Polygon with holes

Polygons with holes, with simply connected brown regions and interior boundaries, including degenerate cases of single vertices and edges, (a,b,f).

An annulus can be approximated by two n-sided boundaries with the same center, but different radius.

In geometry, a polygon with holes is an area-connected planar polygon with one external boundary and one or more interior boundaries (holes).[1] Polygons with holes can be dissected into multiple polygons by adding new edges, so they are not frequently needed.

An ordinary polygon can be called simply-connected, while a polygon-with-holes is multiply-connected. An H-holed-polygon is H-connected.[2]

## Contents

* 1 Degenerate holes
* 2 Boundary orientation
* 3 Conversion to ordinary polygon
* 4 In polyhedra
* 5 See also
* 6 References

## Degenerate holes

Degenerate 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.[3]

## 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.

Example conversion of a single-holed polygon by connecting edges, or dissection

## In polyhedra

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

* (genus 0) with two 1-holed-faces (top and bottom).
V=16, E=20, F=8, H=2.
3-connected

* Toroid (genus 1) with two 1-holed-faces.
V=16, E=24, F=10, H=2.
2-connected

* (genus 0) with one 1-holed-face.
V=16, E=24, F=11, H=1.
2-connected

* (genus 0), with six 1-holed faces.
V=32, E=36, F=12, H=6.
7-connected

* Toroid (genus 5), with six 1-holed faces.
V=40, E=72, F=30, H=6.
2-connected

* Toroid (genus 2) with two 2-holed-faces.
V=24, E=36, F=14, H=4.
3-connected

* Toroid (genus 1) with one 2-holed-face, and one 1-holed-face.
V=24, E=36, F=15, H=3.
3-connected

* (genus 0) with one 2-holed-face.
V=24, E=36, F=16, H=2.
3-connected

* Toroid (genus 1) with two 1-holed-faces.
V=24, E=36, F=14, H=2.
2-connected

* 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 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.

* (genus 1) with two (degenerate point or monogon) 1-holed-faces.
V=10, E=15, F=7, H=2.
2-connected

* (genus 1) with two (degenerate digon) 1-holed-faces.
V=12, E=18, F=8, H=2.
2-connected

## See also

* Prince Rupert's cube -- largest cube that can pass through a unit cube's hole.

## References

1. ↑ "IX.4: Polyhedra with ring-shaped faces", An Introduction To The Geometry Of N Dimensions, Methuen & Co., 1929, pp. 144–145, https://archive.org/details/in.ernet.dli.2015.220891/page/n165
2. ↑ "Chapter 5: Holes", Art Gallery Theorems and Algorithms, International Series of Monographs on Computer Science, 3, Oxford University Press, 1987, pp. 125–145, ISBN 0-19-503965-3, https://www.science.smith.edu/~jorourke/books/ArtGalleryTheorems/Art_Gallery_Chapter_5.pdf
3. ↑ "Art Gallery and Illumination Problems", Handbook of Computational Geometry, Elsevier, 2000, pp. 973–1027, doi:10.1016/b978-044482537-7/50023-1, ISBN 9780444825377

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