quick-vertex

Plot feasible regions of two-variable linear programs in Typst, drawn with CeTZ. One call gives you the shaded region, the boundary lines, the vertices, the objective's level lines and optimization direction, and a vertex table with the optimum highlighted.

It is built for teaching (secondary school / early undergraduate), and it handles the cases a synthetic example set usually forgets: unbounded regions, multiple optima, unbounded objectives, empty regions, strict inequalities and any quadrant. The objective's level lines are named explicitly ("Rectas de nivel" in Spanish); the direction arrow says min Z or max Z.

Usage

#import "@preview/quick-vertex:0.2.0": feasible-region

#feasible-region(
  ((1, 1, 4, "<="), (1, 3, 6, "<=")),
  objective: (3, 4),
  sense: "max",
  labels: ($x + y = 4$, $x + 3y = 6$),
)
// → Optimum (max): Z = 13 at C(3, 1)

This is exactly the first image in the gallery above.

Each constraint is a 4-tuple (a, b, c, op) meaning a·x + b·y op c, where op is one of "<=", ">=", "<", ">". The objective is Z = c1·x + c2·y, given as (c1, c2).

When working from a local checkout, import "lib.typ" instead of the registry package. The Typst Universe submission for 0.2.0 is PR #6100. Registry imports become available after that PR is merged and the publication workflow completes. See the PR for its current review and publication status.

Examples

Minimize with Spanish labels

#import "@preview/quick-vertex:0.2.0": feasible-region
#feasible-region(
  ((1, 1, 4, ">="), (1, 0, 1, ">=")),
  objective: (2, 3), sense: "min", lang: "es",
)
// → Óptimo (min): Z = 8 en A(4, 0)

The purple rectas de nivel represent Z = k. The arrow labeled min Z points toward decreasing objective values. With sense: "max", it points toward increasing values and is labeled max Z.

Level lines without the direction arrow

#import "@preview/quick-vertex:0.2.0": feasible-region
#feasible-region(
  ((1, 1, 4, "<="), (1, 3, 6, "<=")),
  objective: (3, 4), gradient: false,
)

Strict inequality: an unattained supremum

#import "@preview/quick-vertex:0.2.0": feasible-region
#feasible-region(
  ((1, 1, 4, "<"),),
  objective: (1, 1), sense: "max", lang: "es",
)
// → Supremo: Z = 4, no alcanzado

The excluded boundary is dashed and its vertices are hollow. No point is marked as an attained optimum. The same conclusion is shown with table: false.

Example sources: gallery, Spanish minimum, and additional cases.

Parameters

Parameter	Default	Meaning
constraints	(required)	Array of (a, b, c, op). op ∈ "<=", ">=", "<", ">". An optional 5th element sets that line's color: (a, b, c, op, color).
objective	none	(c1, c2) of Z = c1·x + c2·y. Omit to just draw the region. An optional 3rd element sets the objective color: (c1, c2, color).
sense	"max"	Optimization sense: "max" or "min".
gradient	true	Draw the optimization direction arrow, labeled max Z or min Z. For an unattained bound, draw it on the limiting face.
table	true	Show the vertex table; if false, a compact vertex legend. Objective diagnostics are shown in either mode.
first-quadrant	true	Add x ≥ 0, y ≥ 0 implicitly. See the note below.
labels	none	Array of line equations (content), one per constraint, in the same order.
lang	"en"	Language of the rendered labels: "en" or "es".
region-color	blue	Fill / hatch / border color of the feasible region.
size	(6, 4.5)	Canvas size, in CeTZ units — with equal-aspect: true, an upper bound: the drawn extent shrinks in whichever dimension isn't the tightest fit.
margin	1.15	Finite padding factor around the region, at least 1.
equal-aspect	true	Both axes use the same units-per-length scale (textbook square-grid convention; both axes also share one tick step). false stretches each axis independently to fill size exactly, with its own tick step — the optimization arrow stays perpendicular to the level lines either way.

⚠️ first-quadrant is true by default, so x ≥ 0 and y ≥ 0 are added automatically (the usual assumption in school problems). Pass first-quadrant: false to draw only the constraints you write — for example when the region lives in another quadrant.

Note on two "senses". The 4th tuple element is the inequality operator ("<=", …). The sense parameter is the optimization sense ("max"/"min"). They are unrelated.

gradient is the historical name of the arrow flag. Set gradient: false to hide the arrow while keeping the objective's level lines.

