# Talk:Arrangement of lines

What is the relationship between an arrangement of lines and a projective configuration? -- Cheers, Steelpillow (Talk) 12:38, 22 November 2009 (UTC)

I think it would be helpful to add a brief explanation to both articles, and cross-link. -- Cheers, Steelpillow (Talk) 13:03, 22 November 2009 (UTC)

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The article looks very well done, as might be expected from an editor who has published in the field. Unfortunately, I don't have time at present to give a good review. But on the definition section,

It seems light on citations--none for the definition itself?
There could be more explanation of concepts and/or wikilinks. Perhaps a digram to illustrate the concepts?
As an intro section, the definition section uses fairly high-level concepts that may baffle readers coming from a high-school geometry understanding of the topic. For instance, they might wonder what is an unbounded convex polygon, and if they make the association of that with what you are talking about, how is a wedge of the plane going off to infinity even considered a polygon?
Building a pedagogical transition from an elementary-looking problem to state-of the art understanding is often difficult, but the intro section could benefit from a bit more of an on-ramp. {{u|Mark viking}} {Talk} 19:23, 29 January 2024 (UTC)

The entire first paragraph, and its embedded three bullet points, have a single source, because that source was adequate for those definitions. If you think[1] that[1] adding[1] lots[1] of fnords[1] would give readers[1] warm[1] fuzzy[1] feelings[1] then[1] I suppose[1] we could repeat[1] that one footnote[1] multiple[1] times.[1] —David Eppstein (talk) 21:35, 29 January 2024 (UTC)

Hi @David Eppstein I appreciate all the wonderful images you make for WP. For the more elementary reader I suggest that the first image is edited such that there is one example of isomorphic arrangements and a small edit to that arrangement to make it not isomorphic. This is will make it more clear to readers that have not encountered isomorphism.
"Two arrangements are said to be isomorphic or combinatorially equivalent if there is a one-to-one boundary-preserving correspondence between the objects in their associated cell complexes."
Then this quote can indicate the image is expository. 
Also the current caption on the first image does not make sense to a new reader who has never seen the word simplicial. Some blue text is in order?
Czarking0 (talk) 15:06, 2 June 2024 (UTC)

The first image was really created for a different purpose: it shows a complete quadrangle and complete quadrilateral respectively. But also it shows the difference between simple arrangements and non-simple arrangements, a distinction that I think may be more fundamental than isomorphism here.
The caption describes two terms defined inside this same article. —David Eppstein (talk) 16:43, 2 June 2024 (UTC)

I'm wondering if this article should include some practical examples of how "Arrangement of lines" is used in computer graphics and robotics. E.g.,
Computer graphics:

Rendering and ray tracing: In ray tracing, an algorithm traces the path of light as lines (rays) to simulate the way light interacts with objects. Arrangements of lines help in efficiently determining intersections and rendering realistic scenes.
Hidden line removal: In 3D modeling and rendering, arrangements of lines are used to determine which edges of a 3D object are visible and which are hidden from the viewer’s perspective. This technique enhances the clarity and realism of the rendered image.
Robotics:

Path planning: In robotics, line arrangements are needed for path planning, where the robot must navigate through an environment with obstacles. The robot's sensors often detect lines that represent walls or other barriers, and efficient algorithms are needed to find a collision-free path.
Motion planning: For robotic arms, especially in manufacturing, line arrangements help in determining the sequence of movements needed to avoid collisions and achieve precise positioning of the end effector.
I'm not an expert in either mathematics or these applications (so am unsure how relevant this is to the topic) but thought I'd throw it out there in case it might be useful. Esculenta (talk) 14:52, 19 June 2024 (UTC)

For one thing, the situations you describe are three-dimensional, but the arrangements discussed in this article are two-dimensional. And vague allusions to vaguely related practical problems are too common in the research literature, but not helpful here. My understanding is that the main practical connection is through point-line duality, as is already briefly mentioned in connection with one practical problem, the computation of the Theil–Sen estimator. —David Eppstein (talk) 18:23, 19 June 2024 (UTC)
Understood on the 2-D arrangements. How about point-line duality and Voronoi diagrams?
In computational geometry, line arrangements are used in the construction and optimization of Voronoi diagrams, which are extensively used for spatial analysis in geographic information systems (GIS) and computer graphics. For example, algorithms for constructing Voronoi diagrams often use line arrangements to efficiently manage and partition space, optimizing tasks such as nearest-neighbor queries and spatial indexing. Esculenta (talk) 05:13, 20 June 2024 (UTC)
In what sense do you think "line arrangements are used in the construction and optimization of Voronoi diagrams"? "Used", how? What is your published source for them being used in this way? —David Eppstein (talk) 05:39, 20 June 2024 (UTC)
As one example, doi:10.1007/BF01840357. Fortune's sweep line algorithm uses line arrangements to manage the "beach line" (a collection of parabolic arcs that changes dynamically) status and handle events efficiently during the sweep. This process can be viewed as managing the intersections and arrangements of lines and parabolas, which is directly related to the study of line arrangements. This connection between line arrangements and Voronoi diagrams helps optimize nearest-neighbor queries and spatial indexing, and demonstrates a practical application of line arrangements in computational geometry. (some notes here). Esculenta (talk) 15:27, 20 June 2024 (UTC)
Are you an AI? You are writing technical words in an order that suggests that they have some meaning that is not present in the sources you cite. There are no line arrangements in the beach line. It is the boundary of a union of parabolas. —David Eppstein (talk) 18:54, 20 June 2024 (UTC)
My mistake, I conflated line arrangement with the boundary of a union of parabolas, thanks for the correction. Esculenta (talk) 19:36, 20 June 2024 (UTC)

There may be a sufficient number of applications to make a separate page for this, considering the multiple different ways to represent them (ones not shown here, that may crowd this page if added but that are nonetheless important to pseudolines, like the wiring diagram) and the problem of "stretchability" being worth quite a lot. Thoughts? BagLuke (talk) 06:37, 17 February 2025 (UTC)