation percent-encoded via lib/link.ts, which also escapes parens so chat-app linkifiers don't truncate the URL; legacy /#… links still load), so any set of equations is linkable and the address bar is the share mechanism. Agent-facing surface:

• /llms.txt — link format + expression syntax reference (web/public/llms.txt)
• /g/ — share form of a graph link; the worker injects og:/twitter: meta tags (the graph's equations as the title, over the site card)
• /mcp — stateless MCP server (Streamable HTTP) with encode_graph_url (validates rows, returns links), decode_graph_url (decodes links for editing), and show_graph (renders the interactive grapher inside MCP Apps hosts). See MCP Apps integration and testing.
• /.well-known/ard.json (also at the older /.well-known/ai-catalog.json) — Agentic Resource Discovery catalog listing the MCP server, and /mcp/server-card — its MCP Server Card. Both are generated from server.json by worker/discovery.ts.

Usage

pnpm web        # dev server (grapher + worker API)

pnpm test # vitest pnpm typecheck # lib + web + worker pnpm web:build # build to dist-web/ (client + worker) pnpm deploy # build and deploy to Cloudflare

Traffic and MCP usage

pnpm traffic 7     # needs `wrangler login`

Site visitors come from Cloudflare's own zone analytics. The script counts unique IPs per day on each page, using browser user agents only, and lists /mcp POSTs by client. Nothing on the site collects this.

MCP tool calls are the one thing Cloudflare can't see, because it doesn't record request bodies. worker/mcp-usage.ts writes one Analytics Engine point per tools/call: the tool, the client, the graph types, and row and error counts, never equation text. Set CLOUDFLARE_API_TOKEN, with Account Analytics: Read, for the script to include them.

Examples

Basics

• y = x^2 · x^2+y^2=4 · y = tan(x) — 2D curves
• y = sin(2πx) · θ = 1; r = θ x · y = x³ — unicode input: π, Greek-letter names, superscript exponents, subscripts (T₀ ≡ T_0, so a₃ is a sequence term), and ·/×/÷/≤/≥/≠; in the editor, typing \pi, \theta, \nabla, … inserts the symbol, and \ before any function name just drops (\trail → trail)
• z = sin(x)cos(y) · x^2+y^2+z^2=9 — 3D surfaces (automatic when z appears)
• y < x/2 + 1 — inequalities shade their region; strict </> have no border, <=/>= draw the boundary line, and chains like 4 <= x^2 + y^2 <= 9 intersect with an edge per non-strict bound
• y = {x < 0: -x, x >= 0: x^2} — piecewise: cond: value cases tried in order, an optional last bare value is the default; conditions chain like {0 < x < 1: 1, 0}, and a bare condition counts 1 ({x > 0, 5})
• y = {0 < x < 2: x^2} — a domain restriction: with no default, the value is undefined outside the conditions, so nothing is drawn there
• sin(x)cos(y) — a bare expression in x, y is a 2D scalar field, shaded in the row color where positive and its complement where negative. sin(x) is a field too (constant along y): the curve is y = sin(x)
• 2+2, sqrt(a), |A - B| — a bare number draws nothing and reads out = 4 under the row, live with sliders and t; write y = 4 for the line

Sliders and animation

• a = 2 — a named constant with a slider; other equations can use a, and it compiles to a uniform so dragging never rebuilds a shader. b = a^2 + t defines a computed/animated constant
• a := 2, f(x) := x^2 — the same definitions written explicitly. := can also take over e, pi, i, w or t as a value (not a function) for the whole graph (e := 0.6 is a slider; t := 0.5 stops all animation); u and v take an interval (u := interval(0, 2pi)); x, y, z and d stay the language's
• (2, 3) / (3, 12, 0) — points. In 2D, coordinates that are plain numbers or slider names can be dragged on the canvas, and the drag rewrites them: a = 1; b = 2; (a, b) moves both sliders, (2sin(t), 3) only its literal height
• (2cos(t), 2sin(t)) — t is seconds since load, so this point orbits

