# What are Definition, Algorithms and Practical Solutions for Concave Hull?

### Convex Hull

A convex hull of a shape is defined as:

In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X (Wikipedia)

Wikipedia visualizes it nicely using a rubber band analogy, and there are some good algorithms to compute it.

### Concave Hull

A concave hull is visualized using the red line in the image below (the blue line visualizes the convex hull). Intuitively, it is a polygon which embraces all the points, but has less (minimal?) area compared to the convex hull. As a result, the polygon's boundary length is longer.

A concave hull may be the solution for some real-world problems (e.g. finding the reasonable boundary of a city).

I have failed to find a proper definition, algorithm and practical solution for the notion of a Concave Hull. The Grass Wiki has some descriptions and images, and there is a commercial solution in concavehull.com.

Any ideas, algorithms and links?

• In what context to you want to generate concave hulls/alpha shapes? In PostGIS, ArcMap, a web-map, your own software? – fmark Aug 17 '10 at 0:39
• Both PostGIS and my own Python scripts. – Adam Matan Aug 17 '10 at 5:29
• Is there a C++ Linux-compatible version of implementation of the concave hull algorithm ? – Sylv255 May 31 '18 at 16:24
• If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. - From Review – Evil Genius May 31 '18 at 17:35
• Computational Geometry Algorithms Library (CGAL) is a C++ library with Alpha Shapes. It has a Linux download and is licensed as GPL/LGPL for version >=4.0. – klewis May 31 '18 at 18:07

As scw points out, you want an implementation of α-shapes.

Alpha shapes can be considered a generalisation of the convex hull. They were first described in 1981 in:

Edelsbrunner, H.; Kirkpatrick, D.; Seidel, R.; , "On the shape of a set of points in the plane," Information Theory, IEEE Transactions on , vol.29, no.4, pp. 551- 559, Jul 1983

Open source implementations exist for the environments you are interested in:

# PostGIS

If you are using the PostGIS stack, pgRouting's optional Driving Distance extension exposes an alpha shape implementation. I'm not sure if you can vary the alpha parameter, however.

Usage is below:

``````SELECT the_geom AS alpha_shape
FROM
points_as_polygon(
'SELECT id, ST_X(your_geom) AS x, ST_Y(your_geom) AS y FROM your_table');
``````

# Python

There are probably many python modules you could use. I have heard good things about CGAL, a C++ computational geometry library. Python wrappers exist for parts of CGAL, including exposing CGAL's alpha shape implementation to Python.

Be aware that parts of CGAL are licensed under the QPL, which means that if you distribute your program, linked to CGAL, you may need to release it under the QPL. It is fine to keep your code proprietary if you do not redistribute your program code or binaries.

• I can't get the python wrappers of CGAL to compile---it seems that these haven't been supported in a while and no longer work with a recent version of CGAL. – conradlee Jul 24 '11 at 22:16
• @fmark: Second link you posted seems to be dead. – radek Dec 6 '11 at 14:32
• @fmark PostGIS links seem to be dead.. – radek Jun 11 '15 at 13:08
• The α-shapes link is dead. That webpage does not exist anymore. One prefix of the original hyperlink target is www.cs.duke.edu/~edels. That prefix URL is also dead. – Samuel Muldoon May 12 at 0:40

Here is what you are looking for.

You can download and test the program: (in java, under GPL license)

The paper presenting the algorithm is there:

Duckham, M., Kulik, L., Worboys, M.F., Galton, A. (2008) Efficient generation of simple polygons for characterizing the shape of a set of points in the plane. Pattern Recognition v41, 3224-3236

• The paper feels quite funny to me. It is alpha shapes with one added constraint (result must be a simple polygon). Instead of saying this sentence, the algorithm is described newly from the beginning. – emu Apr 29 '20 at 12:41

This seems to be a specific application of alpha shapes, which are from my reading a more general form of this problem. R has the alphahull module, which has excellent documentation on computing alpha shapes. Also check this detailed background on alpha shapes. If you only want to compute convex/concave hulls, check out lasboundary, part of lastools, it scales well and can handle millions of input points.

Finally, this interesting application of alpha shapes by Flickr made the rounds a while ago, showing their utility for aggregating user generated point content:

• OMG the source is written in FORTRAN :-) – Adam Matan Aug 16 '10 at 7:51
• There's the clustr package written in C++ if that suits you better; but be careful with the licensing on CGAL: github.com/straup/Clustr – scw Aug 16 '10 at 8:11
• Nice real-world example. – DavidF Aug 16 '10 at 13:23

There is an implementation of ST_ConcaveHull in PostGIS trunk. http://postgis.net/docs/ST_ConcaveHull.html

• I think this first appears in Version 2.0 of PostGis – Adrian Aug 14 '11 at 14:42

I created a highly-efficient tool, called lasboundary (1,2), that computes a concave hull for LIDAR in LAS/LAZ/SHP/ASCII format and stores the result as a vector boundary polygon in ESRI Shapefile format or a geo-referenced KML file.

