Cartesium reference

Expressions and relations#

Operators are +, -, *, /, ^, =, <, <=, >, and >=. Parentheses group expressions. Numeric juxtaposition such as 2x is multiplication; write * between names or before a grouped expression. Constants are pi, tau (equal to 2*pi), and e.

(x^2 + y^2)^2 = x^2 - y^2

Explore Lemniscate Click to load the interactive graph

r = 5*cos(7*theta)

Explore Polar rose Click to load the interactive graph

Restrictions and piecewise#

Append a condition in braces to restrict a plot:

y = sin(x) {-2*pi <= x <= 2*pi}
(cos(t), sin(t)) {0 <= t <= 2*pi}

A piecewise expression tests branches from left to right. The unlabelled final value is the fallback:

y = {x < -1: -x-1, x <= 1: x^2, x-1}

Strict inequality boundaries are dashed; non-strict boundaries are solid.

y <= sin(a x)
(2*cos(t), 4*sin(t)) {0 + a <= t <= pi + a}
x^2 + y^2 >= 12
a = 1

Explore Hello Click to load the interactive graph

Variables and functions#

Definitions do not draw geometry. A numeric scalar receives slider controls; its minimum, maximum, and step are document settings. Definitions are resolved as dependencies rather than executed top to bottom.

a = 2
b = 3
f(u) = a*sin(b*u)
y = f(x)

Explore Adjustable wave Click to load the interactive graph

Points, polylines, and polygons#

A pair of equal-length sequences plots a point collection:

n = 1:5
(n, n^2)

Point collections do not connect their entries. Use polyline(xs, ys) to connect equal-length coordinate sequences in order, or polygon(xs, ys) to close the final edge and fill the enclosed path:

xs = [0, 3, 4, 2]
ys = [0, 4, 2, -1]
polygon(xs, ys)

polyline requires at least two points and polygon requires at least three. Both preserve sequence order and duplicate vertices. Polygon orientation is therefore retained even though clockwise and counterclockwise fills generally look the same. Arrowheads and directed-path presentation are separate features and are not implied by either operation.

In the 3D calculator, vector(v) draws an arrow from the origin to a finite length-3 sequence v. An anchored form adds a tail point and a displacement: vector([4, 5, 6], [1, 2, 3]) ends at [5, 7, 9]. A zero displacement remains valid and is shown as a point marker. An N×3 matrix draws one arrow per row, and a single row broadcasts when paired with a matrix:

V = [1 0 0; 0 1 0; 0 0 1]
vector([0, 0, 0], V)

Anchored vectors can also be sampled over one to three finite parameter domains:

vector([x, y, 0], [y, -x, 0]) {-5 <= x <= 5} {-5 <= y <= 5}

The 3D calculator samples nine points per declared axis, skips undefined samples, and keeps zero vectors as point markers. Vector collections and fields are fixed whole-scene resources; adaptive density, streamlines, and configurable arrowhead styling are not yet supported.

3D surfaces and solids#

The 3D calculator also supports implicit equalities, single-comparison solid inequalities, and explicit height surfaces:

A 3D inequality must contain exactly one <, <=, >, or >= comparison. The accepted side is sampled as a signed field, and its boundary is rendered as an opaque, double-sided surface. Strict and non-strict comparisons therefore share the same geometric boundary, and a closed solid intentionally looks like the corresponding hollow surface from outside. Chained or brace-restricted inequalities, Boolean constraints, automatic plot-boundary caps, and true volume rendering are not currently supported.

px = [-4, -1, 3, 4, 0]
py = [1, 4, 3, -1, -3]
polygon(px, py)
lx = [-4, -2, 0, 2, 4]
ly = [-3, -1, -2, 1, 0]
polyline(lx, ly)

Explore Connected points Click to load the interactive graph

Derivatives#

Derivatives support arithmetic, powers, smooth elementary functions, and compositions of user-defined functions. This includes the trigonometric, inverse-trigonometric, reciprocal-trigonometric, hyperbolic, exponential, logarithmic, square-root, absolute-value, and constant-degree nthroot functions listed below. atan2 and both forms of log also differentiate structurally.

Expressions are differentiated when the expression model is built, then evaluated through the ordinary compiled numeric path. Prime notation differentiates the function before applying its argument, so f'(x^2) means the value of f' at x^2, not the derivative of f(x^2). Scalar piecewise expressions differentiate each branch while leaving variable-dependent switching boundaries undefined. floor, ceil, round, sign, mod, sequence extrema, reductions, and linear-algebra functions report an unsupported-derivative diagnostic rather than claiming a derivative across their discontinuities or non-scalar operations.