Calculus

• f(x) = x^3 - a x — user-defined functions, inlined symbolically
• f(z) = {re(z) >= 1: 1, f(4 - 3(z^6)^(1/6))} then f(x i - |y|) >= 0 — a tail-recursive function (every self-call a whole case of its {…}) runs as a bounded loop per pixel; this one shades the Koch snowflake
• y = d/dx f(x) / d^2/dx^2 (x^4) — symbolic Leibniz derivatives; works for any single-letter variable, nests, and flows through function definitions: g(x) = d/dx f(x) then y = f(a) + g(a)(x - a) is a live tangent line

Probability

• X ~ Normal(0, a) — a random variable; the row plots its density, and parameters may use sliders. Then P(X < b), P(X > b), or P(-1 < X < 2) shades that area under the density and shows the numeric probability
• Also Uniform(lo, hi), Exponential(rate), Gamma(shape, rate), Beta(a, b), ChiSquared(df), StudentT(df) (or T(5)), LogNormal(mu, sigma), Cauchy(location, scale), Weibull(shape, scale) — exact densities, exact P(…), and median/IQR readouts where heavy tails leave no σ to report
• erf, normalpdf(x, mean, sd), and normalcdf(x, mean, sd) are also plain functions, so y = normalcdf(x, 0, 1) graphs the CDF

Vector fields and ODEs

• (-y, x) — a tuple depending on x, y is a vector field, rendered as animated streamlines via GPU line-integral convolution; t works too: (cos(t)-y, x)
• grad(x^2 + y^2) (or ∇(…)) — the symbolic gradient as a tuple, so it plots as a vector field and works in dot(grad(f), (1, 0))
• dy/dx = x y / y' = sin(x) - y — ODEs plot the slope/direction field (1, f); click the canvas to drop an RK4 integral curve through that point, double-click to clear
• (x', y') = (y, -sin(x)) — a system plots its phase portrait, with the same click-to-trace trajectories

Simulation (states)

• th' = om (angle) with om' = -sin(th) (angular velocity) and th(0) = 3 — a *state*: a prime on a name of your own is d/dt of it, integrated forward by RK4 at a fixed step as the graph animates — see lib/state.ts. Everywhere else th behaves exactly like a constant, uniform and all, so drawing the system is ordinary plotting: (sin(th), -cos(th)) is the bob, (u sin(th), -u cos(th)) the rod. It is the one value in a graph that is not a formula in t, which is what makes a double pendulum — chaotic, no closed form — possible. Initial values get a slider that relaunches the run; ↻ in the panel restarts it
• r' = vel with vel' = -r/|r|^3 and r(0) = (1, 0) — a *vector state*: a derivative that is a 2- or 3-vector integrates componentwise as r_1, r_2(, r_3), and the bare name draws as a moving point and joins point arithmetic — an orbit in two rows
• label((2, 4), "peak") / label(A, "vertex") — text beside a point, in the row's color; the point follows sliders and t like any other
• y = x^2 #e24 — a note that opens with a hex color draws the row in it
• trail(A) — leaves a live motion trail behind a 2D or 3D point. For example, A = (cos(t), sin(t)); trail(A) draws an orbit as it runs; trail((cos(t), sin(t), t/5)) draws a rising helix. Vector states work too. Trails retain up to 30 seconds / 2048 observed positions, reset when the equations or simulation restart, and are local to the current session.
• p(0) = ([0..299]/30, 0, 0) — a *state family*: a list of starting values runs the system once per element (up to 1024), and states coupled to it run along. p then draws a cloud of moving points and mean(p_1) reduces across runs. Started from a tuple, p(0) = (sort([0..299])/30, 0, 0), the runs are in order and p[1] is the first
• p(50..400) — an *orbit*: where the state goes between those times, integrated ahead of time with the live simulation's own steps (lib/orbit.ts), so moving points ride their orbit. A family draws one path per run (p[1](50..400) draws one, when the runs start from a tuple); a scalar state plots against time, th(0..20) being the curve (t, th). With both, the Rössler attractor is a thin band of orbit with particles flowing along it

Custom coordinates and complex roots

• r = sqrt(x^2+y^2); theta = atan2(y,x) defines polar coordinates. (r, theta) = (2, 9pi/4) draws their point, with angles wrapping modulo 2π. Use literal or slider values on the right to drag the point in those coordinates.
• (r, theta) = (3u, 6pi u) traces a three-turn spiral;