Equation.io

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.

• lib/ — tokenizer, shunting-yard parser, symbolic expression core (expr.ts), and a GLSL compiler (glsl.ts) used for plotting.
• web/ — the grapher. 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/<eqs> — share form of a graph link; the worker injects og:/twitter: meta tags and /api/og/<eqs> renders the preview PNG on the CPU (expressions compile to a stack machine — no WebGL in Workers)
• /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.

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

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 and τ, 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}
• 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/density field
• 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
• (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
• 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
• 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.

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; (r', theta') = (r(1-r), 1) draws a polar limit-cycle field.
• 1+2i draws an Argand point; w^3 = 1 draws the three cube roots of unity. Systems use a numerical search in the current view; small solution branches may be missed. Coordinate examples are available in the examples menu.

Matrices

• M = [(a, b), (c, d)] — a 2×2 or 3×3 matrix; det(M), trace(M), the matvec M v, and solve(M, v) (Cramer's rule) expand symbolically at lowering time, see lib/mat.ts. So (x', y') = A (x, y) is a phase portrait with sliders in the entries, and om' = solve(M, f) integrates the double pendulum in the Lagrangian form M(θ)ω′ = f it is derived in

Parametric curves and surfaces

• (2cos(2pi u), 2sin(2pi u), 3u) — parametric curve, u ∈ (0,1)
• (cos(2pi u)(2+cos(2pi v)), sin(2pi u)(2+cos(2pi v)), sin(2pi v)) — parametric surface, u,v ∈ (0,1); per-fragment Newton ray/surface intersection with a glossy specular material

Sequences and data

• a_n = 1/n^2 — a sequence: dots at integer n ≥ 0; the Σ toggle on the row switches to partial sums S_N (this one converges to π²/6)
• a_{n+1} = r a_n (1 - a_n) — a recurrence: draws the map's curve, the diagonal y = x, and the cobweb path from the seed a_0 (define a_0 = 0.2 for a slider, default ½). With x free on the right side, x becomes the parameter axis and the plot is the orbit/bifurcation diagram: a_{n+1} = x a_n (1 - a_n) is the logistic bifurcation
• [3, 1, 4, 1, 5] — a data list: dots at (k, value), k = 1, 2, …; the row's bar toggle draws it as a bar chart. [(1, 2), (3, 4)] is a scatter of points

Regression

• X = [0, 1, 2, 3]; Y = [1, 3, 5, 7]; Y ~ m X + b fits a line. Unbound names m and b become fitted constants; y = m x + b draws the model and (X, Y - (m X + b)) draws its residuals. A fit row reports the coefficients, RMSE, R² (when defined), and observation count.
• Y ~ a X^2 + b X + c fits a polynomial; Y ~ a exp(b X) fits a nonlinear model. Already defined constants stay fixed and changing them refits the other coefficients. Define data and fixed constants above the fit.
• data.height ~ m data.age + b works with CSV columns. Missing/nonfinite data pairs are skipped with a count; mismatched lengths and unidentifiable coefficients are errors. Missing CSVs remain device-local in shared links.
• Fits are static, with at most 8 coefficients and 10,000 observations (2,000 for nonlinear models). Nonlinear fitting uses deterministic starts and reports a local fit; it does not guarantee a global optimum.

Contextual syntax help

The equation editor suggests functions, defined names, and loaded CSV columns as you type, and shows signatures inside function calls. Tab or a click inserts a suggestion; arrow keys select one for Enter to insert. Enter otherwise creates an equation row, Escape dismisses help, and completion is one undoable text edit. Comments and quoted strings do not trigger suggestions.

Number theory and complex analysis

• gcd(a, b) / isprime(n) — number theory; try a_n = isprime(n)
• ln(w-2) - ln(w+2) — complex analysis: i is the imaginary unit and w = x + iy; a complex-valued expression renders the level curves of its imaginary part (field lines) and real part (equipotentials), so complex potentials draw electrostatics directly. re/im/arg/abs/conj bring values back to ℝ, e.g. im(ln(w)) = 1 plots as an ordinary implicit curve

Equations persist in the URL hash. Drag to pan/orbit, wheel to zoom, right-drag (or shift) to pan in 3D, click a color dot to cycle colors. Points and dropped ODE seeds highlight under the cursor and drag with it. The equations panel is a corner-pinned card: flick it — touch anywhere on it, or drag the grip strip along its top edge with a mouse — to send it to any corner, or throw it past any edge to clear the view entirely; it tracks the pointer and leaves along the throw. The y= chip left behind brings it back (tap it, or drag it to pull the panel in), and the chosen corner sticks.

worker/ — the Cloudflare Worker entry: serves the built app and handles /api/* routes.

License

MIT — see LICENSE. The pre-2026 graph.tk code on the legacy branch remains under its original LGPL-3.0 terms; no code from it was reused in the current codebase.

Axis scaling

Option/Alt + drag the 2D canvas to scale each axis independently: horizontal movement scales x and vertical movement scales y, anchored at the initial pointer position. Normal zoom preserves the ratio. Set ratio = 1 in the viewport row to restore equal axis units.

Scaling creates or updates a shareable viewport row: view(x = -10..10, y = -1..1, ratio = 5). The positive ratio is pixels per y unit divided by pixels per x unit; omitted means 1. The bounds are fitted with that ratio preserved, including on different screen sizes.