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Polygon with holes

Polygon with holes is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Polygon with holes rather than just read about it. In short: 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 into multiple polygons by adding new edges, so they are not frequently needed.

Polygon with holes — main illustration
Polygon with holes — illustration

Key takeaways

  • Polygon with holes belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Polygon with holes to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Polygon with holes from memory before moving on to harder problems.

Reference excerpt

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 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.

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 Computational geometry algorithms which process polygons with holes, such as in order to draw their interiors or to perform a point in polygon test, may require that the exterior boundary is distinguished from the interior boundaries by their orientation, the order in which the points are listed; commonly, the exterior boundary is listed in counter-clockwise order, and interior boundaries in clockwise order. This allows the interior area to be defined as left of each edge, regardless of whether that edge is part of an exterior or interior boundary.

Conversion to ordinary polygon A polygon 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.

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

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.

See also Prince Rupert's cube — largest cube that can pass through a unit cube's hole.

References

Illustrations

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).
Polygons with holes, with simply connected brown regions and interior boundaries, including degenerate cases of single vertices and edges, (a,b,f).
Polygon with holes illustration
Polygon with holes illustration
Polygon with holes: Example conversion of a single-holed polygon by connecting edges, or dissection
Example conversion of a single-holed polygon by connecting edges, or dissection
Polygon with holes illustration

Worked examples

Example 1 — a first encounter with Polygon with holes

Start with the simplest possible case. Write down what Polygon with holes claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Polygon with holes before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Polygon with holes ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Polygon with holes

In research
Polygon with holes appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Polygon with holes in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Polygon with holes is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean plane geometry, Polygons, so understanding it makes those chapters shorter.
In everyday life
Look for Polygon with holes outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Polygon with holes in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Polygon with holes means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Polygon with holes out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Polygon with holes in simple terms?

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 into multiple polygons by adding new edges, so they are not frequently needed.

Why does Polygon with holes matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Polygon with holes?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Polygon with holes.

Tags

  • Euclidean plane geometry
  • Polygons

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