Finding edges of a non self-intersecting polygon from given vertices - r

Question

I have the vertices of a non-self-intersecting polygon in 2-D where the x-coordinate is centred longitude and y-coordinate is centred latitude. I want to find the edges of the polygon.
I can plot the vertices and see which vertices are neighbouring and see the edges. But my question is how can I get these edges.
For example, I am considering the sample data:
> data1
vertices lon lat
5 1.133179 1.027886
4 1.094459 1.013952
2 1.055672 1.000000
1 1.000000 1.028578
3 1.038712 1.042541
6 1.116241 1.070438
Sample Plot of the points is
I want to have an array like this
>edges
ind1 ind2
[1,] 5 6
[2,] 1 3
[3,] 3 6
[4,] 1 2
[5,] 2 4
[6,] 4 5
I am interested about this kind of shape of the polygon (with minimum area)
I got this array by using a function ashape of the R-package alphahull. But in this function Euclidean distance is used to find distance between points, which not applicable in my case (since I am considering data on (lon, lat), we can use distHaversine distance function in the package geosphere). And this function giving unsatisfactory result in case if the polygon has large number vertices and have complex shape. This polygon may or may not be convex.
Now all I want is to build an algorithm to find the edges of the non-intersecting polygon with minimum area.
Any help in this direction will be gratefully appreciated.

Answer 1

Algorithm for finding all possible polygons:
generate the convex hull .
Note that any non intersecting polygon must traverse its convex hull in order.
Start with any point on the convex hull
Generate a list of paths from that point to each interior point, and to the next adjacent point on the convex hull
recursively extend each path to each remaining interior point as well as to that first free point on the convex hull
for each segment added to a path reject the path if it self intersects
I'm not going to post the code, but here are all 67 possible polygons for a random set of 8 points.
As one can imagine the set of results blows up quickly with the number of points (eg. n=12 -> ~10000 polygons.. )
here are the polygons with min and max perimeters.

Answer 2

convert points from lon,lat to Cartesian x,y or x,y,z
use spherical or ellipsoidal surface
if the size is small enough you can project (x,y,z) to local surface plane to avoid 3D computing
you can also use lon,lat as x,y but make sure there is no zero crossing (if is then offset that axis by some value until it isn't)
now there are many possible strategies
you did not provide any rule for the shape
so I assume 'minimal' perimeter/size/area and generic concave polygon
you can not go directly to edge lines before you know where is inside and where is outside
I would do this task like this: find polygon based on find holes in 2D point set
modification 1
as you already have all the edge points (at least that is my impression)
so you can make flag for each point from the above algorithm
that will tell you where is inside or outside of polygon
for example take 8 directions (N,NE,E,...) and encode which way is filled and which empty
then on each edge start in the middle of empty direction
and find 2 closest lines to it (in angular terms) that are not intersecting any previous line
and if more available use the smallest ones
gray means inside polygon
make list of all such possible lines (2 per point)
then search for connected loops
beware this modification is not 100% error prone (I do not think that is for concave polygon even possible)
modification 2
use complete polygon from bullet 2
and try to match its edge points to your input edge points
then use the edge lines as in original polygon but with your new points
if some points are skipped then find closest edge line to it and divide it by this point
this should be more safe and accurate then bullet 3.
simple approach
if the above is too much for you then
create list of all possible lines
sort them by size ascending
remove all 'long' lines that are intersecting any 'short' line
what is short or long depends on you
for example first third of lines can be the short ones and the last third the long ones
or make average size and what is < 0.6*avg_size or > 1.2*avg_size ...
or if you have N points then first 2N lines are short the rest is long (2 lines per point)
test all and select the best option for you ...
try to find joined lines
find only lines that are connected once (no more then 2 lines per point)
remove them from list into the final solution list
after this you will have list of possible lines and list of found lines
remove all lines from possible lines that intersect any line in found lines
this should remove any non relevant lines
try to find connections again
take first possible line if found connection move it to the solution list
and go to bullet 5.
if none found continue with next line ...
stop if none line left or none connection found.