Here is an example run:

``````C:\lastools\bin>lasboundary -i SerpentMound.las -o SerpentMound_boundary.shp
reading 3265110 points and computing convex hull for 3265110 points
growing inward towards concave hull (with concavity = 50)
outputting the concave hull
concave hull has 1639 points
``````

Some result pictures are here.

Here is an R function that implements the Alpha Hull model. The output is an sp polygon object. Please see the example in the header. It requires the sp, alphahull and maptools packages.

**Update (01-15-2018) There have been numerous changes to the resulting objects produced by the alphahull package. As such, I needed to rewrite the function. I added a convexHull function to the spatialEco package. However, due to licensing restrictions in the alphahull package this function is only available in the development version on GitHub. The development version can be installed using:

``````library(devtools)
install_github("jeffreyevans/spatialEco")
``````

Here is an example of the functions usage

``````library(sp)
library(spatialEco)
data(meuse)
coordinates(meuse) = ~x+y
a <- convexHull(meuse, alpha=100000)
plot(a)
points(meuse, pch=19)
``````

Convert the resulting SpatialLinesDataFrame to SpatialPolygonsDataFrame

``````library(sf)
a <- sf::st_as_sf(a)
a <- sf::st_polygonize(a)
class( a <- as(a, "Spatial") )
plot(a)
``````

Test multiple alpha values

``````par(mfcol=c(2,2))
for (a in c(500, 1500, 5000, 100000)) {
ch <- convexHull(meuse, alpha = a)
plot(ch)
points(meuse, pch=19)
title( paste0("alpha=", a))
}
``````

• +1 Could you explain how this differs from the alpha shapes package? – whuber Aug 14 '12 at 19:07
• The output of alphahull object is stored as a matrix and must be coerced to a usable sp object. I would consider this a "helper" function to create a polygon that can be exported to a GIS format. This function uses the alphahull package to create the hull matrix object, creates an sp object and then explodes the polygon slot so it is a single-part polygon dataframe object. Nothing is showing up in the package help but there may be newly implemented native coercion to an sp class object that I am not aware of. If this is the case please let me know so I can decommission this function. – Jeffrey Evans Aug 14 '12 at 21:13
• What's the programming language? – Adam Matan Aug 15 '12 at 8:07
• Thanks @JeffreyEvans I've managed to get this working. Could you possibly explain the parameters? I've had a look at the linked jstatsoft paper, but it's pretty impenetrable. – geotheory Dec 20 '16 at 12:30

About R implementation Alpha-Shapes, there's an article about "Converting Alpha-Shapes into SP Objects"

It's based on alphahull, sp and spgrass6 http://casoilresource.lawr.ucdavis.edu/drupal/node/919

JTS (https://github.com/locationtech/jts) provides a Convex Hull implementation. Martin Davies also mentioned having an Alpha Shape algorithm in the works so you might want to check the SVN repository to see if it is in yet if that's what you want.

Speaking about JTS, you can use Geoscript for manipulating JTS library. http://geoscriptblog.blogspot.com/2010/06/unwrapped-jts-with-python.html for an article about convex hull. GeoScript implementations are available in JavaScript, Python, Scala, and Groovy. The official website : http://geoscript.org

Here's a program written in C that computes convex hulls, alpha shapes, Delauney triangluations and Voronoi volumes:

• Hull - Ken Clarkson (2002)

Another convex hull algorithm with C and Java implementations is here:

To add to previous answers for this post, at least as of QGIS 2.6 does have concave hull algorithm

Parameters
Input point layer [vector: point]
put parameter description here

Threshold (0-1, where 1 is equivalent with Convex Hull) [number]
put parameter description here
Default: 0.3

Allow holes [boolean]
put parameter description here
Default: True

Split multipart geometry into singleparts geometries [boolean]
Default: False

Outputs Concave hull [vector]
put output description here

Console usage
processing.runalg('qgis:concavehull', input, alpha, holes, no_multigeometry, output)

Also, Esri has a tool Minimum Bounding Geometry (Data Management)

Which allows you to choose the geometry type, which includes convex hull

There is a new Addon for GRASS GIS 7 available: v.concave.hull. See also http://grasswiki.osgeo.org/wiki/Create_concave_hull

To help with the "proper definition" part of your question; you may have looked at https://en.wikipedia.org/wiki/Convex_hull and gotten what could well be considered a "proper" definition, but found it lacking, so perhaps a more "useful" definition is:

For every point inside a convex hull, a straight line to any point not within the hull will only intersect the hull once.

This is useful because given a point you can construct a line through it and test for that constructed line intersecting segments of the hull.

• No intersection the point is not in the hull.
• One intersection the point is on the hull.
• Two intersections the point is within the hull
• A straight line cannot intersect a convex hull more than twice
• the op is asking about concave hulls, and not convex hulls – winwaed Jun 29 '15 at 2:31