Talk:Alpha shape

Untitled
So is it "alpha shape" or "alpha-shape"? Probably doesn't matter, but it would be nice to have some internal consistency. --Matt Westwood 13:46, 11 September 2011 (UTC)

Edelsbrunner
Are there any references which do not include Edelsbrunner? What is it that is supposed to let me know that this is something of merit, and not just Edelsbrunner's latest mathematical toy? Sanpitch (talk) 03:01, 14 September 2011 (UTC)
 * Yes, about 1750 of them. —David Eppstein (talk) 04:16, 14 September 2011 (UTC)

Definition
According to Edelsbrunner et al: "Given a set S of points in the plane and an arbitrary real number &alpha;, the &alpha;-shape of S is the straight line graph whose vertices are the &alpha;-extreme points and whose edges connect the respective &alpha;-neighbors."

(emphasis mine). Thus the edges of the &alpha;-shape are straight line segments. (There is a related notion, that of the &alpha;-hull, whose edges are the arcs of circles.) Moreover, the images in that article show examples of the &alpha;-hull and &alpha;-shape, where it is clear that one is supposed to have curvilinear edges and the other linear ones. Sławomir Biały (talk) 15:49, 17 September 2011 (UTC)

Examples
The examples should show results of varying alpha. —DIV (137.111.13.4 (talk) 23:08, 2 November 2015 (UTC))

Something wrong...
Read sentence: If α = ∞, then the alpha-shape associated with the finite point set is its ordinary convex hull.

But it is not true! Must be If α = 0,...

In case α = ∞ we have disk of radius = 1/∞ = 0, so does not exist any disk of this radius, containing entire point set...

Jumpow (talk) 20:14, 28 March 2019 (UTC)

On a related note, there are two definitions of α. The definition under Characterisation currently is the one in the original paper - Edelsbrunner et al (1983). Later, in Edelsbrunner & Mucke (1994), he used the definition, where the old α is -1/α in the 1994 paper, as mentioned in footnote 3. I think the new definition of α as a radius is more widely adapted in various implementations, e.g. MATLAB etc. The newer definition will give α = ∞ a convex hull.

Is it worth making the two definitions clearer on the page?

Phoenix1206 (talk) 13:53, 22 March 2020 (UTC)

"empty circle"
"the radius of the smallest empty circle containing the edge or triangle"

What is the meaning of "empty" here? Can't we leave it out? Jpmaterial (talk) 05:55, 6 October 2022 (UTC)
 * It does not contain any of the given finite set of points. —David Eppstein (talk) 06:09, 6 October 2022 (UTC)

Alpha complex
Do people agree that the alpha complex would actually additionally need its own article? It plays an important role in persistent homology, as an alternative to the Vietoris-Rips and Čech complex. Many types of simplicial complexes have their own article, alpha complexes do not.

If there is no disagreement, I would offer myself to create a draft about alpha complexes. Peppey314 (talk) 09:37, 20 March 2024 (UTC)