API

The database can be queried as JSON. Invalid parameters return status 400 with a field error.

Identify accepts at most 25 vertices and denominators up to 25.

Identify

GET /polygons/api/v1/identify?vertices=(-1,-1),(2,-1),(-1,2)

{
  "in_database": true,
  "polygon": {...}
}

Field	Meaning
`in_database`	Whether the polygon is in the database
`polygon`	The convex hull of the given points, described below

The parameter vertices is a comma-separated list of coordinates, which may be fractions like 1/2.

Search

GET /polygons/api/v1/search?k=2&i=1&page=1

{
  "total": 5129,
  "page": 1,
  "pages": 86,
  "page_size": 60,
  "complete": true,
  "incomplete_reason": null,
  "polygons": [{...}, ...]
}

Field	Meaning
`total`	Number of polygons matching the search
`page`, `pages`	Number of this page and of all pages
`page_size`	Maximal number of polygons per page
`complete`	Whether the database contains all polygons matching the search
`incomplete_reason`	Which polygons may be missing, or `null`
`polygons`	The polygons on this page, described below

The parameters are those of the search form, as in the address of a search, and page.

Polygons

{
  "url": "/polygons/polygon/1/3,0,0,3,0,0",
  "vertices": [["3", "0"], ["0", "3"], ["0", "0"]],
  "denominator": 1,
  "interior_lattice_points": 1,
  "boundary_lattice_points": 9,
  "area": "9/2",
  "number_of_vertices": 3,
  "boundary_k_rational_points": 9,
  "ehrhart_period": 1,
  "automorphism_group": "D3",
  "maximal": true,
  "integer_hull_dimension": 2,
  "interior_integer_hull_dimension": 0,
  "ldp": true,
  "thin": null
}

Field	Meaning
`url`	Page of the polygon
`vertices`	Vertices in affine normal form, with coordinates as strings
`denominator`	Smallest k such that kP has integral vertices
`interior_lattice_points`	Number of interior lattice points
`boundary_lattice_points`	Number of boundary lattice points
`area`	Euclidean area, as a string
`number_of_vertices`	Number of vertices
`boundary_k_rational_points`	Number of points of (1/k)ℤ² on the boundary, where k is the denominator
`ehrhart_period`	Period of the Ehrhart quasipolynomial
`automorphism_group`	Group of affine unimodular automorphisms: Zn cyclic, Dn dihedral of order 2n
`maximal`	Whether the polygon is maximal
`integer_hull_dimension`	Dimension of the convex hull of the lattice points, −1 if there are none
`interior_integer_hull_dimension`	Dimension of the convex hull of the interior lattice points, −1 if there are none
`ldp`	Whether the polygon is LDP; `null` unless it has one interior lattice point
`thin`	Whether the polygon is thin; `null` unless it has no interior lattice points