Definite integrals#

Use integral(expression, variable, lower, upper) to evaluate a finite definite integral. For example, y = integral(sin(t), t, 0, x) plots the accumulated area from zero to x. The integration variable is local to the integrand; bounds may use graph variables, sliders, constants, and user-defined functions. Reversing the bounds reverses the sign.

Definite integrals use bounded adaptive numerical quadrature. If the bounds are not finite, the integrand produces a non-finite sampled value, or the requested accuracy cannot be reached within the work limit, the result is undefined. Improper integrals and symbolic antiderivatives are not currently supported.

f(t) = sin(t)
y = f(x)
y = integral(f(t), t, 0, x)
y = 1 - cos(x)

Regression#

Use ~ to fit undefined scalar parameters in a model to finite sequence data:

xs = [1, 2, 3, 4]
ys = [3.1, 4.9, 7.2, 8.8]
ys ~ m*xs + b
f(u) = m*u + b
y = f(x)

Here xs and ys are the data, while the undefined scalars m and b are fitted by least squares. Successful regression rows display their parameters and root-mean-square error. Fitted parameters are document-level definitions, so later variables, functions, and plots can reuse them. An existing assignment, such as b = 0, fixes that value instead of fitting it.

Models linear in every fitted parameter use a direct deterministic solver. This includes polynomial models such as ys ~ a*xs^2 + b*xs + c: the powers of the data do not make the parameters nonlinear.

Genuinely nonlinear models use bounded nonlinear least squares. For example:

ys ~ a*exp(b*xs) + c
ys ~ a*xs^b
ys ~ l/(1 + exp(-k*(xs - x0)))
ys ~ a*sin(b*xs + c) + d

Nonlinear fitting is iterative and can have multiple local solutions, particularly for sinusoidal models. The solver tries a deterministic set of starting points and reports the best finite solution it finds; equivalent parameterizations can therefore display different parameter values for the same curve. A nonlinear relation may fit at most 8 parameters and 20,000 data rows. All regression data must be nonempty and equally sized, and the supplied data must independently determine every fitted parameter.

Sequences and ranges#

Scalar arithmetic broadcasts over sequences. Equal-length sequences support elementwise + and -; .* is elementwise multiplication.

k = 400
radius = [0.7*sqrt(n) for n in 0:k]
angle = [n*pi*(3-sqrt(5)) for n in 0:k]
(radius.*cos(angle), radius.*sin(angle))

Explore Generated phyllotaxis Click to load the interactive graph

Vectors and matrices#

One-dimensional sequences are vectors. In matrix literals, spaces separate columns and semicolons separate rows. Indexing is one-based.

A * [x, y] <= b is the conjunction of the component inequalities and draws their feasible region.

A = [-1 0; 0 -1; 2 1; 1 3]
b = [0, 0, 8, 9]
A * [x, y] <= b
c = [3, 2]
z = 8
y = (z-c[1]*x)/c[2]

Explore Linear-program feasible region Click to load the interactive graph

A = [-1 0; 1 -2; -2 1]
b = [0, 0, 0]
A * [x, y] <= b

Explore Polyhedral convex cone Click to load the interactive graph

Built-in functions#

log(x) is base 10; use ln(x) for the natural logarithm or log(x, base) for another base. Logarithms require a positive argument and a positive base other than one. atan2(y, x) uses the signs of both coordinates to select the angle’s quadrant. nthroot(x, n) requires nonzero n and supports negative x when n is an odd integer. For nonzero m, mod(x, m) returns the Euclidean remainder from zero up to, but not including, abs(m).

min and max accept one or more arguments. atan2, nthroot, mod, and dot take two; log takes one or two; the remaining built-ins take one. Scalar functions broadcast over sequences, pairing equal-length sequence arguments element by element. Unary scalar functions also map over matrices. The built-in constants are pi, tau, and e.

Styling and animation#

Visible curves expose color, width, and opacity. Inequalities also expose fill opacity. Point collections instead expose color, marker opacity, pixel radius, and marker shape: disc, annulus, cross, square, diamond, or triangle.

Variables and function definitions have no plot style and do not consume the automatic color sequence. Numeric variables expose slider bounds, step, playback speed, direction, and repeat behavior.

Graph settings control axes, grids, bounds, equal axis scale, and appearance. Drag the panel edge to resize the expression panel; double-click the edge to reset it. Drag to pan; use the wheel or a pinch gesture to zoom.

Saving, sharing, and embedding#

Snapshot links open as unsaved copies and require no server-side anonymous record. Embedded graphs show a still preview until activated. Only one embed on a page owns a live renderer; Expand shows its read-only expression list and Edit opens an editable copy.

When Spaces is disabled or the user is signed out, browser drafts are labeled Browser only and are not synced. With Spaces enabled, a signed-in Save As creates a mutable document in the account’s personal Space; public listed and unlisted publications remain separate immutable records.