Colors

Every boundary line gets a distinct color automatically — a curated palette first, then generated hues — so colors never repeat, no matter how many constraints you add. Override any of them explicitly:

#feasible-region(
  (
    (1, 1, 4, "<=", red),          // this line in red
    (1, 3, 6, "<="),               // auto color
  ),
  objective: (3, 4, olive),        // objective, optimum and direction arrow in olive
  region-color: rgb("#2b8a3e"),    // the feasible region in green
)

What it handles

Case	Behavior
Bounded region	Filled + hatched polygon.
Unbounded region	Filled correctly; arrows on the open sides + an "unbounded" note.
Unique optimum	Highlighted vertex, optimal level line, direction arrow, table with the optimal row marked.
Multiple optima	Highlights the feasible optimal segment, ray or line. Excluded endpoints remain hollow.
Zero objective (0, 0)	Every feasible point is optimal; highlights the whole region.
No vertices	Handles half-planes, strips, lines and the whole plane without confusing them with an empty region.
Unbounded objective	Detects that no finite max/min exists and says so, instead of inventing a vertex.
Empty region	Labels "Infeasible region".
Strict inequalities (<, >)	Dashed boundary and hollow excluded vertices; reports an unattained supremum/infimum when appropriate.
Any quadrant	first-quadrant: false works (frame, axes and ticks with negatives).

Localization

Rendered labels default to English. Pass lang: "es" for Spanish. Adding a language is a matter of copying one block in the _i18n dictionary in lib.typ and translating it — contributions welcome.

How it works

The analysis is independent of drawing. Constraint normals and the objective are normalized before geometric comparisons. The recession cone is analyzed using constraint tangents and inward normals, without angular sampling. Feasibility is tested independently of the visible frame, including strict systems and regions without vertices. Strict optimal faces are checked for an included point before claiming that a maximum or minimum is attained.

The shading clips the visible frame against each half-plane (Sutherland–Hodgman). Tables list vertices of the closure: a row marked ○ and a hollow plotted point identify an excluded boundary point, which is never marked as an attained optimum. The arrow is perpendicular to the level lines and shows the optimization direction: toward increasing Z, labeled max Z, for maximization; toward decreasing Z, labeled min Z, for minimization. For a linear objective in two variables, these level curves are straight lines. The matching-color legend names them "Level lines: Z = k", or "Rectas de nivel: Z = k" with lang: "es".

Inputs are validated: coefficients must be finite numbers, operators and optimization sense must be supported, colors must be colors, dimensions must be positive, and labels must match the number of constraints. Constant constraints such as (0, 0, 5, "<=") are allowed and checked normally.

Pure analysis

analyze takes constraints, objective, sense and first-quadrant, with the same defaults as feasible-region, and returns data without drawing:

#import "@preview/quick-vertex:0.2.0": analyze
#let result = analyze(
  ((1, 1, 4, "<"),),
  objective: (1, 1),
)
#assert.eq(result.status, "unattained")
#assert(calc.abs(result.value - 4) < 1e-8)

Its fields include feasible, bounded, vertices (closure vertices), recession (feasible directions), witness (a feasible point), status, value, optimum, optimal-indices and closure-indices. Empty regions have witness: none, no recession directions and bounded: true.

status is none (no objective), infeasible, unbounded (objective), unattained, unique, segment, ray, line or all (zero objective). For a finite objective, value is its optimal value or limiting bound; optimum is a point on that face and belongs to the actual region unless the status is unattained. It is none when the objective is absent, the region is empty, or the objective is unbounded. optimal-indices only includes vertices that are actual attained optima. No optimal point is asserted for an empty region, even if its closure has vertices.

Compatibility

• Typst >= 0.14.0
• CeTZ 0.5.2

Verified with Typst 0.14.0 and 0.15.1. Mathematical assertions, an independent exact projection oracle and text/SVG regressions are described in TESTING.md. See CHANGELOG.md for the 0.2.0 changes.

Known limitations

Calculations use floating-point arithmetic with normalized geometric tolerances (1e-10, plus a roundoff allowance for large coordinates). Positive coefficient scaling preserves classification within floating-point accuracy, but near-coincident boundaries and nearly parallel constraints can still be numerically ill-conditioned. Inputs whose normalization overflows are rejected. This is not an exact rational solver. Values are displayed to two decimal places, except small nonzero values which retain their numeric representation.

See ROADMAP.md for remaining presentation limitations and TESTING.md for the mathematical and rendering checks.

License

MIT.