computer science

Planar straight-line graph

Planar straight-line graph is a computer science 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 Planar straight-line graph rather than just read about it. In short: In computational geometry and geometric graph theory, a planar straight-line graph (PSLG), also called a straight-line plane graph or plane straight-line graph, is an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding.

Key takeaways

Planar straight-line graph belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Planar straight-line graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Planar straight-line graph from memory before moving on to harder problems.

Reference excerpt

In computational geometry and geometric graph theory, a planar straight-line graph (PSLG), also called a straight-line plane graph or plane straight-line graph, is an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding. In computational geometry, PSLGs have often been called planar subdivisions, with an assumption or assertion that subdivisions are polygonal rather than having curved boundaries. PSLGs may serve as representations of various maps, e.g., geographical maps in geographical information systems. Special cases of PSLGs are triangulations (polygon triangulation, point-set triangulation). Point-set triangulations are maximal PSLGs in the sense that it is impossible to add straight edges to them while keeping the graph planar. Triangulations have numerous applications in various areas. PSLGs may be seen as a special kind of Euclidean graphs. However, in discussions involving Euclidean graphs, the primary interest is their metric properties, i.e., distances between vertices, while for PSLGs the primary interest is the topological properties. For some graphs, such as Delaunay triangulations, both metric and topological properties are of importance.

Representations There exist three well-known data structures for representing PSLGs, these are the Winged-edge data structure, Halfedge, and Quadedge. The winged-edge data structure is the oldest of the three, but manipulating it often requires complicated case distinctions. This is because edge references do not store the edge direction, and the directions of edges around a face need not be consistent. The halfedge data structure stores both orientations of an edge and links them properly, simplifying operations and the storage scheme. The Quadedge data structure stores both the planar subdivision and its dual simultaneously. Its records consist explicitly only of edge records, four for each edge, and in a simplified form it is suitable for storing PSLGs.

Problems in terms of PSLG Point location. For a query point, find which face of the PSLG it belongs to. Map overlay. Find the overlay of two PSLGs (maps), which is the subdivision of the plane by the two simultaneously embedded PSLGs. In GIS this problem is known as "thematic map overlay".

See also Doubly connected edge list, a data structure to represent a PSLG Local feature size

References

Illustrations

Worked examples

Example 1 — a first encounter with Planar straight-line graph

Start with the simplest possible case. Write down what Planar straight-line graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, 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 Planar straight-line graph 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 Planar straight-line graph 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 Planar straight-line graph

In research: Planar straight-line graph appears in computer science 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 Planar straight-line graph 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: Planar straight-line graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric algorithms, Geometric graphs, Planar graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Planar straight-line graph 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 Planar straight-line graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Planar straight-line graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Planar straight-line graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Planar straight-line graph in simple terms?

Why does Planar straight-line graph matter?

Because it connects several computer science 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 Planar straight-line graph?

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 Planar straight-line graph.