In geometry, a polygon with holes is an area-connected planar polygon with one external boundary and one or more interior boundaries (holes). Polygons with holes can be

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

into multiple polygons by adding new edges, so they are not frequently needed. An ordinary polygon can be called

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...

, while a polygon-with-holes is ''multiply-connected''. An ''H''-holed-polygon is ''H''-''connected''.

Degenerate holes

Degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...

cases may be considered, but a well-formed holed-polygon must have no contact between exterior and interior boundaries, or between interior boundaries. Nondegenerate holes should have 3 or more sides, excluding internal point boundaries ( monogons) and single edge boundaries ( digons).

Boundary orientation

Area fill algorithms in computational lists the external boundary vertices can be listed in counter-clockwise order, and interior boundaries clockwise. This allows the interior area to be defined as ''left'' of each edge.

Conversion to ordinary polygon

A ''polygons with holes'' can be transformed into an ordinary unicursal boundary path by adding (degenerate) connecting double-edges between boundaries, or by dissecting or triangulating it into 2 or more simple polygons. : Holed-polygon-dissections

In polyhedra

''Polygons with holes'' can be seen as faces in polyhedra, like a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

with a smaller cube externally placed on one of its square faces (augmented), with their common surfaces removed. A toroidal polyhedron can also be defined connecting a holed-face to a holed-faced on the opposite side (excavated). The

1-skeleton In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other word ...

(vertices and edges) of a polyhedron with holed-faces is not a connected graph. Each set of connected edges will make a separate polyhedron if their edge-connected holes are replaced with faces. The

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

of hole-faced polyhedron is χ = ''V'' - ''E'' + ''F'' = 2(1-''g'') + ''H'', genus ''g'', for ''V'' vertices, ''E'' edges, ''F'' faces, and ''H'' holes in the faces. ;Examples Polyhedron with holed-faces0.png, (genus 0) with two 1-holed-faces (top and bottom).
V=16, E=20, F=8, H=2.
3-connected Toroidal polyhedron with faces with holes.png, Toroid (genus 1) with two 1-holed-faces.
V=16, E=24, F=10, H=2.
2-connected Polyhedron with holed-faces.png, (genus 0) with one 1-holed-face.
V=16, E=24, F=11, H=1.
2-connected Cubic polyhedron with holed-faces.png, (genus 0), with six 1-holed faces.
V=32, E=36, F=12, H=6.
7-connected Cubic toroidal polyhedron with holed-faces.png, Toroid (genus 5), with six 1-holed faces.
V=40, E=72, F=30, H=6.
2-connected Toroidal polyhedron with faces with 2 holes.png, Toroid (genus 2) with two 2-holed-faces.
V=24, E=36, F=14, H=4.
3-connected Polyhedron with holed-faces2.png, Toroid (genus 1) with one 2-holed-face, and one 1-holed-face.
V=24, E=36, F=15, H=3.
3-connected Toroidal_polyhedron_with_faces_with_2_holes2.png, (genus 0) with one 2-holed-face.
V=24, E=36, F=16, H=2.
3-connected Toroidal polyhedron with faces with 2 holes3.png, Toroid (genus 1) with two 1-holed-faces.
V=24, E=36, F=14, H=2.
2-connected Polyhedron with faces with holes2.png, Toroid (genus 1) with two 1-holed-faces.
V=32, E=48, F=18, H=2.
2-connected ;Examples with degenerate holes A face with a point hole is considered a monogonal hole, adding one vertex, and one edge, and can attached to a degenerate

monogonal hosohedron In geometry, a monogon, also known as a henagon, is a polygon with one edge and one vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388 In Euclidean geometry In ...

hole, like a cylinder hole with zero radius. A face with a degenerate digon hole adds 2 vertices and 2 coinciding edges, where the two edges attach to two coplanar faces, as a dihedron hole. Toroid_cube_with_point_hole.png, (genus 1) with two (degenerate point or monogon) 1-holed-faces.
V=10, E=15, F=7, H=2.
2-connected Toroid_cube_with_edge_slit.png, (genus 1) with two (degenerate digon) 1-holed-faces.
V=12, E=18, F=8, H=2.
2-connected

References

{{reflist Polygons Euclidean plane geometry

Degenerate holes

Boundary orientation

Conversion to ordinary polygon

In polyhedra

See also

References